A008597 -id:A008597

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A121035

Multiples of 15 containing a 15 in their decimal representation.

+10
34

15, 150, 315, 615, 915, 1155, 1215, 1500, 1515, 1530, 1545, 1560, 1575, 1590, 1815, 2115, 2415, 2715, 3015, 3150, 3315, 3615, 3915, 4155, 4215, 4515, 4815, 5115, 5415, 5715, 6015, 6150, 6315, 6615, 6915, 7155, 7215, 7515, 7815, 8115, 8415, 8715, 9015

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Paolo Xausa, Table of n, a(n) for n = 1..10000

Index entries for 10-automatic sequences.

FORMULA

a(n) ~ 15n. - Charles R Greathouse IV, Nov 02 2022

MATHEMATICA

Select[15*Range[1000], StringContainsQ[IntegerString[#], "15"] &] (* Paolo Xausa, Feb 25 2024 *)

PROG

(PARI) is(n)=if(n%15, return(0)); while(n>14, if(n%100==15, return(1)); n\=10); 0 \\ Charles R Greathouse IV, Feb 12 2017

CROSSREFS

Cf. A121041, A008597, A011531, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121036, A121037, A121038, A121039, A121040.

KEYWORD

nonn,base,easy

AUTHOR

Reinhard Zumkeller, Jul 21 2006, Jul 26 2006

STATUS

approved

A008598

Multiples of 16.

+10
27

0, 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240, 256, 272, 288, 304, 320, 336, 352, 368, 384, 400, 416, 432, 448, 464, 480, 496, 512, 528, 544, 560, 576, 592, 608, 624, 640, 656, 672, 688, 704, 720, 736, 752, 768, 784, 800, 816, 832

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

If X is an n-set and Y_i (i=1,2,3,4) mutually disjoint 2-subsets of X then a(n-6) is equal to the number of 5-subsets of X intersecting each Y_i (i=1,2,3,4). - Milan Janjic, Aug 26 2007

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 328

Milan Janjic, Two Enumerative Functions

Tanya Khovanova, Recursive Sequences

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

Leo Tavares, Illustration: Square Block Star Frames

Index entries for linear recurrences with constant coefficients, signature (2,-1).

FORMULA

a(n) = Sum_{k=1..8n} (i^k+1)*(i^(8n-k)+1), where i=sqrt(-1). - Bruno Berselli, Mar 19 2012

G.f.: 16*x/(x-1)^2. - Vincenzo Librandi, Jun 10 2013

a(n) = A014641(n) - A185212(n). - Leo Tavares, May 24 2022

MAPLE

A008598:=n->16*n; seq(A008598(n), n=0..100); # Wesley Ivan Hurt, Nov 13 2013

MATHEMATICA

Range[0, 1000, 16] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)

CoefficientList[Series[16 x / (x - 1)^2, {x, 0, 60}], x] (* Vincenzo Librandi Jun 10 2013 *)

PROG

(PARI) a(n)=16*n \\ Charles R Greathouse IV, Sep 24 2015

CROSSREFS

Cf. A008596, A008597.

Cf. A014641, A185212.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A249674

a(n) = 30*n.

+10
24

0, 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, 390, 420, 450, 480, 510, 540, 570, 600, 630, 660, 690, 720, 750, 780, 810, 840, 870, 900, 930, 960, 990, 1020, 1050, 1080, 1110, 1140, 1170, 1200, 1230, 1260, 1290, 1320, 1350, 1380, 1410, 1440

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Numbers divisible by 2, 3 and 5. - Robert Israel, Nov 19 2014

a(n) is the maximum score of a 10-pin n-frame bowling game and the maximum score of an n-pin 10-frame bowling game, given the rules: a strike is worth the number of pins in each frame plus the number of pins knocked down by the next two balls (except in the last frame), a spare is worth the number of pins in each frame plus the number of pins knocked down by the next ball (except in the last frame), and if a strike or spare is earned in the last frame then the player must continue to throw balls until they have thrown 3 balls in the last frame. - Iain Fox, Mar 02 2018

LINKS

Iain Fox, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Bowling

Wikipedia, Ten-pin bowling

Index entries for linear recurrences with constant coefficients, signature (2,-1).

FORMULA

G.f.: 30*x/(x-1)^2; a(n) = 2*a(n-1) - a(n-2). - Wesley Ivan Hurt, Nov 18 2014

a(n) = 2*A008597(n) = 3*A008592(n) = 5*A008588(n) = 6*A008587(n) = 10*A008585(n) = 15*A005843(n). - Omar E. Pol, Nov 24 2014

EXAMPLE

a(7) = 7 * 30 = 210.

MAPLE

A249674:=n->30*n: seq(A249674(n), n=0..50); # Wesley Ivan Hurt, Nov 18 2014

MATHEMATICA

30*Range[0, 59] (* Alonso del Arte, Nov 18 2014 *)

PROG

(Magma) [30*n : n in [0..50]]; // Wesley Ivan Hurt, Nov 18 2014

(PARI) vector(100, n, 30*(n-1)) \\ Derek Orr, Nov 18 2014

(PARI) first(n) = Vec(30*x/(x-1)^2 + O(x^n), -n) \\ Iain Fox, Mar 02 2018

CROSSREFS

Cf. A008585, A008592, A050519.

KEYWORD

nonn,easy

AUTHOR

Kaylan Purisima, Nov 03 2014

STATUS

approved

A008599

Multiples of 17.

+10
17

0, 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255, 272, 289, 306, 323, 340, 357, 374, 391, 408, 425, 442, 459, 476, 493, 510, 527, 544, 561, 578, 595, 612, 629, 646, 663, 680, 697, 714, 731, 748, 765, 782, 799, 816, 833, 850, 867, 884

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Ivan Panchenko, Table of n, a(n) for n = 0..200

Tanya Khovanova, Recursive Sequences

Michael Penn, Divisibility by 17: The Lesser Known Rule, YouTube video, 2023.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 329

Index entries for linear recurrences with constant coefficients, signature (2, -1).

FORMULA

(floor(a(n)/10) - 5*(a(n) mod 10)) == 0 (mod 17), see A076311. - Reinhard Zumkeller, Oct 06 2002

From Vincenzo Librandi, Dec 24 2010: (Start)

a(n) = 17*n = 2*a(n-1) - a(n-2).

G.f.: 17x/(x-1)^2. (End)

MATHEMATICA

Range[0, 1003, 17] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)

PROG

(PARI) a(n)=17*n \\ Charles R Greathouse IV, Oct 07 2015

(Scala) for (n <- 0 to 1003 by 17) print(n + ", ") // Alonso del Arte, Jun 17 2018

CROSSREFS

Cf. A008597, A008598, A244630.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A141419

Triangle read by rows: T(n, k) = A000217(n) - A000217(n - k) with 1 <= k <= n.

+10
16

1, 2, 3, 3, 5, 6, 4, 7, 9, 10, 5, 9, 12, 14, 15, 6, 11, 15, 18, 20, 21, 7, 13, 18, 22, 25, 27, 28, 8, 15, 21, 26, 30, 33, 35, 36, 9, 17, 24, 30, 35, 39, 42, 44, 45, 10, 19, 27, 34, 40, 45, 49, 52, 54, 55

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

As a rectangle, the accumulation array of A051340.

From Clark Kimberling, Feb 05 2011: (Start)

Here all the weights are divided by two where they aren't in Cahn.

As a rectangle, A141419 is in the accumulation chain

... < A051340 < A141419 < A185874 < A185875 < A185876 < ...

(See A144112 for the definition of accumulation array.)

row 1: A000027

col 1: A000217

diag (1,5,...): A000326 (pentagonal numbers)

diag (2,7,...): A005449 (second pentagonal numbers)

diag (3,9,...): A045943 (triangular matchstick numbers)

diag (4,11,...): A115067

diag (5,13,...): A140090

diag (6,15,...): A140091

diag (7,17,...): A059845

diag (8,19,...): A140672

(End)

Let N=2*n+1 and k=1,2,...,n. Let A_{N,n-1} = [0,...,0,1; 0,...,0,1,1; ...; 0,1,...,1; 1,...,1], an n X n unit-primitive matrix (see [Jeffery]). Let M_n=[A_{N,n-1}]^4. Then t(n,k)=[M_n]_(1,k), that is, the n-th row of the triangle is given by the first row of M_n. - L. Edson Jeffery, Nov 20 2011

Conjecture. Let N=2*n+1 and k=1,...,n. Let A_{N,0}, A_{N,1}, ..., A_{N,n-1} be the n X n unit-primitive matrices (again see [Jeffery]) associated with N, and define the Chebyshev polynomials of the second kind by the recurrence U_0(x) = 1, U_1(x) = 2*x and U_r(x) = 2*x*U_(r-1)(x) - U_(r-2)(x) (r>1). Define the column vectors V_(k-1) = (U_(k-1)(cos(Pi/N)), U_(k-1)(cos(3*Pi/N)), ..., U_(k-1)(cos((2*n-1)*Pi/N)))^T, where T denotes matrix transpose. Let S_N = [V_0, V_1, ..., V_(n-1)] be the n X n matrix formed by taking V_(k-1) as column k-1. Let X_N = [S_N]^T*S_N, and let [X_N]_(i,j) denote the entry in row i and column j of X_N, i,j in {0,...,n-1}. Then t(n,k) = [X_N]_(k-1,k-1), and row n of the triangle is given by the main diagonal entries of X_N. Remarks: Hence t(n,k) is the sum of squares t(n,k) = sum[m=1,...,n (U_(k-1)(cos((2*m-1)*Pi/N)))^2]. Finally, this sequence is related to A057059, since X_N = [sum_{m=1,...,n} A057059(n,m)*A_{N,m-1}] is also an integral linear combination of unit-primitive matrices from the N-th set. - L. Edson Jeffery, Jan 20 2012

Row sums: n*(n+1)*(2*n+1)/6. - L. Edson Jeffery, Jan 25 2013

n-th row = partial sums of n-th row of A004736. - Reinhard Zumkeller, Aug 04 2014

T(n,k) is the number of distinct sums made by at most k elements in {1, 2, ... n}, for 1 <= k <= n, e.g., T(6,2) = the number of distinct sums made by at most 2 elements in {1,2,3,4,5,6}. The sums range from 1, to 5+6=11. So there are 11 distinct sums. - Derek Orr, Nov 26 2014

A number n occurs in this sequence A001227(n) times, the number of odd divisors of n, see A209260. - Hartmut F. W. Hoft, Apr 14 2016

Conjecture: 2*n + 1 is composite if and only if gcd(t(n,m),m) != 1, for some m. - L. Edson Jeffery, Jan 30 2018

From Peter Munn, Aug 21 2019 in respect of the sequence read as a triangle: (Start)

A number m can be found in column k if and only if A286013(m, k) is nonzero, in which case m occurs in column k on row A286013(m, k).

The first occurrence of m is in row A212652(m) column A109814(m), which is the rightmost column in which m occurs. This occurrence determines where m appears in A209260. The last occurrence of m is in row m column 1.

Viewed as a sequence of rows, consider the subsequences (of rows) that contain every positive integer. The lexicographically latest of these subsequences consists of the rows with row numbers in A270877; this is the only one that contains its own row numbers only once.

(End)

REFERENCES

R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

LINKS

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 24.

Carlton Gamer, David W. Roeder, and John J. Watkins, Trapezoidal Numbers, Mathematics Magazine 58:2 (1985), pp. 108-110.

L. E. Jeffery, Unit-primitive matrices

M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.

FORMULA

t(n,m) = m*(2*n - m + 1)/2.

t(n,m) = A000217(n) - A000217(n-m). - L. Edson Jeffery, Jan 16 2013

Let v = d*h with h odd be an integer factorization, then v = t(d+(h-1)/2, h) if h+1 <= 2*d, and v = t(d+(h-1)/2, 2*d) if h+1 > 2*d; see A209260. - Hartmut F. W. Hoft, Apr 14 2016

G.f.: y*(-x + y)/((-1 + x)^2*(-1 + y)^3). - Stefano Spezia, Oct 14 2018

T(n, 2) = A060747(n) for n > 1. T(n, 3) = A008585(n - 1) for n > 2. T(n, 4) = A016825(n - 2) for n > 3. T(n, 5) = A008587(n - 2) for n > 4. T(n, 6) = A016945(n - 3) for n > 5. T(n, 7) = A008589(n - 3) for n > 6. T(n, 8) = A017113(n - 4) for n > 7.r n > 5. T(n, 7) = A008589(n - 3) for n > 6. T(n, 8) = A017113(n - 4) for n > 7. T(n, 9) = A008591(n - 4) for n > 8. T(n, 10) = A017329(n - 5) for n > 9. T(n, 11) = A008593(n - 5) for n > 10. T(n, 12) = A017593(n - 6) for n > 11. T(n, 13) = A008595(n - 6) for n > 12. T(n, 14) = A147587(n - 7) for n > 13. T(n, 15) = A008597(n - 7) for n > 14. T(n, 16) = A051062(n - 8) for n > 15. T(n, 17) = A008599(n - 8) for n > 16. - Stefano Spezia, Oct 14 2018

T(2*n-k, k) = A070543(n, k). - Peter Munn, Aug 21 2019

EXAMPLE

As a triangle:

2, 3,

3, 5, 6,

4, 7, 9, 10,

5, 9, 12, 14, 15,

6, 11, 15, 18, 20, 21,

7, 13, 18, 22, 25, 27, 28,

8, 15, 21, 26, 30, 33, 35, 36,

9, 17, 24, 30, 35, 39, 42, 44, 45,

10, 19, 27, 34, 40, 45, 49, 52, 54, 55;

As a rectangle:

1 2 3 4 5 6 7 8 9 10

3 5 7 9 11 13 15 17 19 21

6 9 12 15 18 21 24 27 30 33

10 14 18 22 26 30 34 38 42 46

15 20 25 30 35 40 45 50 55 60

21 27 33 39 45 51 57 63 69 75

28 35 42 49 56 63 70 77 84 91

36 44 52 60 68 76 84 92 100 108

45 54 63 72 81 90 99 108 117 126

55 65 75 85 95 105 115 125 135 145

Since the odd divisors of 15 are 1, 3, 5 and 15, number 15 appears four times in the triangle at t(3+(5-1)/2, 5) in column 5 since 5+1 <= 2*3, t(5+(3-1)/2, 3), t(1+(15-1)/2, 2*1) in column 2 since 15+1 > 2*1, and t(15+(1-1)/2, 1). - Hartmut F. W. Hoft, Apr 14 2016

MAPLE

a:=(n, k)->k*n-binomial(k, 2): seq(seq(a(n, k), k=1..n), n=1..12); # Muniru A Asiru, Oct 14 2018

MATHEMATICA

T[n_, m_] = m*(2*n - m + 1)/2; a = Table[Table[T[n, m], {m, 1, n}], {n, 1, 10}]; Flatten[a]

PROG

(Haskell)

a141419 n k = k * (2 * n - k + 1) `div` 2

a141419_row n = a141419_tabl !! (n-1)

a141419_tabl = map (scanl1 (+)) a004736_tabl

-- Reinhard Zumkeller, Aug 04 2014

CROSSREFS

Cf. A000330 (row sums), A004736, A057059, A070543.

A144112, A051340, A141419, A185874, A185875, A185876 are accumulation chain related.

A141418 is a variant.

Cf. A001227, A209260. - Hartmut F. W. Hoft, Apr 14 2016

A109814, A212652, A270877, A286013 relate to where each natural number appears in this sequence.

A000027, A000217, A000326, A005449, A045943, A059845, A115067, A140090, A140091, A140672 are rows, columns or diagonals - refer to comments.

KEYWORD

nonn,tabl,easy

AUTHOR

Roger L. Bagula, Aug 05 2008

EXTENSIONS

Simpler name by Stefano Spezia, Oct 14 2018

STATUS

approved

A344330

Sides s of squares that can be tiled with squares of two different sizes so that the number of large or small squares is the same.

+10
11

10, 15, 20, 30, 40, 45, 50, 60, 65, 68, 70, 75, 78, 80, 90, 100, 105, 110, 120, 130, 135, 136, 140, 150, 156, 160, 165, 170, 175, 180, 190, 195, 200, 204, 210, 220, 222, 225, 230, 234, 240, 250, 255, 260, 270, 272, 280, 285, 290, 300, 310, 312, 315, 320, 325, 330, 340, 345, 350, 360, 369, 370

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

This sequence is a generalization of the 4th problem proposed for the pupils in grade 6 during the 19th Mathematical Festival at Moscow in 2008.

Some notations: s = side of the tiled square, a = side of small squares, b = side of large squares, and z = number of small squares = number of large squares.

Side s of such tiled squares must satisfy the Diophantine equation s^2 = z * (a^2+b^2).

There are two types of solutions. See A344331 for type 1 and A344332 for type 2.

If q is a term, k * q is another term for k > 1.

REFERENCES

Ivan Yashchenko, Invitation to a Mathematical Festival, pp. 10 and 102, MSRI, Mathematical Circles Library, 2013.

LINKS

Table of n, a(n) for n=1..62.

Index to sequences related to Olympiads.

EXAMPLE

-> Example of type 1:

Square 10 x 10 with a = 1, b = 2, s = 10, z = 20.

___ ___ _ ___ ___ _

| | |_| | |_|

|___|___|_|___|___|_|

| | |_| | |_| with 10 elementary 2 x 5 rectangles

|___|___|_|___|___|_|

| | |_| | |_| ___ ___ _

|___|___|_|___|___|_| | | |_|

| | |_| | |_| |___|___|_|

|___|___|_|___|___|_|

| | |_| | |_|

|___|___|_|___|___|_|

-> Example of type 2:

Square 15 x 15 with a = 3, b = 4, s = 15, z = 9.

________ ________ ________ _____

| | | | |

| | | |_____|

|_______ |________|________| |

| | | | |

| | | |_____|

| | | | |

|________|________|________| |

| | | |_____|

| | | | |

|_____ __|___ ____|_ ______|_____|

| | | | | |

|_____|______|______|______|_____|

Remarks:

- With terms as 10, 20, ... we only obtain sides of squares of type 1:

10 is a term of this type because the square 10 X 10 only can be tiled with 20 squares of size 1 X 1 and 20 squares of size 2 X 2 (see first example),

20 is another term of this type because the square 20 X 20 only can be tiled with 80 squares of size 1 x 1 and 80 squares of size 2 x 2.

- With terms as 15, 65, ... we only obtain sides of squares of type 2:

15 is a term of this type because the square 15 X 15 only can be tiled with 9 squares of size 3 X 3 and 9 squares of size 4 X 4 (see second example),

65 is another term of this type because the square 65 X 65 only can be tiled with 25 squares of size 5 X 5 and 25 squares of size 12 X 12.

- With terms as 30, 60, ... we obtain both sides of squares of type 1 and of type 2:

30 is a term of type 1 because the square 30 X 30 can be tiled with 180 squares of size 1 X 1 and 180 squares of size 2 X 2, but,

30 is also a term of type 2 because the square 30 X 30 can be tiled with 9 squares of size 6 X 6 and 9 squares of size 8 X 8.

PROG

(PARI) pts(lim) = my(v=List(), m2, s2, h2, h); for(middle=4, lim-1, m2=middle^2; for(small=1, middle, s2=small^2; if(issquare(h2=m2+s2, &h), if(h>lim, break); listput(v, [small, middle, h])))); vecsort(Vec(v)); \\ A009000

isdp4(s) = my(k=1, x); while(((x=k^4 - (k-1)^4) <= s), if (x == s, return (1)); k++); return(0);

isokp2(s) = {if (!isdp4(s), return(0)); if (s % 2, my(vp = pts(s)); for (i=1, #vp, my(vpi = vp[i], a = vpi[1], b = vpi[2], c = vpi[3]); if (a*c/(c-b) == s, return(1)); ); ); }

isok2(s) = {if (isokp2(s), return (1)); fordiv(s, d, if ((d>1) || (d<s), if (isokp2(s/d), return (1)))); }

isokp1(s) = {if (!(s % 2) && ispower(s/2, 4), return (0)); my(m = sqrtnint(s, 3)); vecsearch(setbinop((x, y)->x*y*(x^2+y^2), [1..m]), s); }

isok1(s) = {if (isokp1(s), return (1)); fordiv(s, d, if ((d>1) || (d<s), if (isokp1(s/d), return (1)))); }

isok(s) = isok1(s) || isok2(s); \\ Michel Marcus, Jun 04 2021

CROSSREFS

Subsequences: A008592 \ {0}, A008597 \ {0}, A034262 \ {0,1}.

Cf. A344331, A344332, A344333, A344334.

KEYWORD

nonn

AUTHOR

Bernard Schott, May 15 2021

EXTENSIONS

Corrected by Michel Marcus, May 18 2021

Incorrect term 145 removed by Michel Marcus, Jun 04 2021

STATUS

approved

A094053

Triangle read by rows: T(n,k) = k*(n-k), 1 <= k <= n.

+10
9

0, 1, 0, 2, 2, 0, 3, 4, 3, 0, 4, 6, 6, 4, 0, 5, 8, 9, 8, 5, 0, 6, 10, 12, 12, 10, 6, 0, 7, 12, 15, 16, 15, 12, 7, 0, 8, 14, 18, 20, 20, 18, 14, 8, 0, 9, 16, 21, 24, 25, 24, 21, 16, 9, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 0, 12

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

T(n,k) = A003991(n-1,k) for 1 <= k < n;

T(n,k) = T(n,n-1-k) for k < n;

T(n,1) = n-1; T(n,n) = 0; T(n,2) = A005843(n-2) for n > 1;

T(n,3) = A008585(n-3) for n>2; T(n,4) = A008586(n-4) for n > 3;

T(n,5) = A008587(n-5) for n>4; T(n,6) = A008588(n-6) for n > 5;

T(n,7) = A008589(n-7) for n>6; T(n,8) = A008590(n-8) for n > 7;

T(n,9) = A008591(n-9) for n>8; T(n,10) = A008592(n-10) for n > 9;

T(n,11) = A008593(n-11) for n>10; T(n,12) = A008594(n-12) for n > 11;

T(n,13) = A008595(n-13) for n>12; T(n,14) = A008596(n-14) for n > 13;

T(n,15) = A008597(n-15) for n>14; T(n,16) = A008598(n-16) for n > 15;

T(n,17) = A008599(n-17) for n>16; T(n,18) = A008600(n-18) for n > 17;

T(n,19) = A008601(n-19) for n>18; T(n,20) = A008602(n-20) for n > 19;

Row sums give A000292; triangle sums give A000332;

All numbers m > 0 occur A000005(m) times;

A002378(n) = T(A005408(n),n+1) = n*(n+1).

k-th columns are arithmetic progressions with step k, starting with 0. If a zero is prefixed to the sequence, then we get a new table where the columns are again arithmetic progressions with step k, but starting with k, k=0,1,2,...: 1st column = (0,0,0,...), 2nd column = (1,2,3,...), 3rd column = (2,4,6,8,...), etc. - M. F. Hasler, Feb 02 2013

Construct the infinite-dimensional matrix representation of angular momentum operators (J_1,J_2,J_3) in the Jordan-Schwinger form (cf. Harter, Klee, Schwinger). The triangle terms T(n,k) = T(2j,j+m) satisfy: (1/2)T(2j,j+m)^(1/2) = <j,m|J_1|j,m+1> = <j,m+1|J_1|j,m> = i <j,m|J_2|j,m+1> = -i <j,m+1|J_2|j,m>. Matrices for J_1 and J_2 are sparse. These equalities determine the only nonzero entries. - Bradley Klee, Jan 29 2016

T(n+1,k+1) is the number of degrees of freedom of a k-dimensional affine subspace within an n-dimensional vector space. This is most readily interpreted geometrically: e.g. in 3 dimensions a line (1-dimensional subspace) has T(4,2) = 4 degrees of freedom and a plane has T(4,3) = 3. T(n+1,1) = n indicates that points in n dimensions have n degrees of freedom. T(n,n) = 0 for any n as all n-dimensional spaces in an n-dimensional space are equivalent. - Daniel Leary, Apr 29 2020

LINKS

Table of n, a(n) for n=1..79.

W. Harter, Principles of Symmetry, Dynamics, Spectroscopy, Wiley, 1993, Ch. 5, page 345-346.

B. Klee, Quantum Angular Momentum Matrices, Wolfram Demonstrations Project, 2016.

J. Schwinger, On Angular Momentum , Cambridge: Harvard University, Nuclear Development Associates, Inc., 1952.

EXAMPLE

From M. F. Hasler, Feb 02 2013: (Start)

Triangle begins:

1, 0;

2, 2, 0;

3, 4, 3, 0;

4, 6, 6, 4, 0;

5, 8, 9, 8, 5, 0;

(...)

If an additional 0 was added at the beginning, this would become:

0, 1;

0, 2, 2;

0, 3, 4; 3;

0, 4, 6, 6, 4;

0, 5, 8, 9, 8, 5;

... (End)

MATHEMATICA

Flatten[Table[(j - m) (j + m + 1), {j, 0, 10, 1/2}, {m, -j, j}]] (* Bradley Klee, Jan 29 2016 *)

PROG

(Magma) /* As triangle */ [[k*(n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jan 30 2016

(PARI) {for(n=1, 13, for(k=1, n, print1(k*(n - k), " "); ); print(); ); } \\ Indranil Ghosh, Mar 12 2017

CROSSREFS

J_3: A114327; J_1^2, J_2^2: A141387, A268759.

Cf. A000005, A002378, A003991, A005408.

Cf. A000292 (row sums), A000332 (triangle sums).

T(n,k) for values of k:

A005843 (k=2), A008585 (k=3), A008586 (k=4), A008587 (k=5), A008588 (k=6), A008589 (k=7), A008590 (k=8), A008591 (k=9), A008592 (k=10), A008593 (k=11), A008594 (k=12), A008595 (k=13), A008596 (k=14), A008597 (k=15), A008598 (k=16), A008599 (k=17), A008600 (k=18), A008601 (k=19), A008602 (k=20).

KEYWORD

nonn,tabl

AUTHOR

Reinhard Zumkeller, May 31 2004

STATUS

approved

A139606

a(n) = 15*n + 6.

+10
8

6, 21, 36, 51, 66, 81, 96, 111, 126, 141, 156, 171, 186, 201, 216, 231, 246, 261, 276, 291, 306, 321, 336, 351, 366, 381, 396, 411, 426, 441, 456, 471, 486, 501, 516, 531, 546, 561, 576, 591, 606, 621, 636, 651, 666, 681, 696, 711, 726, 741, 756, 771, 786

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Numbers of the 6th column of positive numbers in the square array of nonnegative and polygonal numbers A139600.

6th transversal numbers (or 6-transversal numbers): (A000217(6)-6)*n + 6.

REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189. - From N. J. A. Sloane, Dec 01 2012

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

Index to sequences related to polygonal numbers

Index entries for linear recurrences with constant coefficients, signature (2,-1).

FORMULA

a(n) = A057145(n+2,6).

G.f.: 3*(2+3*x)/(x-1)^2 . - R. J. Mathar, Jul 28 2016

From Elmo R. Oliveira, Apr 12 2024: (Start)

E.g.f.: 3*exp(x)*(2 + 5*x).

a(n) = 3*A016873(n) = A008597(n) + 6.

a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

MATHEMATICA

Range[6, 1000, 15] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)

15*Range[0, 60]+6 (* or *) LinearRecurrence[{2, -1}, {6, 21}, 60] (* Harvey P. Dale, Aug 09 2019 *)

PROG

(PARI) a(n)=15*n+6 \\ Charles R Greathouse IV, Oct 05 2011

(Magma) [15*n+6 : n in [0..60]]; // Vincenzo Librandi, Oct 06 2011

CROSSREFS

Cf. A067076, A144562, A153238.

Cf. A000217, A008597, A016873, A057145, A139600.

KEYWORD

easy,nonn

AUTHOR

Omar E. Pol, Apr 27 2008

STATUS

approved

A169823

Multiples of 60.

+10
5

0, 60, 120, 180, 240, 300, 360, 420, 480, 540, 600, 660, 720, 780, 840, 900, 960, 1020, 1080, 1140, 1200, 1260, 1320, 1380, 1440, 1500, 1560, 1620, 1680, 1740, 1800, 1860, 1920, 1980, 2040, 2100, 2160, 2220, 2280, 2340, 2400, 2460, 2520, 2580, 2640, 2700

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Numbers that are divisible by all of 1, 2, 3, 4, 5, 6.

LINKS

Table of n, a(n) for n=0..45.

Erich Friedman, What's Special About This Number? (See the entry for "60")

Index entries for linear recurrences with constant coefficients, signature (2,-1).

FORMULA

From Elmo R. Oliveira, Apr 16 2024: (Start)

G.f.: 60*x/(x-1)^2.

E.g.f.: 60*x*exp(x).

a(n) = 60*n = 2*a(n-1) - a(n-2) for n >= 2.

a(n) = 2*A249674(n) = 3*A008602(n) = 4*A008597(n) = 5*A008594(n) = 6*A008592(n) = 10*A008588(n) = 12*A008587(n) = 15*A008586(n) = 20*A008585(n) = 30*A005843(n) = 60*A001477(n) = A169827(n)/14 = A169825(n)/7. (End)

MATHEMATICA

Range[0, 2700, 60] (* Vladimir Joseph Stephan Orlovsky, Jul 12 2011 *)

PROG

(PARI) a(n)=60*n \\ Charles R Greathouse IV, Mar 19 2017

CROSSREFS

Cf. A001477, A005843, A008585, A008586, A008587, A008588, A008592, A008594.

Cf. A008597, A008602, A169825, A169827, A249674.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, May 29 2010

STATUS

approved

A212773

Amounts (in cents) of coins in denominations 1, 5, 10, 25, and 50 (cents) which can consist of equal numbers of coins of all denominations present when two or more denominations are used (or none are used: term 0).

+10
5

0, 6, 11, 12, 15, 16, 18, 22, 24, 26, 30, 31, 32, 33, 35, 36, 40, 41, 42, 44, 45, 48, 51, 52, 54, 55, 56, 60, 61, 62, 64, 65, 66, 70, 72, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 90, 91, 93, 96, 99, 102, 104, 105, 108, 110, 112, 114, 120, 121, 122, 123, 124

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Nonnegative multiples of each of 6, 11, 15, 16, 26, 31, 35, 40, 41, 51, 56, 61, 65, 76, 81, 85, 86, and 91.

All products of terms are terms.

LINKS

Rick L. Shepherd, Table of n, a(n) for n = 1..10000

Index entries for sequences related to making change.

EXAMPLE

4 is not a term because it is not an appropriate multiple. Also 4 = 4*1 cannot be represented with more than one denomination of coin. Similarly 7 is not a term; although 7 = 7*1 = 2*1 + 1*5 does have a representation in terms of two denominations, 1 and 5, there are unequal numbers of each.

a(11) = 30 is a term because it is a multiple of 6. 30 = 5*1 + 5*5 = 2*5 + 2*10 = 1*5 + 1*25, so five coins each of denominations 1 and 5, two each of 5 and 10, or one each of 5 and 25 totals 30.

The term 34924118340711600 (5 times the LCM of the numbers in the first comment, so also divisible by 75) is the smallest which can be expressed in 26 such ways, one for each possible combination of two or more of these five coin denominations. (It also can be expressed as a multiple of each of these five alone of course.)

PROG

(PARI)

{c = 0; n = -1; until(c==10000, n++;

if(n%6==0 || n%11==0 || n%15==0 || n%16==0 || n%26==0 ||

n%31==0 || n%35==0 || n%40==0 || n%41==0 || n%51==0 ||

n%56==0 || n%61==0 || n%65==0 || n%76==0 || n%81==0 ||

n%85==0 || n%86==0 || n%91==0,

c++; write("b212773.txt", c, " ", n)))}

CROSSREFS

Cf. A212774, A008588, A008593, A008597, A008598, A135631.

KEYWORD

nonn

AUTHOR

Rick L. Shepherd, May 26 2012

STATUS

approved

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