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

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)

A016936 a(n) = (6*n + 2)^4.

Original entry on oeis.org

16, 4096, 38416, 160000, 456976, 1048576, 2085136, 3748096, 6250000, 9834496, 14776336, 21381376, 29986576, 40960000, 54700816, 71639296, 92236816, 116985856, 146410000, 181063936, 221533456, 268435456, 322417936, 384160000, 454371856, 533794816, 623201296
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016780, A016933, A016934, A016935.

Programs

  • Magma
    [(6*n+2)^4: n in [0..30]]; // Vincenzo Librandi, May 04 2011
  • Mathematica
    (6*Range[0,30]+2)^4 (* or *) LinearRecurrence[{5,-10,10,-5,1},{16,4096,38416,160000,456976},30] (* Harvey P. Dale, Aug 22 2012 *)

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, Aug 22 2012
From Amiram Eldar, Mar 29 2022: (Start)
a(n) = A016933(n)^4 = A016934(n)^2.
a(n) = 16*A016780(n).
Sum_{n>=0} 1/a(n) = PolyGamma(3, 1/3)/7776. (End)

A016937 a(n) = (6*n + 2)^5.

Original entry on oeis.org

32, 32768, 537824, 3200000, 11881376, 33554432, 79235168, 164916224, 312500000, 550731776, 916132832, 1453933568, 2219006624, 3276800000, 4704270176, 6590815232, 9039207968, 12166529024, 16105100000, 21003416576, 27027081632, 34359738368, 43204003424, 53782400000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016781, A016933, A016934, A016935, A016936.

Programs

  • Magma
    [(6*n+2)^5: n in [0..30]]; // Vincenzo Librandi, May 04 2011
  • Mathematica
    (6*Range[0,20]+2)^5 (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{32,32768,537824,3200000,11881376,33554432},20] (* Harvey P. Dale, Dec 13 2012 *)

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, Dec 13 2012
From Amiram Eldar, Mar 29 2022: (Start)
a(n) = A016933(n)^5.
a(n) = 32*A016781(n).
Sum_{n>=0} 1/a(n) = Pi^5/(11664*sqrt(3)) + 121*zeta(5)/7776. (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

A016938 a(n) = (6*n + 2)^6.

Original entry on oeis.org

64, 262144, 7529536, 64000000, 308915776, 1073741824, 3010936384, 7256313856, 15625000000, 30840979456, 56800235584, 98867482624, 164206490176, 262144000000, 404567235136, 606355001344, 885842380864, 1265319018496, 1771561000000, 2436396322816, 3297303959104
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016782, A016933, A016934, A016935, A016936, A016937.

Programs

Formula

From Amiram Eldar, Mar 29 2022: (Start)
a(n) = A016933(n)^6 = A016934(n)^3 = A016935(n)^2.
a(n) = 64*A016782(n).
Sum_{n>=0} 1/a(n) = PolyGamma(5, 1/3)/5598720. (End)

A016939 a(n) = (6n+2)^7.

Original entry on oeis.org

128, 2097152, 105413504, 1280000000, 8031810176, 34359738368, 114415582592, 319277809664, 781250000000, 1727094849536, 3521614606208, 6722988818432, 12151280273024, 20971520000000, 34792782221696, 55784660123648, 86812553324672, 131593177923584, 194871710000000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016783, A016933, A016934, A016935, A016936, A016937, A016938.

Programs

Formula

a(n) = 128*A016783(n). - R. J. Mathar, May 07 2008
G.f.: 128*(1 + 16376*x + 692499*x^2 + 3870352*x^3 + 4890287*x^4 + 1475736*x^5 + 77101*x^6 + 128*x^7)/(1 - x)^8. - Ilya Gutkovskiy, Jun 16 2016
From Amiram Eldar, Mar 29 2022: (Start)
a(n) = A016933(n)^7.
Sum_{n>=0} 1/a(n) = 7*Pi^7/(3149280*sqrt(3)) + 1093*zeta(7)/279936. (End)

A016940 a(n) = (6*n + 2)^8.

Original entry on oeis.org

256, 16777216, 1475789056, 25600000000, 208827064576, 1099511627776, 4347792138496, 14048223625216, 39062500000000, 96717311574016, 218340105584896, 457163239653376, 899194740203776, 1677721600000000, 2992179271065856, 5132188731375616, 8507630225817856
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016784, A016933, A016934, A016935, A016936, A016937, A016938, A016939.

Programs

  • Magma
    [(6*n+2)^8: n in [0..20]]; // Vincenzo Librandi, May 04 2011
  • Mathematica
    (6*Range[0,20]+2)^8 (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{256,16777216,1475789056,25600000000,208827064576,1099511627776,4347792138496,14048223625216,39062500000000},20] (* Harvey P. Dale, Sep 06 2020 *)

Formula

From Amiram Eldar, Mar 29 2022: (Start)
a(n) = A016933(n)^8 = A016934(n)^4 = A016936(n)^2.
a(n) = 2^8*A016784(n).
Sum_{n>=0} 1/a(n) = PolyGamma(7, 1/3)/8465264640. (End)

A016941 a(n) = (6*n + 2)^9.

Original entry on oeis.org

512, 134217728, 20661046784, 512000000000, 5429503678976, 35184372088832, 165216101262848, 618121839509504, 1953125000000000, 5416169448144896, 13537086546263552, 31087100296429568, 66540410775079424, 134217728000000000, 257327417311663616, 472161363286556672
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016785, A016933, A016934, A016935, A016936, A016937, A016938, A016939, A016940.

Programs

  • Magma
    [(6*n+2)^9: n in [0..25]]; // Vincenzo Librandi, May 05 2011
  • Mathematica
    (6*Range[0,20]+2)^9 (* or *) LinearRecurrence[ {10,-45,120,-210,252,-210,120,-45,10,-1},{512,134217728,20661046784,512000000000,5429503678976,35184372088832,165216101262848,618121839509504,1953125000000000,5416169448144896},20] (* Harvey P. Dale, Sep 21 2013 *)

Formula

a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10). - Harvey P. Dale, Sep 21 2013
From Amiram Eldar, Mar 29 2022: (Start)
a(n) = A016933(n)^9 = A016935(n)^3.
a(n) = 2^9*A016785(n).
Sum_{n>=0} 1/a(n) = 809*Pi^9/(14285134080*sqrt(3)) + 9841*zeta(9)/10077696. (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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