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Search: a057780 -id:a057780
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A218864
Numbers of the form 9*k^2 + 8*k, k an integer.
+10
37
0, 1, 17, 20, 52, 57, 105, 112, 176, 185, 265, 276, 372, 385, 497, 512, 640, 657, 801, 820, 980, 1001, 1177, 1200, 1392, 1417, 1625, 1652, 1876, 1905, 2145, 2176, 2432, 2465, 2737, 2772, 3060, 3097, 3401, 3440, 3760, 3801, 4137, 4180, 4532, 4577, 4945, 4992
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Numbers m such that 9*m + 16 is a square. - Vincenzo Librandi, Apr 07 2013
Equivalently, integers of the form h*(h + 8)/9 (nonnegative values of h are listed in A090570). - Bruno Berselli, Jul 15 2016
Generalized 20-gonal (or icosagonal) numbers: r*(9*r - 8) with r = 0, +1, -1, +2, -2, +3, -3, ... - Omar E. Pol, Jun 06 2018
Partial sums of A317316. - Omar E. Pol, Jul 28 2018
Exponents in expansion of Product_{n >= 1} (1 + x^(18*n-17))*(1 + x^(18*n-1))*(1 - x^(18*n)) = 1 + x + x^17 + x^20 + x^52 + .... - Peter Bala, Dec 10 2020
LINKS
Jason Kimberley, Table of n, a(n) for n = 1..2000
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math., Vol. 274, No. 1-3 (2004), pp. 9-24. See C(q).
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
FORMULA
a(n) = (18*n*(n - 1) - 7*(-1)^n*(2*n - 1) - 7)/8. - Bruno Berselli, Nov 13 2012
G.f.: x*(1 + 16*x + x^2)/((1 + x)^2*(1 - x)^3). - Bruno Berselli, Nov 14 2012
Sum_{n>=2} 1/a(n) = (9 + 8*Pi*cot(Pi/9))/64. - Amiram Eldar, Feb 28 2022
MATHEMATICA
Array[(18 # (# - 1) - 7 (-1)^#*(2 # - 1) - 7)/8 &, 48] (* or *)
CoefficientList[Series[x (1 + 16 x + x^2)/((1 + x)^2*(1 - x)^3), {x, 0, 47}], x] (* Michael De Vlieger, Jun 06 2018 *)
PROG
(Magma) a:=func<n | 9*n^2+8*n>; [0]cat[a(n*m): m in [-1, 1], n in [1..20]];
CROSSREFS
Characteristic function is A205987.
Numbers of the form 9*m^2+k*m, for integer n: A016766 (k=0), A132355 (k=2), A185039 (k=4), A057780 (k=6), this sequence (k=8).
Cf. A074377 (numbers m such that 16*m+9 is a square).
Cf. A317316.
For similar sequences of numbers m such that 9*m+i is a square, see list in A266956.
Cf. sequences of the form m*(m+i)/(i+1) listed in A274978. [Bruno Berselli, Jul 25 2016]
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), this sequence (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
KEYWORD
nonn,easy
AUTHOR
Jason Kimberley, Nov 08 2012
STATUS
approved
A016766
a(n) = (3*n)^2.
+10
22
0, 9, 36, 81, 144, 225, 324, 441, 576, 729, 900, 1089, 1296, 1521, 1764, 2025, 2304, 2601, 2916, 3249, 3600, 3969, 4356, 4761, 5184, 5625, 6084, 6561, 7056, 7569, 8100, 8649, 9216, 9801, 10404, 11025, 11664, 12321, 12996, 13689, 14400, 15129, 15876
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Number of edges of the complete tripartite graph of order 6n, K_n, n, 4n. - Roberto E. Martinez II, Jan 07 2002
Area of a square with side 3n. - Wesley Ivan Hurt, Sep 24 2014
Right-hand side of the binomial coefficient identity Sum_{k = 0..3*n} (-1)^(n+k+1)* binomial(3*n,k)*binomial(3*n + k,k)*(3*n - k) = a(n). - Peter Bala, Jan 12 2022
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..200
John Elias, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = 9*n^2 = 9 * A000290(n). - Omar E. Pol, Dec 11 2008
a(n) = 3 * A033428(n). - Omar E. Pol, Dec 13 2008
a(n) = a(n-1) + 9*(2*n-1) for n>0, a(0)=0. - Vincenzo Librandi, Nov 19 2010
From Wesley Ivan Hurt, Sep 24 2014: (Start)
G.f.: 9*x*(1 + x)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n>3.
a(n) = A000290(A008585(n)). (End)
From Amiram Eldar, Jan 25 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/54.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/108.
Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/3)/(Pi/3).
Product_{n>=1} (1 - 1/a(n)) = sinh(Pi/2)/(Pi/2) = 3*sqrt(3)/(2*Pi) (A086089). (End)
a(n) = A051624(n) + 8*A000217(n). In general, if P(k,n) = the k-th n-gonal number, then (k*n)^2 = P(k^2 + 3,n) + (k^2 - 1)*A000217(n). - Charlie Marion, Mar 09 2022
MAPLE
A016766:=n->(3*n)^2: seq(A016766(n), n=0..50); # Wesley Ivan Hurt, Sep 24 2014
MATHEMATICA
(3Range[0, 49])^2 (* Alonso del Arte, Sep 24 2014 *)
PROG
(Maxima) A016766(n):=(3*n)^2$
makelist(A016766(n), n, 0, 20); /* Martin Ettl, Nov 12 2012 */
(Magma) [(3*n)^2 : n in [0..50]]; // Wesley Ivan Hurt, Sep 24 2014
(PARI) a(n)=9*n^2 \\ Charles R Greathouse IV, Sep 28 2015
CROSSREFS
Numbers of the form 9n^2 + kn, for integer n: this sequence (k = 0), A132355 (k = 2), A185039 (k = 4), A057780 (k = 6), A218864 (k = 8). - Jason Kimberley, Nov 09 2012
Cf. A086089.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Zerinvary Lajos, May 30 2006
STATUS
approved
A132355
Numbers of the form 9*h^2 + 2*h, for h an integer.
+10
14
0, 7, 11, 32, 40, 75, 87, 136, 152, 215, 235, 312, 336, 427, 455, 560, 592, 711, 747, 880, 920, 1067, 1111, 1272, 1320, 1495, 1547, 1736, 1792, 1995, 2055, 2272, 2336, 2567, 2635, 2880, 2952, 3211, 3287, 3560, 3640, 3927, 4011, 4312, 4400, 4715, 4807
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
X values of solutions to the equation 9*X^3 + X^2 = Y^2.
The set of all m such that 9*m + 1 is a perfect square. - Gary Detlefs, Feb 22 2010
The concatenation of any term with 11..11 (1 repeated an even number of times, see A099814) belongs to the list. Example: 87 is a term, so also 8711, 871111, 87111111, 871111111111, ... are terms of this sequence. - Bruno Berselli, May 15 2017
LINKS
Jason Kimberley, Table of n, a(n) for n = 1..2108
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math., Vol. 274, No. 1-3 (2004), pp. 9-24. See A(q).
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
FORMULA
a(2*k) = k*(9*k-2), a(2*k+1) = k*(9*k+2).
a(n) = n^2 - n + 5*floor(n/2)^2. - Gary Detlefs, Feb 23 2010
From R. J. Mathar, Mar 17 2010: (Start)
a(n) = +a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5).
G.f.: x^2*(7 + 4*x + 7*x^2)/((1 + x)^2*(1 - x)^3). (End)
a(n) = (2*n - 1 + (-1)^n)*(9*(2*n - 1) + (-1)^n)/16. - Luce ETIENNE, Sep 13 2014
Sum_{n>=2} 1/a(n) = 9/4 - cot(2*Pi/9)*Pi/2. - Amiram Eldar, Mar 15 2022
MAPLE
readlib(issqr); for n from 0 to 3560 do if(issqr(9*n+1)) then print(n) fi od; # Gary Detlefs, Feb 22 2010
seq(n^2+n+5*ceil(n/2)^2, n=0..39); # Gary Detlefs, Feb 23 2010
MATHEMATICA
f[n_]:=IntegerQ[Sqrt[1+9*n]]; Select[Range[0, 8! ], f[ # ]&] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
Sort[Table[9n^2+2n, {n, -30, 30}]] (* Harvey P. Dale, Dec 06 2013 *)
PROG
(Magma) a:=func<n | 9*n^2+2*n>; [0] cat [a(n*m): m in [-1, 1], n in [1..25]]; // Jason Kimberley, Nov 08 2012
(PARI) a(n)=n^2-n+5*(n\2)^2 \\ Charles R Greathouse IV, Sep 28 2015
CROSSREFS
A205808 is the characteristic function.
Cf. A000217, A001082, A002378, A005563, A028347, A036666, A046092, A054000, A056220, A062717, A087475, A132209, A010701, A056020.
Numbers of the form 9*n^2+k*n, for integer n: A016766 (k=0), this sequence (k=2), A185039 (k=4), A057780 (k=6), A218864 (k=8). - Jason Kimberley, Nov 09 2012
For similar sequences of numbers m such that 9*m+k is a square, see list in A266956.
KEYWORD
nonn,easy
AUTHOR
Mohamed Bouhamida, Nov 08 2007
EXTENSIONS
Simpler definition and minor edits from N. J. A. Sloane, Feb 03 2012
Since this is a list, offset changed to 1 and formulas translated by Jason Kimberley, Nov 18 2012
STATUS
approved
A185039
Numbers of the form 9*m^2 + 4*m, m an integer.
+10
9
0, 5, 13, 28, 44, 69, 93, 128, 160, 205, 245, 300, 348, 413, 469, 544, 608, 693, 765, 860, 940, 1045, 1133, 1248, 1344, 1469, 1573, 1708, 1820, 1965, 2085, 2240, 2368, 2533, 2669, 2844, 2988, 3173, 3325, 3520, 3680, 3885, 4053, 4268, 4444, 4669, 4853, 5088
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Also, numbers m such that 9*m+4 is a square. After 0, therefore, there are no squares in this sequence. - Bruno Berselli, Jan 07 2016
LINKS
Bruno Berselli, Table of n, a(n) for n = 1..1000
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math. 274 (2004), no. 1-3, 9-24. See B(q).
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
FORMULA
From Bruno Berselli, Feb 04 2012: (Start)
G.f.: x*(5+8*x+5*x^2)/((x+1)^2*(1-x)^3).
a(n) = a(-n+1) = (18*n*(n-1)+(2*n-1)*(-1)^n+1)/8 = A004526(n)*A156638(n). (End).
MATHEMATICA
CoefficientList[Series[x*(5+8*x+5*x^2)/((x+1)^2*(1-x)^3), {x, 0, 50}], x] (* G. C. Greubel, Jun 20 2017 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 5, 13, 28, 44}, 50] (* Harvey P. Dale, Jan 23 2018 *)
PROG
(Magma) [0] cat &cat[[9*n^2-4*n, 9*n^2+4*n]: n in [1..32]]; // Bruno Berselli, Feb 04 2011
(PARI) x='x+O('x^50); Vec(x*(5+8*x+5*x^2)/((x+1)^2*(1-x)^3)) \\ G. C. Greubel, Jun 20 2017
CROSSREFS
Characteristic function is A205809.
Numbers of the form 9*n^2+k*n, for integer n: A016766 (k=0), A132355 (k=2), this sequence (k=4), A057780 (k=6), A218864 (k=8). [Jason Kimberley, Nov 08 2012]
For similar sequences of numbers m such that 9*m+k is a square, see list in A266956.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Feb 04 2012
STATUS
approved
A033684
1 iff n is a square not divisible by 3.
+10
4
0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
a(n)=1 iff n-1 is in the list A057780. - Jason Kimberley, Nov 13 2012
REFERENCES
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 105, Eq. (40).
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..65536
Index entries for characteristic functions.
FORMULA
Essentially the series psi_3(z)=(1/2)(theta_3(z/9)-theta_3(z)).
a(n) * A000035(n) = A033683(n).
Multiplicative with a(p^e) = 1 if 2 divides e and p != 3, 0 otherwise. - Mitch Harris, Jun 09 2005
Dirichlet g.f.: zeta(2*s)*(1-3^(-2*s)). - R. J. Mathar, Mar 10 2011
a(n) = A010052(n)*A011655(n). - Antti Karttunen, Sep 13 2017
Sum_{k=1..n} a(k) ~ 2*sqrt(n)/3. - Amiram Eldar, Jan 14 2024
MAPLE
A033684 := proc(n)
if issqr(n) then
if n mod 3 = 0 then
0;
else
1;
end if;
else
0;
end if;
end proc:
seq(A033684(n), n=0..80) ; # R. J. Mathar, Oct 07 2011
MATHEMATICA
Table[If[IntegerQ[Sqrt[n]]&&Mod[n, 3]!=0, 1, 0], {n, 0, 130}] (* Harvey P. Dale, Oct 19 2018 *)
PROG
(PARI) A033684(n) = (issquare(n)&&(n%3)); \\ Antti Karttunen, Sep 13 2017
CROSSREFS
Cf. A010052, A011655, A033683, A057780.
KEYWORD
nonn,easy,mult
AUTHOR
N. J. A. Sloane
EXTENSIONS
Data-section extended up to a(121) by Antti Karttunen, Sep 13 2017
STATUS
approved
page 1

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