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.

Previous Showing 31-40 of 40 results.

A128991 a(n) = (n^3 - n^2)*8^n.

Original entry on oeis.org

0, 256, 9216, 196608, 3276800, 47185920, 616562688, 7516192768, 86973087744, 966367641600, 10393820856320, 108851651149824, 1114904790564864, 11206222510292992, 110830772079820800, 1080863910568919040
Offset: 1

Views

Author

Mohammad K. Azarian, Apr 30 2007

Keywords

Links

Crossrefs

Cf. A128796, A036289.

Programs

  • Magma
    [(n^3-n^2)*8^n: n in [1..25]]; /* or */ I:=[0,256,9216,196608]; [n le 4 select I[n] else 32*Self(n-1)-384*Self(n-2)+2048*Self(n-3)-4096*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 12 2013
  • Mathematica
    CoefficientList[Series[256 x (1 + 4 x)/(1 - 8 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 12 2013 *)
LinearRecurrence[{32,-384,2048,-4096},{0,256,9216,196608},30] (* Harvey P. Dale, Aug 25 2013 *)

Formula

G.f.: 256*x^2*(1+4*x)/(1-8*x)^4. - Vincenzo Librandi, Feb 12 2013
a(n) = 32*a(n-1)-384*a(n-2)+2048*a(n-3)-4096*a(n-4). - Vincenzo Librandi, Feb 12 2013

Extensions

Offset corrected by Mohammad K. Azarian, Nov 20 2008

A128992 a(n) = (n^3 - n^2)*9^n.

Original entry on oeis.org

0, 324, 13122, 314928, 5904900, 95659380, 1406192886, 19284931008, 251048476872, 3138105960900, 37971082126890, 447368385785904, 5154903899851212, 58290067175240628, 648557066098144350
Offset: 1

Views

Author

Mohammad K. Azarian, Apr 30 2007

Keywords

Links

Crossrefs

Cf. A128796, A036289.

Programs

  • Magma
    [(n^3-n^2)*9^n: n in [1..25]]; /* or */ I:=[0,324,13122,314928]; [n le 4 select I[n] else 36*Self(n-1)-486*Self(n-2)+2916*Self(n-3)-6561*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 12 2013
  • Mathematica
    CoefficientList[Series[162 x (2 + 9 x)/(1 - 9 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 12 2013 *)

Formula

G.f.: 162*x^2*(2+9*x)/(1-9*x)^4. - Vincenzo Librandi, Feb 12 2013
a(n) = 36*a(n-1)-486*a(n-2)+2916*a(n-3)-6561*a(n-4). - Vincenzo Librandi, Feb 12 2013

Extensions

Offset corrected by Mohammad K. Azarian, Nov 20 2008

A129003 a(n) = (n^3 + n^2)*3^n.

Original entry on oeis.org

6, 108, 972, 6480, 36450, 183708, 857304, 3779136, 15943230, 64953900, 257217444, 994857552, 3772168218, 14061928860, 51656065200, 187339329792, 671787127926, 2384960530284, 8391527791740, 29288988968400, 101486346775506
Offset: 1

Views

Author

Mohammad K. Azarian, May 01 2007

Keywords

Links

Crossrefs

Cf. A036216, A128796, A036289.

Programs

  • Magma
    [(n^3+n^2)*3^n: n in [1..25]]; // Vincenzo Librandi, Feb 12 2013
  • Magma
    I:=[6,108,972,6480]; [n le 4 select I[n] else 12*Self(n-1)-54*Self(n-2)+108*Self(n-3)-81*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 12 2013
  • Mathematica
    LinearRecurrence[{12, -54, 108, -81}, {6, 108, 972, 6480}, 30] (* Vincenzo Librandi, Feb 12 2013 *)

Formula

G.f.: 6*x*(1+6*x)/(1-3*x)^4. - R. J. Mathar, Dec 19 2008
a(n) = 12*a(n-1)-54*a(n-2)+108*a(n-3)-81*a(n-4). - Vincenzo Librandi, Feb 12 2013
a(n) = 6*(A036216(n-1)+6*A036216(n-2)), with A036216(-1)=0. - Bruno Berselli, Feb 12 2013

A129004 a(n) = (n^3 + n^2)*4^n.

Original entry on oeis.org

8, 192, 2304, 20480, 153600, 1032192, 6422528, 37748736, 212336640, 1153433600, 6090129408, 31406948352, 158779572224, 789200240640, 3865470566400, 18691697672192, 89369679495168, 423037098786816, 1984618488135680, 9235897673318400, 42669847250731008, 195836215046438912
Offset: 1

Views

Author

Mohammad K. Azarian, May 01 2007

Keywords

Links

Crossrefs

Cf. A036289, A038846, A128796.

Programs

  • Magma
    [(n^3+n^2)*4^n: n in [1..25]]; // Vincenzo Librandi, Feb 12 2013
  • Mathematica
    Table[(n^3+n^2)4^n,{n, 20}] (* or *) LinearRecurrence[{16,-96,256,-256}, {8,192,2304,20480},20] (* Harvey P. Dale, May 12 2011 *)

Formula

G.f.: 8*x*(1+8*x)/(1-4*x)^4. - R. J. Mathar, Dec 19 2008
a(1)=8, a(2)=192, a(3)=2304, a(4)=20480, a(n)=16*a(n-1)-96*a(n-2)+ 256*a(n-3)-256*a(n-4). - Harvey P. Dale, May 12 2011
a(n) = 8*(A038846(n-1)+8*A038846(n-2)), with A038846(-1)=0. - Bruno Berselli, Feb 12 2013
E.g.f.: 8*exp(4*x)*x*(1 + 8*x + 8*x^2). - Stefano Spezia, Sep 02 2024

A129005 a(n) = (n^3 + n^2)*5^n.

Original entry on oeis.org

0, 10, 300, 4500, 50000, 468750, 3937500, 30625000, 225000000, 1582031250, 10742187500, 70898437500, 457031250000, 2888183593750, 17944335937500, 109863281250000, 664062500000000, 3968811035156250, 23483276367187500
Offset: 0

Views

Author

Mohammad K. Azarian, May 01 2007

Keywords

Links

Crossrefs

Cf. A036289, A081143, A128796.

Programs

  • Magma
    [(n^3+n^2)*5^n: n in [0..25]]; // Vincenzo Librandi, Feb 12 2013
  • Magma
    I:=[0,10,300,4500]; [n le 4 select I[n] else 20*Self(n-1)-150*Self(n-2)+500*Self(n-3)-625*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 12 2013
  • Mathematica
    CoefficientList[Series[10 x (1 + 10 x)/(1 - 5 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 12 2013 *)
Table[(n^3+n^2)5^n,{n,0,30}] (* or *) LinearRecurrence[{20,-150,500,-625},{0,10,300,4500},30] (* Harvey P. Dale, May 15 2022 *)
  • PARI
    A129005(n)=5^n*(1+n)*n^2 \\ - M. F. Hasler, Feb 12 2013

Formula

G.f.: 10*x*(1 + 10*x)/(1 - 5*x)^4. - Vincenzo Librandi, Feb 12 2013
a(n) = 20*a(n-1)-150*a(n-2)+500*a(n-3)-625*a(n-4). - Vincenzo Librandi, Feb 12 2013
a(n) = 10*A081143(n+2)+100*A081143(n+1). - Bruno Berselli, Feb 13 2013

Extensions

Initial term a(0)=0 added by M. F. Hasler, Feb 12 2013

A129006 a(n) = (n^3 + n^2)*6^n.

Original entry on oeis.org

12, 432, 7776, 103680, 1166400, 11757312, 109734912, 967458816, 8162933760, 66512793600, 526781325312, 4074936532992, 30901602041856, 230390642442240, 1692665944473600, 12277470317248512, 88052482431516672
Offset: 1

Views

Author

Mohammad K. Azarian, May 01 2007

Keywords

Links

Crossrefs

Cf. A128796, A036289.

Programs

  • Magma
    [(n^3+n^2)*6^n: n in [1..25]]; // Vincenzo Librandi, Feb 12 2013
  • Magma
    I:=[12,432,7776,103680]; [n le 4 select I[n] else 24*Self(n-1)-216*Self(n-2)+864*Self(n-3)-1296*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 12 2013
  • Mathematica
    CoefficientList[Series[12 (1 + 12 x)/(1 - 6 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 12 2013 *)
Table[(n^3+n^2)6^n,{n,20}] (* or *) LinearRecurrence[{24,-216,864,-1296},{12,432,7776,103680},20] (* Harvey P. Dale, Aug 16 2014 *)

Formula

G.f.: 12*x*(1+12*x)/(1-6*x)^4. - Vincenzo Librandi, Feb 12 2013
a(n) = 24*a(n-1)-216*a(n-2)+864*a(n-3)-1296*a(n-4). - Vincenzo Librandi, Feb 12 2013

A129007 (n^3+n^2)*7^n.

Original entry on oeis.org

14, 588, 12348, 192080, 2521050, 29647548, 322828856, 3320525376, 32686421670, 310722773900, 2871078430836, 25910889640272, 229239398622962, 1993975834176060, 17091221435794800, 144629713838903552
Offset: 1

Views

Author

Mohammad K. Azarian, May 01 2007

Keywords

Links

Crossrefs

Cf. A128796, A036289.

Programs

  • Magma
    [(n^3+n^2)*7^n: n in [1..25]]; // Vincenzo Librandi, Feb 12 2013
  • Mathematica
    Table[(n^3+n^2)7^n,{n,30}] (* or *) LinearRecurrence[{28,-294,1372,-2401},{14,588,12348,192080},30] (* Harvey P. Dale, Nov 21 2012 *)
CoefficientList[Series[14 (1 + 14 x)/(1 - 7 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 12 2013 *)

Formula

a(1)=14, a(2)=588, a(3)=12348, a(4)=192080, a(n)=28*a(n-1)-294*a(n-2)+ 1372*a(n-3)-2401*a(n-4). - Harvey P. Dale, Nov 21 2012
G.f.: 14*x*(1+14*x)/(1-7*x)^4. - Vincenzo Librandi, Feb 12 2013

A129008 a(n) = (n^3 + n^2)*8^n.

Original entry on oeis.org

16, 768, 18432, 327680, 4915200, 66060288, 822083584, 9663676416, 108716359680, 1181116006400, 12472585027584, 128642860449792, 1300722255659008, 12930256742645760, 126663739519795200, 1224979098644774912
Offset: 1

Views

Author

Mohammad K. Azarian, May 01 2007

Keywords

Links

Crossrefs

Cf. A128796, A036289.

Programs

  • Magma
    [(n^3+n^2)*8^n: n in [1..25]]; // Vincenzo Librandi, Feb 12 2013
  • Magma
    I:=[16, 768, 18432, 327680]; [n le 4 select I[n] else 32*Self(n-1)-384*Self(n-2)+2048*Self(n-3)-4096*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 12 2013
  • Mathematica
    CoefficientList[Series[16 (1 + 16 x)/(1 - 8 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 12 2013 *)
LinearRecurrence[{32,-384,2048,-4096},{16,768,18432,327680},20] (* Harvey P. Dale, Jun 24 2025 *)

Formula

G.f.: 16*x*(1 + 16*x)/(1 - 8*x)^4. - Vincenzo Librandi, Feb 12 2013
a(n) = 32*a(n-1)-384*a(n-2)+2048*a(n-3)-4096*a(n-4). - Vincenzo Librandi, Feb 12 2013

A129009 a(n) = (n^3 + n^2)*9^n.

Original entry on oeis.org

18, 972, 26244, 524880, 8857350, 133923132, 1874923848, 24794911296, 313810596090, 3835462841100, 45565298552268, 528708092292432, 6014054549826414, 67257769817585340, 741208075540736400
Offset: 1

Views

Author

Mohammad K. Azarian, May 01 2007

Keywords

Links

Crossrefs

Cf. A128796, A036289.

Programs

  • Magma
    [(n^3+n^2)*9^n: n in [1..25]]; // Vincenzo Librandi, Feb 12 2013
  • Magma
    I:=[18,972,26244,524880]; [n le 4 select I[n] else 36*Self(n-1)-486*Self(n-2)+2916*Self(n-3)-6561*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 12 2013
  • Mathematica
    LinearRecurrence[{36, -486, 2916, -6561}, {18, 972, 26244, 524880}, 30] (* Vincenzo Librandi, Feb 12 2013 *)

Formula

G.f.: 18*x*(1+18*x)/(1-9*x)^4. - R. J. Mathar, Dec 19 2008
a(n) = 36*a(n-1)-486*a(n-2)+2916*a(n-3)-6561*a(n-4). - Vincenzo Librandi, Feb 12 2013

A276041 Exponential convolution of odd numbers (A005408) with themselves.

Original entry on oeis.org

1, 6, 28, 104, 336, 992, 2752, 7296, 18688, 46592, 113664, 272384, 643072, 1499136, 3457024, 7897088, 17891328, 40239104, 89915392, 199753728, 441450496, 970981376, 2126512128, 4638900224, 10083106816, 21843935232, 47177531392, 101602820096, 218238025728
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 17 2016

Keywords

Links

Crossrefs

Cf. A000079, A002061, A005408, A053755, A128796 (exponential convolution of even numbers with themselves).

Programs

  • Mathematica
    LinearRecurrence[{6, -12, 8}, {1, 6, 28}, 29]
Table[2^n (n^2 + n + 1), {n, 0, 28}]

Formula

O.g.f.: (1 + 4*x^2)/(1 - 2*x)^3.
E.g.f.: (1 + 2*x)^2*exp(2*x).
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3).
a(n) = 2^n*(n^2 + n + 1).
a(n) = A000079(n)*A002061(n+1).
Binomial transform of A053755.
Previous Showing 31-40 of 40 results.

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

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