cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 13 results. Next

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Magma
    [(6*n+2)^3: n in [0..50]]; // Vincenzo Librandi, May 04 2011
  • Mathematica
    (6*Range[0,30]+2)^3 (* or *) LinearRecurrence[{4,-6,4,-1},{8,512,2744,8000},30] (* Harvey P. Dale, Aug 23 2019 *)

Formula

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

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

Views

Author

N. J. A. Sloane

Keywords

Examples

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

Links

Crossrefs

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

Programs

Formula

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

Views

Author

N. J. A. Sloane

Keywords

Examples

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

Links

Crossrefs

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

Programs

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

Formula

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)

A016960 a(n) = (6*n + 4)^4.

Original entry on oeis.org

256, 10000, 65536, 234256, 614656, 1336336, 2560000, 4477456, 7311616, 11316496, 16777216, 24010000, 33362176, 45212176, 59969536, 78074896, 100000000, 126247696, 157351936, 193877776, 236421376, 285610000, 342102016, 406586896, 479785216, 562448656, 655360000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016792, A016957, A016958, A016959.
Subsequence of A000583.

Programs

  • Magma
    [(6*n+4)^4: n in [0..40]]; // Vincenzo Librandi, May 06 2011
  • Mathematica
    (6*Range[0,20]+4)^4 (* or *) LinearRecurrence[{5,-10,10,-5,1},{256,10000,65536,234256,614656},30] (* Harvey P. Dale, Sep 23 2013 *)

Formula

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Sep 23 2013
G.f.: 16*(16+545*x+1131*x^2+251*x^3+x^4)/(1-x)^5. - Harvey P. Dale, Aug 21 2021
From Amiram Eldar, Mar 31 2022: (Start)
a(n) = A016957(n)^4 = A016958(n)^2.
a(n) = 16*A016792(n).
Sum_{n>=0} 1/a(n) = PolyGamma(3, 2/3)/7776. (End)

A016961 a(n) = (6*n + 4)^5.

Original entry on oeis.org

1024, 100000, 1048576, 5153632, 17210368, 45435424, 102400000, 205962976, 380204032, 656356768, 1073741824, 1680700000, 2535525376, 3707398432, 5277319168, 7339040224, 10000000000, 13382255776, 17623416832, 22877577568, 29316250624, 37129300000, 46525874176
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016793, A016957, A016958, A016959, A016960.
Subsequence of A000584.

Programs

  • Magma
    [(6*n+4)^5: n in [0..30]]; // Vincenzo Librandi, May 06 2011
  • Mathematica
    a[n_] := (6*n + 4)^5; Array[a, 20, 0] (* Amiram Eldar, Mar 31 2022 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{1024,100000,1048576,5153632,17210368,45435424},30] (* Harvey P. Dale, Jan 01 2025 *)

Formula

From Amiram Eldar, Mar 31 2022: (Start)
a(n) = A016957(n)^5.
a(n) = 32*A016793(n).
Sum_{n>=0} 1/a(n) = 121*zeta(5)/7776 - Pi^5/(11664*sqrt(3)). (End)

A016911 a(n) = (6*n)^3.

Original entry on oeis.org

0, 216, 1728, 5832, 13824, 27000, 46656, 74088, 110592, 157464, 216000, 287496, 373248, 474552, 592704, 729000, 884736, 1061208, 1259712, 1481544, 1728000, 2000376, 2299968, 2628072, 2985984, 3375000, 3796416, 4251528, 4741632, 5268024, 5832000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Volume of a cube with side 6*n. - Wesley Ivan Hurt, Jul 05 2014

Examples

			a(1) = (6*1)^3 = 216.

Links

Crossrefs

Cf. A000578, A002117, A016923, A016935, A016947, A016959, A016971.
Cf. similar sequences listed in A244725.

Programs

  • Magma
    [(6*n)^3: n in [0..40]]; // Vincenzo Librandi, May 03 2011
  • Magma
    I:=[0,216,1728,5832]; [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]]; // Vincenzo Librandi, Jul 05 2014
  • Maple
    A016911:=n->216*n^3: seq(A016911(n), n=0..40); # Wesley Ivan Hurt, Jul 05 2014
  • Mathematica
    Table[216 n^3, {n, 0, 40}] (* or *) CoefficientList[Series[216 x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 05 2014 *)

Formula

G.f.: 216*x*(1 + 4*x + x^2)/(1 - x)^4. - Vincenzo Librandi, Jul 05 2014
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. - Vincenzo Librandi, Jul 05 2014
a(n) = 216 * A000578(n). - Wesley Ivan Hurt, Jul 05 2014
Sum_{n>=1} 1/a(n) = zeta(3)/216. - Amiram Eldar, Oct 02 2020

A016962 a(n) = (6*n + 4)^6.

Original entry on oeis.org

4096, 1000000, 16777216, 113379904, 481890304, 1544804416, 4096000000, 9474296896, 19770609664, 38068692544, 68719476736, 117649000000, 192699928576, 304006671424, 464404086784, 689869781056, 1000000000000, 1418519112256, 1973822685184, 2699554153024, 3635215077376
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016794, A016957, A016958, A016959, A016960, A016961.
Subsequence of A001014.

Programs

  • Magma
    [(6*n+4)^6: n in [0..25]]; // Vincenzo Librandi, May 06 2011
  • Mathematica
    (6Range[0,20]+4)^6 (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{4096,1000000,16777216,113379904,481890304,1544804416,4096000000},20] (* Harvey P. Dale, Aug 08 2019 *)

Formula

From Amiram Eldar, Mar 31 2022: (Start)
a(n) = A016957(n)^6 = A016958(n)^3 = A016959(n)^2.
a(n) = 64*A016794(n).
Sum_{n>=0} 1/a(n) = PolyGamma(5, 2/3)/5598720. (End)

A016963 a(n) = (6*n + 4)^7.

Original entry on oeis.org

16384, 10000000, 268435456, 2494357888, 13492928512, 52523350144, 163840000000, 435817657216, 1028071702528, 2207984167552, 4398046511104, 8235430000000, 14645194571776, 24928547056768, 40867559636992, 64847759419264, 100000000000000, 150363025899136, 221068140740608
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016795, A016957, A016958, A016959, A016960, A016961, A016962.
Subsequence of A001015.

Programs

  • Magma
    [(6*n+4)^7: n in [0..20]]; // Vincenzo Librandi, May 07 2011
  • Mathematica
    (6*Range[0,20]+4)^7 (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{16384,10000000,268435456,2494357888,13492928512,52523350144,163840000000,435817657216},20] (* Harvey P. Dale, Mar 03 2018 *)

Formula

From Amiram Eldar, Mar 31 2022: (Start)
a(n) = A016957(n)^7.
a(n) = 2^7*A016795(n).
Sum_{n>=0} 1/a(n) = 1093*zeta(7)/279936 - 7*Pi^7/(3149280*sqrt(3)). (End)

A016964 a(n) = (6*n + 4)^8.

Original entry on oeis.org

65536, 100000000, 4294967296, 54875873536, 377801998336, 1785793904896, 6553600000000, 20047612231936, 53459728531456, 128063081718016, 281474976710656, 576480100000000, 1113034787454976, 2044140858654976, 3596345248055296, 6095689385410816, 10000000000000000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016796, A016957, A016958, A016959, A016960, A016961, A016962, A016963.
Subsequence of A001016.

Programs

  • Magma
    [(6*n+4)^8: n in [0..20]]; // Vincenzo Librandi, May 07 2011
  • Mathematica
    (6*Range[0,20]+4)^8 (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{65536,100000000,4294967296,54875873536,377801998336,1785793904896,6553600000000,20047612231936,53459728531456},20] (* Harvey P. Dale, Dec 26 2018 *)

Formula

From Amiram Eldar, Mar 31 2022: (Start)
a(n) = A016957(n)^8 = A016958(n)^4 = A016960(n)^2.
a(n) = 2^8*A016796(n).
Sum_{n>=0} 1/a(n) = PolyGamma(7, 2/3)/8465264640. (End)

A016965 a(n) = (6*n + 4)^9.

Original entry on oeis.org

262144, 1000000000, 68719476736, 1207269217792, 10578455953408, 60716992766464, 262144000000000, 922190162669056, 2779905883635712, 7427658739644928, 18014398509481984, 40353607000000000, 84590643846578176, 167619550409708032, 316478381828866048, 572994802228616704
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016797, A016957, A016958, A016959, A016960, A016961, A016962, A016963, A016964.
Subsequence of A001017.

Programs

  • Magma
    [(6*n+4)^9: n in [0..20]]; // Vincenzo Librandi, May 07 2011
  • Mathematica
    (6*Range[0,20]+4)^9 (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{262144,1000000000,68719476736,1207269217792,10578455953408,60716992766464,262144000000000,922190162669056,2779905883635712,7427658739644928},20] (* Harvey P. Dale, Mar 04 2016 *)

Formula

From Amiram Eldar, Mar 31 2022: (Start)
a(n) = A016957(n)^9 = A016958(n)^3.
a(n) = 2^9*A016797(n).
Sum_{n>=0} 1/a(n) = 9841*zeta(9)/10077696 - 809*Pi^9/(14285134080*sqrt(3)). (End)
Showing 1-10 of 13 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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