A016779 -id:A016779 - cp's OEIS frontend

A016787 a(n) = (3*n + 1)^11.

1, 4194304, 1977326743, 100000000000, 1792160394037, 17592186044416, 116490258898219, 584318301411328, 2384185791015625, 8293509467471872, 25408476896404831, 70188843638032384, 177917621779460413, 419430400000000000, 929293739471222707, 1951354384207722496

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A016777, A016778, A016779, A016780, A016781, A016782, A016783, A016784, A016785, A016786.

Subsequence of A008455.

[(3*n+1)^11: n in [0..20]]; // Vincenzo Librandi, Sep 29 2011

Table[(3*n + 1)^11, {n, 0, 30}] (* Amiram Eldar, Mar 30 2022 *)

From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016777(n)^11.
Sum_{n>=0} 1/a(n) = 7388*Pi^11/(2511058725*sqrt(3)) + 88573*zeta(11)/177147. (End)
a(n) = 12*a(n-1) - 66*a(n-2) + 220*a(n-3) - 495*a(n-4) + 792*a(n-5) - 924*a(n-6) + 792*a(n-7) - 495*a(n-8) + 220*a(n-9) - 66*a(n-10) + 12*a(n-11) - a(n-12). - Wesley Ivan Hurt, Apr 12 2023

A118719 Cubes for which the digital root is also a cube.

Original entry on oeis.org

0, 1, 8, 64, 125, 343, 512, 1000, 1331, 2197, 2744, 4096, 4913, 6859, 8000, 10648, 12167, 15625, 17576, 21952, 24389, 29791, 32768, 39304, 42875, 50653, 54872, 64000, 68921, 79507, 85184, 97336, 103823, 117649, 125000, 140608, 148877

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), May 21 2006

All cubes have a digital root 1,8 or 9. (except for the number 0) So this sequence contains all cubes with a digital root which is not 9.

This sequence is 0 union A016779 union A016791.

			64 is in the sequence because (1) it is a cube and (2) the digital root 1 is also a cube.

Vincenzo Librandi, Table of n, a(n) for n = 1..3000

Cf. A000578, A002117, A010888, A116978.

[0] cat [(6*n+(-1)^n-9)^3 div 64: n in [2..37]];  // Bruno Berselli, May 05 2011

Join[{0}, Table[(3*k + {1, 2})^3, {k, 0, 15}] // Flatten] (* Amiram Eldar, Dec 19 2020 *)

a010888(n)=if(n, (n-1)%9+1)
lista(nn) = {for (n=0, nn, if (ispower(a010888(n^3), 3), print1(n^3, ", ")););} \\ Michel Marcus, Feb 18 2015

a(n) = (floor(3*n/2)-2)^3 for n >= 2. - Nathaniel Johnston, May 05 2011
G.f.: x^2*(1+7*x+53*x^2+40*x^3+53*x^4+7*x^5+x^6)/((1+x)^3*(1-x)^4).  a(n) = A001651(n-1)^3 for n>1. - Bruno Berselli, May 05 2011
Sum_{n>=2} 1/a(n) = 26*zeta(3)/27. - Amiram Eldar, Dec 19 2020

A016786 a(n) = (3*n+1)^10.

Original entry on oeis.org

1, 1048576, 282475249, 10000000000, 137858491849, 1099511627776, 6131066257801, 26559922791424, 95367431640625, 296196766695424, 819628286980801, 2064377754059776, 4808584372417849, 10485760000000000, 21611482313284249, 42420747482776576, 79792266297612001

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A008454, A016777, A016778, A016779, A016780, A016781, A016782, A016783, A016784, A016785.

[(3*n+1)^10: n in [0..20]]; // Vincenzo Librandi, Sep 29 2011

Table[(3n+1)^10,{n,0,100}] (* Mohammad K. Azarian, Jun 15 2016 *)
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,1048576,282475249,10000000000,137858491849,1099511627776,6131066257801,26559922791424,95367431640625,296196766695424,819628286980801},20] (* Harvey P. Dale, May 14 2019 *)

a(n) = A008454(A016777(n)). - Michel Marcus, Jun 15 2016

Sum_{n>=0} 1/a(n) = PolyGamma(9, 1/3)/21427701120. - Amiram Eldar, Mar 29 2022

A377557 Decimal expansion of 2Pi^3/(81sqrt(3)) + 13*zeta(3)/27.

Original entry on oeis.org

1, 0, 2, 0, 7, 8, 0, 0, 4, 4, 4, 3, 3, 3, 6, 3, 1, 0, 2, 8, 2, 3, 2, 5, 4, 7, 3, 9, 9, 0, 3, 9, 8, 1, 8, 2, 5, 3, 5, 3, 4, 1, 0, 9, 3, 7, 5, 1, 9, 0, 6, 9, 6, 6, 9, 7, 3, 5, 7, 2, 0, 7, 5, 2, 5, 3, 9, 1, 4, 6, 5, 9, 9, 2, 6, 5, 6, 2, 7, 1, 5, 5, 4, 4, 9, 8, 0, 6, 7, 2, 0, 3, 4, 2, 6, 7, 6, 1, 3, 7

Offset: 1

Stefano Spezia, Nov 01 2024

			1.0207800444333631028232547399039818253534109375...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 42.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 52.

Cf. A002117, A002194, A016777, A016779, A091925, A233091, A377558, A377559, A377560.

RealDigits[2Pi^3/(81Sqrt[3])+13Zeta[3]/27,10,100][[1]]

Equals Sum_{k>=0} 1/(3*k + 1)^3 (see Finch).
Equals -psi''(1/3)/54 (see Shamos).
Equals hypergeom([1/3, 1/3, 1/3, 1], [4/3, 4/3, 4/3], 1). - R. J. Mathar, Jul 14 2025

A016788 a(n) = (3*n+1)^12.

Original entry on oeis.org

1, 16777216, 13841287201, 1000000000000, 23298085122481, 281474976710656, 2213314919066161, 12855002631049216, 59604644775390625, 232218265089212416, 787662783788549761, 2386420683693101056, 6582952005840035281, 16777216000000000000, 39959630797262576401

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A008456, A016777, A016778, A016779, A016780, A016781, A016782, A016783, A016784, A016785, A016786, A016787.

Subsequence of A008456.

[(3*n+1)^12: n in [0..20]]; // Vincenzo Librandi, Sep 29 2011

Table[(3n+1)^12,{n,0,100}] (* Mohammad K. Azarian, Jun 15 2016 *)

a(n) = A008456(A016777(n)). - Michel Marcus, Jun 16 2016

Sum_{n>=0} 1/a(n) = PolyGamma(11, 1/3)/21213424108800. - Amiram Eldar, Mar 30 2022

A017199 a(n) = (9*n + 3)^3.

Original entry on oeis.org

27, 1728, 9261, 27000, 59319, 110592, 185193, 287496, 421875, 592704, 804357, 1061208, 1367631, 1728000, 2146689, 2628072, 3176523, 3796416, 4492125, 5268024, 6128487, 7077888, 8120601, 9261000, 10503459, 11852352, 13312053, 14886936, 16581375, 18399744, 20346417

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002117, A016779, A017197.

[(9*n+3)^3: n in [0..30]]; // Vincenzo Librandi, Jul 23 2011

(9Range[0,30]+3)^3 (* or *) LinearRecurrence[{4,-6,4,-1},{27,1728,9261,27000},30] (* Harvey P. Dale, Jun 20 2024 *)

G.f.: 27*(1 + 60*x + 93*x^2 + 8*x^3)/ (x-1)^4. - R. J. Mathar, Aug 01 2014
a(n) = 27*A016779(n). - R. J. Mathar, Aug 01 2014
From Amiram Eldar, Oct 03 2024: (Start)
a(n) = A017197(n)^3.
Sum_{n>=0} 1/a(n) = 2*Pi^3/(2187*sqrt(3)) + 13*zeta(3)/729. (End)

A017571 a(n) = (12n+4)^3.

Original entry on oeis.org

64, 4096, 21952, 64000, 140608, 262144, 438976, 681472, 1000000, 1404928, 1906624, 2515456, 3241792, 4096000, 5088448, 6229504, 7529536, 8998912, 10648000, 12487168, 14526784, 16777216, 19248832, 21952000, 24897088, 28094464, 31554496, 35287552, 39304000, 43614208

Offset: 0

N. J. A. Sloane

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578, A002117, A016779, A017569.

a[n_] := (12*n+4)^3; Array[a, 30, 0] (* Amiram Eldar, Jul 14 2024 *)
LinearRecurrence[{4,-6,4,-1},{64,4096,21952,64000},30] (* Harvey P. Dale, Sep 06 2024 *)

From Amiram Eldar, Jul 14 2024: (Start)
a(n) = A000578(A017569(n)) = A017569(n)^3.
a(n) = 64 * A016779(n).
Sum_{n>=0} 1/a(n) = Pi^3/(2592*sqrt(3)) + 13*zeta(3)/1728. (End)

More terms from Amiram Eldar, Jul 14 2024

A229212 Square array of numerators of t(n,k) = (1+1/(k*n))^n, read by descending antidiagonals.

Original entry on oeis.org

2, 3, 9, 4, 25, 64, 5, 49, 343, 625, 6, 81, 1000, 6561, 7776, 7, 121, 2197, 28561, 161051, 117649, 8, 169, 4096, 83521, 1048576, 4826809, 2097152, 9, 225, 6859, 194481, 4084101, 47045881, 170859375, 43046721, 10, 289

Offset: 1

Jean-François Alcover, Sep 16 2013

Limit(t(n,k), n -> infinity) = exp(1/k).
1st row = A020725
2nd row = A016754
3rd row = A016779
4th row = A016816
5th row = A016865
1st column = A000169
2nd column = A085527

			Table of fractions begins:
   2,       3/2,        4/3,         5/4, ...
  9/4,     25/16,      49/36,       81/64, ...
64/27,   343/216,   1000/729,    2197/1728, ...
625/256, 6561/4096, 28561/20736, 83521/65536, ...
...
Table of numerators begins:
2,      3,     4,     5, ...
9,     25,    49,    81, ...
64,   343,  1000,  2197, ...
625, 6561, 28561, 83521, ...
...
Triangle of antidiagonals begins:
2;
3, 9;
4, 25, 64;
5, 49, 343, 625;
...

Cf. A229213(denominators), A016754, A016779, A016816, A016865, A000169, A085527.

t[n_, k_] := (1+1/(k*n))^n; Table[t[n-k+1, k], {n, 1, 9}, {k, n, 1, -1}] // Flatten // Numerator

cp's OEIS Frontend

A016787 a(n) = (3*n + 1)^11.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A118719 Cubes for which the digital root is also a cube.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A016786 a(n) = (3*n+1)^10.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A377557 Decimal expansion of 2*Pi^3/(81*sqrt(3)) + 13*zeta(3)/27.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A016788 a(n) = (3*n+1)^12.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A017199 a(n) = (9*n + 3)^3.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A017571 a(n) = (12n+4)^3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A229212 Square array of numerators of t(n,k) = (1+1/(k*n))^n, read by descending antidiagonals.

Views

Author

Keywords

Comments

A377557 Decimal expansion of 2Pi^3/(81sqrt(3)) + 13*zeta(3)/27.