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 40 results. Next

A128960 a(n) = (n^3 - n)*2^n.

Original entry on oeis.org

0, 24, 192, 960, 3840, 13440, 43008, 129024, 368640, 1013760, 2703360, 7028736, 17891328, 44728320, 110100480, 267386880, 641728512, 1524105216, 3586129920, 8367636480, 19377684480, 44568674304, 101871255552, 231525580800, 523449139200, 1177760563200, 2638183661568
Offset: 1

Views

Author

Mohammad K. Azarian, Apr 28 2007

Keywords

Links

Crossrefs

Cf. A000079, A007531, A036289, A128796.
Cf. A128961, A128962, A128963, A128964, A128965, A128967, A128969.

Programs

  • Magma
    [(n^3-n)*2^n: n in [1..25]]; /* or */ I:=[0,24,192,960]; [n le 4 select I[n] else 8*Self(n-1)-24*Self(n-2)+32*Self(n-3)-16*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 12 2013
  • Mathematica
    CoefficientList[Series[24 x/(1 - 2 x)^4, {x, 0, 30}], x] (* or *) LinearRecurrence[{8, -24, 32, -16}, {0, 24, 192, 960}, 30] (* Vincenzo Librandi, Feb 12 2013 *)
  • PARI
    a(n)=(n^3-n)<Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: 24*x^2/(1-2*x)^4. - Vincenzo Librandi, Feb 12 2013
a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4). - Vincenzo Librandi, Feb 12 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A000079(n).
Sum_{n>=2} 1/a(n) = (2*log(2)-1)/8.
Sum_{n>=2} (-1)^n/a(n) = (3/2)^2*log(3/2) - 7/8. (End)

Extensions

Offset corrected by Mohammad K. Azarian, Nov 19 2008

A129002 a(n) = (n^3 + n^2)*2^n.

Original entry on oeis.org

4, 48, 288, 1280, 4800, 16128, 50176, 147456, 414720, 1126400, 2973696, 7667712, 19382272, 48168960, 117964800, 285212672, 681836544, 1613758464, 3785359360, 8808038400, 20346568704, 46690992128, 106501767168, 241591910400
Offset: 1

Views

Author

Mohammad K. Azarian, May 01 2007

Keywords

Comments

Number of paths along four vertices contained within the n+1 dimensional hypercube graph. - Ben Eck, Mar 30 2022

Links

Crossrefs

Cf. A128796, A036289.

Programs

  • Magma
    [(n^3+n^2)*2^n: n in [1..25]]; // Vincenzo Librandi, Feb 12 2013
  • Magma
    I:=[4, 48, 288, 1280]; [n le 4 select I[n] else 8*Self(n-1)-24*Self(n-2)+32*Self(n-3)-16*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 12 2013
  • Mathematica
    CoefficientList[Series[4 (1 + 4 x)/(1 - 2 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 12 2013 *)
LinearRecurrence[{8,-24,32,-16},{4,48,288,1280},30] (* Harvey P. Dale, Aug 21 2021 *)
  • PARI
    a(n)=(n^3+n^2)<Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: 4x*(1+4*x)/(1-2*x)^4. - Vincenzo Librandi, Feb 12 2013
a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4). - Vincenzo Librandi, Feb 12 2013
Sum_{n>=1} 1/a(n) = Pi^2/12 - 1 + log(2) - log(2)^2/2. - Amiram Eldar, Aug 05 2020

A128074 a(n) = (n^3+n)*9^n.

Original entry on oeis.org

0, 18, 810, 21870, 446148, 7676370, 117979902, 1674039150, 22384294920, 285916320882, 3521652245010, 42113381995278, 491427393476940, 5617523480607090, 63094193590782438, 697970937800860110
Offset: 0

Views

Author

Mohammad K. Azarian, May 02 2007

Keywords

Links

Crossrefs

Cf. A128796; A128960; A128985; A129002.

Programs

  • Magma
    [(n^3 + n) * 9^n: n in [1..20]]; // Vincenzo Librandi, Feb 22 2012
  • Mathematica
    Table[(n^3+n)9^n,{n,20}] (* or *) LinearRecurrence[{36,-486,2916,-6561}, {18,810,21870,446148},20] (* Harvey P. Dale, Jun 16 2011 *)
  • PARI
    A128074(n)=(n^3+n)*9^n \\ M. F. Hasler, Oct 06 2014

Formula

a(1)=18, a(2)=810, a(3)=21870, a(4)=446148, a(n)=36*a(n-1)- 486*a(n-2)+ 2916*a(n-3)-6561*a(n-4). - Harvey P. Dale, Jun 16 2011
G.f.: 18*x*(1+9*x+81*x^2)/(1-9*x)^4. - Harvey P. Dale, Jun 16 2011

Extensions

Extended to a(0)=0 by M. F. Hasler, Oct 06 2014

A128964 a(n) = (n^3-n)*6^n.

Original entry on oeis.org

0, 216, 5184, 77760, 933120, 9797760, 94058496, 846526464, 7255941120, 59861514240, 478892113920, 3735358488576, 28524555730944, 213934167982080, 1579821548175360, 11510128422420480, 82872924641427456, 590469588070170624, 4168020621671792640, 29176144351702548480
Offset: 1

Views

Author

Mohammad K. Azarian, Apr 28 2007

Keywords

Links

Crossrefs

Cf. A000400, A007531, A036289, A081144, A128796.
Cf. A128960, A128961, A128962, A128963, A128965, A128967, A128969.

Programs

  • Magma
    [(n^3-n)*6^n: n in [1..25]]; // Vincenzo Librandi, Feb 11 2013
  • Magma
    I:=[0, 216, 5184, 77760]; [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 11 2013
  • Mathematica
    CoefficientList[Series[216 x/(1 - 6 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 11 2013 *)

Formula

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 216*x^2/(1-6*x)^4.
a(n) = 216*A081144(n+1). (End)
a(n) = 24*a(n-1) - 216*a(n-2) + 864*a(n-3) - 1296*a(n-4). - Vincenzo Librandi, Feb 11 2013
From Amiram Eldar, Jan 04 2022: (Start)
Sum_{n>=2} 1/a(n) = 25*log(6/5)/12 - 3/8.
Sum_{n>=2} (-1)^n/a(n) = 49*log(7/6)/12 - 5/8. (End)
a(n) = A007531(n+1)*A000400(n). - Amiram Eldar, Oct 02 2022

Extensions

Corrected offset. - Mohammad K. Azarian, Nov 20 2008

A128985 a(n) = (n^3 - n^2)*2^n.

Original entry on oeis.org

0, 16, 144, 768, 3200, 11520, 37632, 114688, 331776, 921600, 2478080, 6488064, 16613376, 41746432, 103219200, 251658240, 606076928, 1443889152, 3406823424, 7969177600, 18496880640, 42630905856, 97626619904, 222264557568
Offset: 1

Views

Author

Mohammad K. Azarian, Apr 30 2007

Keywords

Links

Crossrefs

Cf. A128796, A036289.

Programs

  • Magma
    [(n^3-n^2)*2^n: n in [1..25]]; // Vincenzo Librandi, Feb 11 2013
  • Magma
    I:=[0, 16, 144, 768]; [n le 4 select I[n] else 8*Self(n-1) - 24*Self(n-2) + 32*Self(n-3) - 16*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 11 2013
  • Mathematica
    CoefficientList[Series[16 x (1+x)/(1 - 2 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 11 2013 *)
Table[(n^3-n^2)2^n,{n,30}] (* or *) LinearRecurrence[{8,-24,32,-16},{0,16,144,768},30] (* Harvey P. Dale, Jul 06 2014  *)
  • PARI
    a(n)=(n^3-n^2)<Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: 16*x^2*(1 + x)/(1 - 2*x)^4. - Vincenzo Librandi, Feb 11 2013
a(n) = 8*a(n-1) -24*a(n-2) +32*a(n-3) -16*a(n-4). - Vincenzo Librandi, Feb 11 2013

Extensions

Offset corrected by Mohammad K. Azarian, Nov 19 2008

A128961 a(n) = (n^3 - n)*3^n.

Original entry on oeis.org

0, 54, 648, 4860, 29160, 153090, 734832, 3306744, 14171760, 58458510, 233834040, 911952756, 3482001432, 13057505370, 48212327520, 175630621680, 632270238048, 2252462723046, 7949868434280, 27824539519980, 96653663595720, 333455139405234, 1143274763675088
Offset: 1

Views

Author

Mohammad K. Azarian, Apr 28 2007

Keywords

Links

Crossrefs

Cf. A000244, A007531, A036289, A128796.
Cf. A128960, A128962, A128963, A128964, A128965, A128967, A128969.

Programs

  • Magma
    [(n^3-n)*3^n: n in [1..25]]; // Vincenzo Librandi, Feb 12 2013
  • Magma
    I:=[0,54,648,4860]; [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}, {0, 54, 648, 4860}, 30] (* or *) CoefficientList[Series[54 x/(1 - 3 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 12 2013 *)

Formula

G.f.: 54*x^2/(1-3*x)^4. - Vincenzo Librandi, Feb 12 2013
a(n) = 12*a(n-1) - 54*a(n-2) + 108*a(n-3) - 81*a(n-4). - Vincenzo Librandi, Feb 12 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A000244(n).
Sum_{n>=2} 1/a(n) = (2/3)*log(3/2) - 1/4.
Sum_{n>=2} (-1)^n/a(n) = (8/3)*log(4/3) - 3/4. (End)

Extensions

Offset corrected by Mohammad K. Azarian, Nov 20 2008

A128962 a(n) = (n^3 - n)*4^n.

Original entry on oeis.org

0, 96, 1536, 15360, 122880, 860160, 5505024, 33030144, 188743680, 1038090240, 5536481280, 28789702656, 146565758976, 732828794880, 3607772528640, 17523466567680, 84112639524864, 399535037743104, 1880164883496960, 8774102789652480, 40637949762600960
Offset: 1

Views

Author

Mohammad K. Azarian, Apr 28 2007

Keywords

Links

Crossrefs

Cf. A000302, A007531, A036289, A128796.
Cf. A128960, A128961, A128963, A128964, A128965, A128967, A128969.

Programs

  • Magma
    [(n^3-n)*4^n: n in [1..20]]; // Vincenzo Librandi, Feb 09 2013
  • Mathematica
    CoefficientList[Series[96 x / (1-4 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 09 2013 *)
Table[(n^3-n)4^n,{n,20}] (* or *) LinearRecurrence[{16,-96,256,-256},{0,96,1536,15360},20] (* Harvey P. Dale, Dec 31 2018 *)

Formula

G.f.: 96*x^2/(1-4*x)^4. - Vincenzo Librandi, Feb 09 2013
a(n) = 16*a(n-1) - 96*a(n-2) + 256*a(n-3) - 256*a(n-4). - Vincenzo Librandi, Feb 09 2013
a(n) = 96*A038846(n-2) for n>1. - Bruno Berselli, Feb 10 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A000302(n).
Sum_{n>=2} 1/a(n) = (9/8)*log(4/3) - 5/16.
Sum_{n>=2} (-1)^n/a(n) = (25/8)*log(5/4) - 11/16. (End)

Extensions

Offset corrected by Mohammad K. Azarian, Nov 20 2008

A128963 a(n) = (n^3 - n)*5^n.

Original entry on oeis.org

0, 150, 3000, 37500, 375000, 3281250, 26250000, 196875000, 1406250000, 9667968750, 64453125000, 418945312500, 2666015625000, 16662597656250, 102539062500000, 622558593750000, 3735351562500000, 22178649902343750, 130462646484375000, 761032104492187500
Offset: 1

Views

Author

Mohammad K. Azarian, Apr 28 2007

Keywords

Links

Crossrefs

Cf. A000351, A007531, A036289, A081143, A128796.
Cf. A128960, A128961, A128962, A128964, A128965, A128967, A128969.

Programs

  • Magma
    [(n^3-n)*5^n: n in [1..25]]; // Vincenzo Librandi, Feb 12 2013
  • Mathematica
    Table[(n^3-n)5^n,{n,20}] (* or *) LinearRecurrence[{20,-150,500,-625},{0,150,3000,37500},20] (* Harvey P. Dale, Jul 22 2012 *)
CoefficientList[Series[150 x/(1 - 5 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 12 2013 *)

Formula

a(1)=0, a(2)=150, a(3)=3000, a(4)=37500, a(n)=20*a(n-1)-150*a(n-2)+ 500*a(n-3)- 625*a(n-4). - Harvey P. Dale, Jul 22 2012
G.f.: 150*x^2/(1 - 5*x)^4. - Vincenzo Librandi, Feb 12 2013
a(n) = 150*A081143(n+1). - Bruno Berselli, Feb 12 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A000351(n).
Sum_{n>=2} 1/a(n) = (8/5)*log(5/4) - 7/20.
Sum_{n>=2} (-1)^n/a(n) = (18/5)*log(6/5) - 13/20. (End)

Extensions

Offset corrected by Mohammad K. Azarian, Nov 20 2008

A128965 a(n) = (n^3 - n)*7^n.

Original entry on oeis.org

0, 294, 8232, 144060, 2016840, 24706290, 276710448, 2905459704, 29054597040, 279650496510, 2610071300760, 23751648836916, 211605598728888, 1851548988877770, 15951806673408480, 135590356723972080, 1138958996481365472, 9467596658251350486, 77968443067952298120
Offset: 1

Views

Author

Mohammad K. Azarian, Apr 28 2007

Keywords

Links

Crossrefs

Cf. A000420, A007531, A036289, A128796, A140107.
Cf. A128960, A128961, A128962, A128963, A128964, A128967, A128969.

Programs

  • Magma
    [(n^3 - n)*7^n: n in [1..25]]; // Vincenzo Librandi, Feb 11 2013
  • Mathematica
    LinearRecurrence[{28, -294, 1372, -2401}, {0, 294, 8232, 144060}, 30] (* Vincenzo Librandi, Feb 11 2013 *)
Table[(n^3-n)7^n,{n,20}] (* Harvey P. Dale, May 14 2020 *)

Formula

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 294x^2/(1-7x)^4.
a(n) = 294*A140107(n-2). (End)
a(n) = 28*a(n-1) - 294*a(n-2) + 1372*a(n-3) - 2401*a(n-4). - Vincenzo Librandi, Feb 11 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A000420(n).
Sum_{n>=2} 1/a(n) = (18/7)*log(7/6) - 11/28.
Sum_{n>=2} (-1)^n/a(n) = (32/7)*log(8/7) - 17/28. (End)

Extensions

Offset corrected by Mohammad K. Azarian, Nov 20 2008

A128967 a(n) = (n^3-n)*8^n.

Original entry on oeis.org

0, 384, 12288, 245760, 3932160, 55050240, 704643072, 8455716864, 96636764160, 1063004405760, 11338713661440, 117922622078976, 1200666697531392, 12006666975313920, 118219490218475520, 1148417904979476480, 11024811887802974208, 104735712934128254976
Offset: 1

Views

Author

Mohammad K. Azarian, Apr 28 2007

Keywords

Links

Crossrefs

Cf. A001018, A007531, A036289, A128796, A140802.
Cf. A128960, A128961, A128962, A128963, A128964, A128965, A128969.

Programs

  • Magma
    [(n^3 - n)*8^n: n in [1..25]]; // Vincenzo Librandi, Feb 11 2013
  • Mathematica
    LinearRecurrence[{32, -384, 2048, -4096}, {0, 384, 12288, 245760}, 30] (* Vincenzo Librandi, Feb 11 2013 *)

Formula

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 384x^2/(1-8x)^4.
a(n) = 384*A140802(n-2). (End)
a(n) = 32*a(n-1) - 384*a(n-2) + 2048*a(n-3) - 4096*a(n-4). - Vincenzo Librandi, Feb 11 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A001018(n).
Sum_{n>=2} 1/a(n) = (49/16)*log(8/7) - 13/32.
Sum_{n>=2} (-1)^n/a(n) = (81/16)*log(9/8) - 19/32. (End)

Extensions

Corrected the offset. - Mohammad K. Azarian, Nov 20 2008
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