A016911 -id:A016911 - cp's OEIS frontend

A016935 a(n) = (6*n + 2)^3.

8, 512, 2744, 8000, 17576, 32768, 54872, 85184, 125000, 175616, 238328, 314432, 405224, 512000, 636056, 778688, 941192, 1124864, 1331000, 1560896, 1815848, 2097152, 2406104, 2744000, 3112136

Offset: 0

N. J. A. Sloane

The generating function is 8 times the g.f. of A016779. - R. J. Mathar, May 07 2008

			a(1) = (6*1 + 2)^3 = 8^3 = 512.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002117, A016779, A016911, A016923, A016933, A016959, A016971.

[(6*n+2)^3: n in [0..50]]; // Vincenzo Librandi, May 04 2011

(6*Range[0,30]+2)^3 (* or *) LinearRecurrence[{4,-6,4,-1},{8,512,2744,8000},30] (* Harvey P. Dale, Aug 23 2019 *)

a(n) = 8*A016779(n). - R. J. Mathar, May 07 2008
Sum_{n>=0} 1/a(n) = Pi^3 / (324*sqrt(3)) + 13*zeta(3)/216. - Amiram Eldar, Oct 02 2020
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Oct 02 2020
G.f.: 8*(1+60*x+93*x^2+8*x^3)/(-1+x)^4. - Wesley Ivan Hurt, Oct 02 2020

A016959 a(n) = (6*n + 4)^3.

Original entry on oeis.org

64, 1000, 4096, 10648, 21952, 39304, 64000, 97336, 140608, 195112, 262144, 343000, 438976, 551368, 681472, 830584, 1000000, 1191016, 1404928, 1643032, 1906624, 2197000, 2515456, 2863288, 3241792

Offset: 0

N. J. A. Sloane

			a(0) = (6*0 + 4)^3 = 4^3 = 64.

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002117, A016911, A016923, A016935, A016947, A016971.

[(6*n+4)^3: n in [0..40]]; // Vincenzo Librandi, May 06 2011

CoefficientList[Series[8*(x^3 + 60*x^2 + 93*x + 8)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jan 27 2013 *)
(6*Range[0,30]+4)^3 (* or *) LinearRecurrence[{4,-6,4,-1},{64,1000,4096,10648},30] (* Harvey P. Dale, Nov 22 2018 *)

G.f.: 8*(x^3 + 60*x^2 + 93*x + 8)/(1-x)^4. - Vincenzo Librandi, Jan 27 2013
Sum_{n>=0} 1/a(n) = -Pi^3 / (324*sqrt(3)) + 13*zeta(3)/216. - Amiram Eldar, Oct 02 2020
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Oct 02 2020

A016971 a(n) = (6*n + 5)^3.

Original entry on oeis.org

125, 1331, 4913, 12167, 24389, 42875, 68921, 103823, 148877, 205379, 274625, 357911, 456533, 571787, 704969, 857375, 1030301, 1225043, 1442897, 1685159, 1953125, 2248091, 2571353, 2924207, 3307949

Offset: 0

N. J. A. Sloane

			a(0) = (6*0 + 5)^3 = 5^3 = 125.

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

Cf. A002117, A016911, A016923, A016935, A016947, A016959.

[(6*n+5)^3: n in [0..40]]; // Vincenzo Librandi, May 07 2011

Table[(6*n + 5)^3, {n, 0, 25}] (* Amiram Eldar, Oct 02 2020 *)

Sum_{n>=0} 1/a(n) = -Pi^3/(36*sqrt(3)) + 91*zeta(3)/216. - Amiram Eldar, Oct 02 2020
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Oct 02 2020
a(n) = (125+831*x+339*x^2+x^3)/(-1+x)^4. - Wesley Ivan Hurt, Oct 02 2020

A016947 a(n) = (6*n + 3)^3.

Original entry on oeis.org

27, 729, 3375, 9261, 19683, 35937, 59319, 91125, 132651, 185193, 250047, 328509, 421875, 531441, 658503, 804357, 970299, 1157625, 1367631, 1601613, 1860867, 2146689, 2460375, 2803221, 3176523, 3581577, 4019679, 4492125, 5000211, 5545233, 6128487, 6751269

Offset: 0

N. J. A. Sloane

			a(0) = (6*0 + 3)^3 = 3^3 = 27.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578, A002117, A016755, A016911, A016923, A016935, A016945, A016946, A016959, A016971.

[(6*n+3)^3: n in [0..50]]; // Vincenzo Librandi, May 05 2011

Table[(6*n + 3)^3, {n, 0, 25}] (* Amiram Eldar, Oct 02 2020 *)
LinearRecurrence[{4,-6,4,-1},{27,729,3375,9261},40] (* Harvey P. Dale, Jul 02 2025 *)

Sum_{n>=0} 1/a(n) = 7*zeta(3)/216. - Amiram Eldar, Oct 02 2020
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Oct 02 2020
G.f.: 27*(1+x)*(1+22*x+x^2)/(-1+x)^4. - Wesley Ivan Hurt, Oct 02 2020
From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016945(n)^3.
a(n) = 3^3*A016755(n).
Sum_{n>=0} (-1)^n/a(n) = Pi^3/864. (End)

A244725 a(n) = 5*n^3.

Original entry on oeis.org

0, 5, 40, 135, 320, 625, 1080, 1715, 2560, 3645, 5000, 6655, 8640, 10985, 13720, 16875, 20480, 24565, 29160, 34295, 40000, 46305, 53240, 60835, 69120, 78125, 87880, 98415, 109760, 121945, 135000, 148955, 163840, 179685, 196520, 214375, 233280, 253265

Offset: 0

Vincenzo Librandi, Jul 05 2014

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

Cf. similar sequences of the type k*n^3: A000578 (k=1), A033431 (k=2), A117642 (k=3), A033430 (k=4), this sequence (k=5), A244726 (k=6), A244727 (k=7), A016743 (k=8), A244728 (k=9), A244729 (k=10), A016767 (k=27), A016803 (k=64), A016851 (k=125), A016911 (k=216), A016983 (k=343), A017067 (k=512), A017163 (k=729), A017271 (k=1000), A017391 (k=1331), A017523 (k=1728).

Magma
```
[5*n^3: n in [0..40]];
```

I:=[0,5,40,135]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]];

Table[5 n^3, {n, 0, 40}] (* or *) CoefficientList[Series[5 x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 40}], x]

a(n)=5*n^3 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 5*x*(1 + 4*x + x^2)/(1 - x)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3.

A305728 Numbers of the form 216*p^3, where p is a Pythagorean prime (A002144).

Original entry on oeis.org

27000, 474552, 1061208, 5268024, 10941048, 14886936, 32157432, 49027896, 84027672, 152273304, 197137368, 222545016, 279726264, 311665752, 555412248, 714516984, 835896888, 1118386872, 1280824056, 1552836312, 1651400568, 2593941624, 2732256792, 3023464536, 3666512088

Offset: 1

Bruno Berselli, Jun 22 2018

nonn

No term can be written as x^2 + y^2 + z^9.

Colin Barker, Table of n, a(n) for n = 1..1000
W. Jagy and I. Kaplansky, Sums of Squares, Cubes and Higher Powers, Experimental Mathematics, vol. 4 (1995), p. 171 (see Theorem under Higher powers).

Cf. A002144, A016911, A080169.

[216*p^3: p in PrimesUpTo(300) | IsOne(p mod 4)];

P := select(p -> isprime(p), [seq(n, n=5..1000, 4)]):
seq((6*p)^3, p in P); # Peter Luschny, Jun 22 2018

P = Select[Range[5, 300, 4], PrimeQ];
A305728 = (6P)^3 (* Jean-François Alcover, Jun 22 2018 *)

first(n) = my(res=List()); forprime(p=5, oo, if(p%4 == 1, listput(res,(6*p)^3); n--; if(n==0, return(res)))) \\ David A. Corneth, Jun 27 2018

cp's OEIS Frontend

A016935 a(n) = (6*n + 2)^3.

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A016959 a(n) = (6*n + 4)^3.

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A016971 a(n) = (6*n + 5)^3.

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A016947 a(n) = (6*n + 3)^3.

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A244725 a(n) = 5*n^3.

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A305728 Numbers of the form 216*p^3, where p is a Pythagorean prime (A002144).

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