A004942 -id:A004942

A195819

Multiples of 29.

+10
8

0, 29, 58, 87, 116, 145, 174, 203, 232, 261, 290, 319, 348, 377, 406, 435, 464, 493, 522, 551, 580, 609, 638, 667, 696, 725, 754, 783, 812, 841, 870, 899, 928, 957, 986, 1015, 1044, 1073, 1102, 1131, 1160, 1189, 1218, 1247, 1276, 1305, 1334

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

OFFSET

0,2

COMMENTS

Length of hypotenuses on the main diagonal of the Pythagorean spiral whose edges have length A195033 and whose vertices are the numbers A195034, if n >= 1.

LINKS

Table of n, a(n) for n=0..46.

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

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

FORMULA

a(n) = 29*n.

From Elmo R. Oliveira, Mar 21 2024: (Start)

G.f.: 29*x/(x-1)^2.

E.g.f.: 29*x*exp(x).

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

MATHEMATICA

29Range[0, 50] (* Harvey P. Dale, Oct 25 2011 *)

CROSSREFS

Cf. A004922, A004942, A008605, A010868, A135631, A195033, A195034.

KEYWORD

nonn,easy

AUTHOR

Omar E. Pol, Oct 12 2011

STATUS

approved

A004939

a(n) = round(n*phi^4), where phi is the golden ratio, A001622.

+10
6

0, 7, 14, 21, 27, 34, 41, 48, 55, 62, 69, 75, 82, 89, 96, 103, 110, 117, 123, 130, 137, 144, 151, 158, 164, 171, 178, 185, 192, 199, 206, 212, 219, 226, 233, 240, 247, 254, 260, 267, 274, 281, 288, 295, 302, 308

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

OFFSET

0,2

LINKS

Paolo Xausa, Table of n, a(n) for n = 0..10000

MATHEMATICA

Round[Range[0, 100]GoldenRatio^4] (* Paolo Xausa, Oct 28 2023 *)

PROG

(Magma) [Round(n*(7+3*Sqrt(5))/2): n in [0..80]]; // G. C. Greubel, Dec 04 2023

(SageMath) [round(golden_ratio^4*n) for n in range(81)] # G. C. Greubel, Dec 04 2023

CROSSREFS

Cf. A001622, A004940, A004941, A004942, A004943.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

A004940

a(n) = round(n*phi^5), where phi is the golden ratio, A001622.

+10
5

0, 11, 22, 33, 44, 55, 67, 78, 89, 100, 111, 122, 133, 144, 155, 166, 177, 189, 200, 211, 222, 233, 244, 255, 266, 277, 288, 299, 311, 322, 333, 344, 355, 366, 377, 388, 399, 410, 421, 433, 444, 455, 466, 477

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

OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..10000

MATHEMATICA

With[{c=GoldenRatio^5}, Round[c*Range[0, 50]]] (* Harvey P. Dale, Nov 28 2021 *)

PROG

(Magma) [Round(n*(11+5*Sqrt(5))/2): n in [0..80]]; // G. C. Greubel, Dec 04 2023

(SageMath) [round(golden_ratio^5*n) for n in range(81)] # G. C. Greubel, Dec 04 2023

CROSSREFS

Cf. A001622, A004939, A004941, A004942, A004943.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

A004941

a(n) = round(n*phi^6), where phi is the golden ratio, A001622.

+10
5

0, 18, 36, 54, 72, 90, 108, 126, 144, 161, 179, 197, 215, 233, 251, 269, 287, 305, 323, 341, 359, 377, 395, 413, 431, 449, 467, 484, 502, 520, 538, 556, 574, 592, 610, 628, 646, 664, 682, 700, 718, 736, 754, 772

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

OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..10000

MATHEMATICA

Round[Range[0, 50]GoldenRatio^6] (* Harvey P. Dale, Jul 21 2020 *)

PROG

(Magma) [Round(n*(9+4*Sqrt(5))): n in [0..80]]; // G. C. Greubel, Jan 23 2024

(SageMath) [round(golden_ratio^6*n) for n in range(81)] # G. C. Greubel, Jan 23 2024

CROSSREFS

Cf. A001622, A004939, A004940, A004942, A004943.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

A004943

a(n) = round(n*phi^8), where phi is the golden ratio, A001622.

+10
5

0, 47, 94, 141, 188, 235, 282, 329, 376, 423, 470, 517, 564, 611, 658, 705, 752, 799, 846, 893, 940, 987, 1034, 1081, 1127, 1174, 1221, 1268, 1315, 1362, 1409, 1456, 1503, 1550, 1597, 1644, 1691, 1738, 1785

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

OFFSET

0,2

COMMENTS

a(n+48) - a(n+47) - a(n+1) + a(n) = 0 for 0 <= n <= 1055, but not for n = 1056. - Robert Israel, Oct 18 2023

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

MAPLE

p8:= simplify(((1+sqrt(5))/2)^8);

seq(round(n*p8), n=0..100); # Robert Israel, Oct 18 2023

MATHEMATICA

Round[GoldenRatio^8 Range[0, 40]] (* Harvey P. Dale, Sep 18 2023 *)

CROSSREFS

Cf. A001622, A004939, A004940, A004941, A004942.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

A249079

a(n) = 29*n + floor( n/29 ) + 0^( 1-floor( (14+(n mod 29))/29 ) ).

+10
3

0, 29, 58, 87, 116, 145, 174, 203, 232, 261, 290, 319, 348, 377, 406, 436, 465, 494, 523, 552, 581, 610, 639, 668, 697, 726, 755, 784, 813, 842, 871, 900, 929, 958, 987, 1016, 1045, 1074, 1103, 1132, 1161, 1190, 1219, 1248, 1278, 1307, 1336

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

OFFSET

0,2

COMMENTS

This is an approximation to A004942 (Nearest integer to n*phi^7, where phi is the golden ratio, A001622).

LINKS

Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000

Ron Knott, Fibonacci numbers

Eric Weisstein's World of Mathematics, Golden Ratio

Wikipedia, Golden ratio

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

EXAMPLE

n= 0, 29*n+floor(0.0) +0^(1-floor(0.48))= 0 +0 +0 = 0 (n/29=0,0^1=0).

n=14, 29*n+floor(0.48)+0^(1-floor(0.97))= 406 +0 +0 = 406 (0^1=0).

n=15, 29*n+floor(0.52)+0^(1-floor(1.0)) = 435 +0 +1 = 436 (0^0=1).

n=28, 29*n+floor(0.97)+0^(1-floor(1.45))= 812 +0 +1 = 813 (0^0=1).

n=29, 29*n+floor(1.0) +0^(1-floor(0.48))= 841 +1 +0 = 842 (n/29*1,0^1=0).

n=43, 29*n+floor(1.48)+0^(1-floor(0.97))= 1247 +1 +0 = 1248 (0^1=0).

n=44, 29*n+floor(1.52)+0^(1-floor(1.0)) = 1276 +1 +1 = 1278 (0^0=1).

n=58, 29*n+floor(2.0) +0^(1-floor(0.48))= 1682 +2 +0 = 1684 (n/29*2,0^1=0).

n=85, 29*n+floor(2.93)+0^(1-floor(1.41))= 2465 +2 +1 = 2468 (0^0=1).

n=86, 29*n+floor(2.97)+0^(1-floor(1.45))= 2494 +2 +1 = 2497 (0^0=1).

n=87, 29*n+floor(3.0) +0^(1-floor(0.48))= 2523 +3 +0 = 2526 (n/29*3,0^0=0).

PROG

(Python)

for n in range(101):

print(29*n+n//29+0**(1-(14+n%29)//29), end=', ')

(Python)

def A249079(n):

a, b = divmod(n, 29)

return 29*n+a+int(b>=15) # Chai Wah Wu, Jul 27 2022

(PARI) a(n) = 29*n + n\29 + 0^(1 - (14+(n % 29))\29); \\ Michel Marcus, Oct 25 2014

(Magma) [29*n + Floor(n/29) + 0^(1-Floor((14+(n mod 29))/29)) : n in [0..50]]; // Vincenzo Librandi, Nov 05 2014

CROSSREFS

Cf. A001622 (phi), A195819 (29*n).

Cf. A004942 (round(n*phi^7)), A004922 (floor(n*phi^7)), A004962 (ceiling(n*phi^7)).

KEYWORD

nonn

AUTHOR

Karl V. Keller, Jr., Oct 20 2014

STATUS

approved

A248739

a(n) = 29*n + ceiling(n/29).

+10
2

0, 30, 59, 88, 117, 146, 175, 204, 233, 262, 291, 320, 349, 378, 407, 436, 465, 494, 523, 552, 581, 610, 639, 668, 697, 726, 755, 784, 813, 842, 872, 901, 930, 959, 988, 1017, 1046, 1075, 1104, 1133, 1162, 1191, 1220, 1249, 1278, 1307, 1336, 1365, 1394, 1423

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

OFFSET

0,2

COMMENTS

This is an approximation to A004962 (ceiling of n*phi^7, where phi is the golden ratio, A001622).

LINKS

Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000

Ron Knott, Fibonacci numbers and the golden section

Eric Weisstein's World of Mathematics, Golden Ratio

Wikipedia, Golden ratio

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

FORMULA

a(n) = 29*n + ceiling(n/29).

a(n) = A004962(n) for n < 871. - Joerg Arndt, Oct 18 2014

EXAMPLE

For n = 10, 29n + ceiling(n/29) = 290 + ceiling(0.3) = 290 + 1 = 291.

MAPLE

A248739:=n->29*n+ceil(n/29): seq(A248739(n), n=0..50); # Wesley Ivan Hurt, Oct 14 2014

MATHEMATICA

Table[29 n + Ceiling[n/29], {n, 0, 60}] (* Vincenzo Librandi, Oct 13 2014 *)

PROG

(Python)

from math import *

for n in range(0, 101):

..print n, (29*n+ceil(n/29.0))

(Magma) [29*n + Ceiling(n/29): n in [0..60]]; // Vincenzo Librandi, Oct 13 2014

CROSSREFS

Cf. A001622 (phi), A195819 (29*n).

Cf. A004922 (floor(n*phi^7)), A004962 (ceiling(n*phi^7)), A004942 (round(n*phi^7)).

KEYWORD

nonn,easy

AUTHOR

Karl V. Keller, Jr., Oct 13 2014

STATUS

approved

A248786

a(n) = 29*n + floor(n/29) + 0^n - 0^(n mod 29).

+10
2

0, 29, 58, 87, 116, 145, 174, 203, 232, 261, 290, 319, 348, 377, 406, 435, 464, 493, 522, 551, 580, 609, 638, 667, 696, 725, 754, 783, 812, 841, 871, 900, 929, 958, 987, 1016, 1045, 1074, 1103, 1132, 1161, 1190, 1219, 1248

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

OFFSET

0,2

COMMENTS

This is an approximation to A004922 (floor of n*phi^7, where phi is the golden ratio, A001622).

The "+ 0^n - 0^(n mod 29)" corrects a(n), for n=0 and multiples of 29. (See examples below.)

LINKS

Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000

Ron Knott, Fibonacci numbers

Eric Weisstein's World of Mathematics, Golden Ratio

Wikipedia, Golden ratio

EXAMPLE

For n = 0, 29*n + floor(0.0) + 0^0 - 0^(0) = 0 + 0 + 1 - 1 = 0 (n=29*0).

For n = 28, 29*n + floor(0.97) + 0^28 - 0^(28)= 812 + 0 + 0 - 0 = 812.

For n = 29, 29*n + floor(1.0) + 0^29 - 0^(0) = 841 + 1 + 0 - 1 = 841 (n=29*1).

For n = 31, 29*n + floor(1.1) + 0^31 - 0^(2) = 899 + 1 + 0 - 0 = 900.

For n = 87, 29*n + floor(3.0) + 0^87 - 0^(0) = 2523 + 3 + 0 - 1 = 2525 (n=29*3).

PROG

(Python)

from math import *

from decimal import *

getcontext().prec = 100

for n in range(0, 101):

..print n, (29*n+floor(n/29.0))+ 0**n-0**(n%29)

(Python)

def A248786(n):

a, b = divmod(n, 29)

return 29*n+a-int(not b) if n else 0 # Chai Wah Wu, Jul 27 2022

(Magma) [(29*n+Floor(n/29))+ 0^n-0^(n mod 29): n in [0..60]]; // Vincenzo Librandi, Oct 14 2014

(PARI) a(n) = 29*n+ n\29 + 0^n - 0^(n % 29); \\ Michel Marcus, Oct 14 2014

CROSSREFS

Cf. A001622 (phi), A195819 (29*n).

Cf. A004922 (floor(n*phi^7)), A004962 (ceiling(n*phi^7)), A004942 (round(n*phi^7)).

KEYWORD

nonn,easy

AUTHOR

Karl V. Keller, Jr., Oct 14 2014

STATUS

approved

A256278

a(0)=1, a(1)=2, a(n) = 31*a(n-1) - 29*a(n-2).

+10
1

1, 2, 33, 965, 28958, 869713, 26121321, 784539274, 23563199185, 707707535789, 21255600833094, 638400107288033, 19173990901769297, 575880114843495250, 17296237823997043137, 519482849213446974997, 15602377428720941973934, 468608697663159238917041

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

OFFSET

0,2

COMMENTS

The sequence A084330 is a(0)=0, a(1)=1, a(n)=31a(n-1)-29a(n-2), and the ratio A084330(n+1)/a(n) converges to phi^7 (~29.034441853748633...), where phi is the golden ratio (A001622).

The continued fraction for phi^7 is {29,{29}}, and 29 occurs in the following approximations for n*phi^7: A248786 (29*n+floor(n/29)+0^n-0^(n mod 29)) for A004922 (floor(n*phi^7)), A249079 (29*n+floor(n/29)+0^(1-floor((14+(n mod 29))/29)) for A004942 (round(n*phi^7)), and A248739 (29*n+ceiling(n/29)) for A004962 (ceiling(n*phi^7)).

LINKS

Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000

Eric W. Weisstein, From MathWorld--A Wolfram Web Resource, Golden Ratio

Eric W. Weisstein, From MathWorld--A Wolfram Web Resource, Golden Ratio Conjugate

Index entries for linear recurrences with constant coefficients, signature (31,-29).

FORMULA

G.f.: (1-29*x)/(29*x^2-31*x+1). - Vincenzo Librandi, Jun 03 2015

EXAMPLE

For n=3, 31*a(2)-29*a(1) = 31*(33)-29*(2) = 1023-58 = 965.

MAPLE

a:= n-> (<<0|1>, <-29|31>>^n. <<1, 2>>)[1, 1]:

seq(a(n), n=0..23); # Alois P. Heinz, Dec 22 2023

MATHEMATICA

LinearRecurrence[{31, -29}, {1, 2}, 50] (* or *) CoefficientList[Series[(1 - 29 x)/(29 x^2 - 31 x + 1), {x, 0, 33}], x] (* Vincenzo Librandi, Jun 03 2015 *)

PROG

(Python)

print(1, end=', ')

print(2, end=', ')

an = [1, 2]

for n in range(2, 26):

print(31*an[n-1]-29*an[n-2], end=', ')

an.append(31*an[n-1]-29*an[n-2])

(Magma) I:=[1, 2]; [n le 2 select I[n] else 31*Self(n-1)-29*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 03 2015

CROSSREFS

Cf. A001622, A195819 (29*n), A084330, A004922, A004942, A004962, A248786, A249079, A248739.

KEYWORD

nonn,easy

AUTHOR

Karl V. Keller, Jr., Jun 02 2015

STATUS

approved