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 17 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

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)

A016924 a(n) = (6*n + 1)^4.

Original entry on oeis.org

1, 2401, 28561, 130321, 390625, 923521, 1874161, 3418801, 5764801, 9150625, 13845841, 20151121, 28398241, 38950081, 52200625, 68574961, 88529281, 112550881, 141158161, 174900625, 214358881, 260144641, 312900721, 373301041, 442050625, 519885601, 607573201, 705911761
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016921, A016922, A016923, A016925, A016926.

Programs

Formula

From Amiram Eldar, Mar 28 2022: (Start)
a(n) = A016921(n)^4 = A016922(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(3, 1/6)/7776. (End)

A016925 a(n) = (6*n + 1)^5.

Original entry on oeis.org

1, 16807, 371293, 2476099, 9765625, 28629151, 69343957, 147008443, 282475249, 503284375, 844596301, 1350125107, 2073071593, 3077056399, 4437053125, 6240321451, 8587340257, 11592740743, 15386239549, 20113571875, 25937424601, 33038369407, 41615795893, 51888844699
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016921, A016922, A016923, A016924, A016926.

Programs

Formula

From Amiram Eldar, Mar 28 2022: (Start)
a(n) = A016921(n)^5.
Sum_{n>=0} 1/a(n) = ((1-1/2^5)*(1-1/3^5)*zeta(5) + 11*(Pi/3)^5/(8*sqrt(3)))/2 (Štofka, 2013). (End)

A016926 a(n) = (6*n + 1)^6.

Original entry on oeis.org

1, 117649, 4826809, 47045881, 244140625, 887503681, 2565726409, 6321363049, 13841287201, 27680640625, 51520374361, 90458382169, 151334226289, 243087455521, 377149515625, 567869252041, 832972004929, 1194052296529, 1677100110841, 2313060765625, 3138428376721
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016921, A016922, A016923, A016924, A016925.

Programs

  • Magma
    [(6*n+1)^6: n in [0..40]]; // Vincenzo Librandi, May 04 2011
  • Mathematica
    (6*Range[0,20]+1)^6 (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,117649,4826809,47045881,244140625,887503681,2565726409},20] (* Harvey P. Dale, Aug 19 2019 *)

Formula

From Amiram Eldar, Mar 28 2022: (Start)
a(n) = A016921(n)^6 = A016922(n)^3 = A016923(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(5, 1/6)/5598720. (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

A016927 a(n) = (6*n + 1)^7.

Original entry on oeis.org

1, 823543, 62748517, 893871739, 6103515625, 27512614111, 94931877133, 271818611107, 678223072849, 1522435234375, 3142742836021, 6060711605323, 11047398519097, 19203908986159, 32057708828125, 51676101935731, 80798284478113, 122987386542487, 182803912081669
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016921, A016922, A016923, A016924, A016925, A016926.

Programs

  • Magma
    [(6*n+1)^7: n in [0..40]]; // Vincenzo Librandi, May 04 2011
  • Mathematica
    (6*Range[0,20]+1)^7 (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,823543,62748517,893871739,6103515625,27512614111,94931877133,271818611107},20] (* Harvey P. Dale, May 12 2015 *)

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, May 12 2015
From Amiram Eldar, Mar 28 2022: (Start)
a(n) = A016921(n)^7.
Sum_{n>=0} 1/a(n) = 301*Pi^7/(1049760*sqrt(3)) + 138811*zeta(7)/279936. (End)

A016928 a(n) = (6*n + 1)^8.

Original entry on oeis.org

1, 5764801, 815730721, 16983563041, 152587890625, 852891037441, 3512479453921, 11688200277601, 33232930569601, 83733937890625, 191707312997281, 406067677556641, 806460091894081, 1517108809906561, 2724905250390625, 4702525276151521, 7837433594376961
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016921, A016922, A016923, A016924, A016925, A016926, A016927.

Programs

Formula

From Amiram Eldar, Mar 28 2022: (Start)
a(n) = A016921(n)^8 = A016922(n)^4 = A016924(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(7, 1/6)/8465264640. (End)
Showing 1-10 of 17 results. Next

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

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