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

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

Views

Author

N. J. A. Sloane

Keywords

Examples

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

Links

Crossrefs

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

Programs

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

Formula

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

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)

A016972 a(n) = (6*n + 5)^4.

Original entry on oeis.org

625, 14641, 83521, 279841, 707281, 1500625, 2825761, 4879681, 7890481, 12117361, 17850625, 25411681, 35153041, 47458321, 62742241, 81450625, 104060401, 131079601, 163047361, 200533921, 244140625, 294499921, 352275361, 418161601, 492884401, 577200625, 671898241
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016969, A016970, A016971.
Subsequence of A000583.

Programs

Formula

From Chai Wah Wu, Mar 20 2017: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
G.f.: (-x^4 - 2396*x^3 - 16566*x^2 - 11516*x - 625)/(x - 1)^5. (End)
From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^4 = A016970(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(3, 5/6)/7776. (End)

A016973 a(n) = (6*n + 5)^5.

Original entry on oeis.org

3125, 161051, 1419857, 6436343, 20511149, 52521875, 115856201, 229345007, 418195493, 714924299, 1160290625, 1804229351, 2706784157, 3939040643, 5584059449, 7737809375, 10510100501, 14025517307, 18424351793, 23863536599, 30517578125, 38579489651, 48261724457
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016969, A016970, A016971, A016972.
Subsequence of A000584.

Programs

  • Magma
    [(6*n+5)^5: n in [0..30]]; // Vincenzo Librandi, May 07 2011
  • Mathematica
    (6*Range[0,20]+5)^5 (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{3125,161051,1419857,6436343,20511149,52521875},20] (* Harvey P. Dale, Sep 24 2014 *)

Formula

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Harvey P. Dale, Sep 24 2014
G.f.: (3125 + 142301*x + 500426*x^2 + 270466*x^3 + 16801*x^4 + x^5)/(-1+x)^6. - Harvey P. Dale, Aug 13 2021
From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^5.
Sum_{n>=0} 1/a(n) = 3751*zeta(5)/7776 - 11*Pi^5/(3888*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

A016974 a(n) = (6*n + 5)^6.

Original entry on oeis.org

15625, 1771561, 24137569, 148035889, 594823321, 1838265625, 4750104241, 10779215329, 22164361129, 42180533641, 75418890625, 128100283921, 208422380089, 326940373369, 496981290961, 735091890625, 1061520150601, 1500730351849, 2081951752609, 2839760855281, 3814697265625
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016969 (6*n +5), A016970, A016971, A016972, A016973.
Subsequence of A001014 (n^6).

Programs

  • Magma
    [(6*n+5)^6: n in [0..25]]; // Vincenzo Librandi, May 10 2011
  • Mathematica
    (6Range[0,20]+5)^6 (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{15625,1771561,24137569,148035889,594823321,1838265625,4750104241},30] (* Harvey P. Dale, Apr 24 2025 *)

Formula

From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^6 = A016970(n)^3 = A016971(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(5, 5/6)/5598720. (End)

A016975 a(n) = (6*n + 5)^7.

Original entry on oeis.org

78125, 19487171, 410338673, 3404825447, 17249876309, 64339296875, 194754273881, 506623120463, 1174711139837, 2488651484819, 4902227890625, 9095120158391, 16048523266853, 27136050989627, 44231334895529, 69833729609375, 107213535210701, 160578147647843, 235260548044817
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016969 (6*n + 5), A016970, A016971, A016972, A016973, A016974.
Subsequence of A001015 (n^7).

Programs

  • Magma
    [(6*n+5)^7: n in [0..25]]; // Vincenzo Librandi, May 11 2011
  • Mathematica
    (6Range[0,20]+5)^7 (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{78125,19487171,410338673,3404825447,17249876309,64339296875,194754273881,506623120463},20] (* Harvey P. Dale, Jan 30 2013 *)

Formula

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - Harvey P. Dale, Jan 30 2013
From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^7.
Sum_{n>=0} 1/a(n) = 138811*zeta(7)/279936 - 301*Pi^7/(1049760*sqrt(3)). (End)

A016976 a(n) = (6*n + 5)^8.

Original entry on oeis.org

390625, 214358881, 6975757441, 78310985281, 500246412961, 2251875390625, 7984925229121, 23811286661761, 62259690411361, 146830437604321, 318644812890625, 645753531245761, 1235736291547681, 2252292232139041, 3936588805702081, 6634204312890625, 10828567056280801
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016969 (6*n + 5), A016970, A016971, A016972, A016973, A016974, A016975.
Subsequence of A001016 (n^8).

Programs

Formula

From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^8 = A016970(n)^4 = A016972(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(7, 5/6)/8465264640. (End)

A016977 a(n) = (6*n + 5)^9.

Original entry on oeis.org

1953125, 2357947691, 118587876497, 1801152661463, 14507145975869, 78815638671875, 327381934393961, 1119130473102767, 3299763591802133, 8662995818654939, 20711912837890625, 45848500718449031, 95151694449171437, 186940255267540403, 350356403707485209, 630249409724609375
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016969, A016970, A016971, A016972, A016973, A016974, A016975, A016976.
Subsequence of A001017.

Programs

Formula

From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^9 = A016971(n)^3.
Sum_{n>=0} 1/a(n) = 5028751*zeta(9)/10077696 - 15371*Pi^9/(529079040*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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