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 15 results. Next

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

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

A128969 a(n) = (n^3 - n)*9^n.

Original entry on oeis.org

0, 486, 17496, 393660, 7085880, 111602610, 1607077584, 21695547384, 278942752080, 3451916556990, 41422998683880, 484649084601396, 5551434969070536, 62453643402043530, 691794203838020640, 7560322370515511280, 81651481601567521824, 872650209616752889494
Offset: 1

Views

Author

Mohammad K. Azarian, Apr 28 2007

Keywords

Links

Crossrefs

Cf. A001019, A007531, A036289, A038291, A128796.
Cf. A128960, A128961, A128962, A128963, A128964, A128965, A128967.

Programs

  • Magma
    [(n^3-n)*9^n: n in [0..25]]; // Vincenzo Librandi, Feb 11 2013
  • Magma
    I:=[0, 486, 17496, 393660]; [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 11 2013
  • Mathematica
    CoefficientList[Series[486 x/(1 - 9 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 11 2013 *)

Formula

From R. J. Mathar, Dec 19 2008 (Start)
G.f.: 486x^2/(1-9x)^4.
a(n) = 486*A038291(n+1,3). (End)
a(n) = 36*a(n-1) - 486*a(n-2) + 2916*a(n-3) - 6561*a(n-4). - Vincenzo Librandi, Feb 11 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A001019(n).
Sum_{n>=2} 1/a(n) = (32/9)*log(9/8) - 5/12.
Sum_{n>=2} (-1)^n/a(n) = (50/9)*log(10/9) - 7/12. (End)

Extensions

Offset corrected by Mohammad K. Azarian, Nov 20 2008

A119635 a(n) = n*(1 + n^2)*2^n.

Original entry on oeis.org

4, 40, 240, 1088, 4160, 14208, 44800, 133120, 377856, 1034240, 2748416, 7127040, 18104320, 45187072, 111083520, 269484032, 646184960, 1533542400, 3606052864, 8409579520, 19465764864, 44753223680, 102257131520, 232330887168
Offset: 1

Views

Author

Mohammad K. Azarian, May 02 2007

Keywords

Links

Crossrefs

Cf. A128796, A128960, A128985, A129002.

Programs

  • GAP
    List([1..30],n->n*(n^2+1)*2^n); # Muniru A Asiru, Mar 04 2019
  • Magma
    [(n^3 + n)*2^n: n in [1..30]]; // Vincenzo Librandi, Feb 22 2013
  • Maple
    [(n^3+n)*2^n$n=1..30]; # Muniru A Asiru, Mar 04 2019
  • Mathematica
    Table[(n^3 + n)*2^n, {n, 30}] (* or *) CoefficientList[Series[4(1 +2x + 4x^2)/(1-2x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 22 2013 *)
  • PARI
    {a(n) = n*(1+n^2)*2^n}; \\ G. C. Greubel, Mar 04 2019
  • Sage
    [n*(1+n^2)*2^n for n in (1..30)] # G. C. Greubel, Mar 04 2019

Formula

G.f.: 4*x*(1 + 2*x + 4*x^2)/(1 - 2*x)^4. - Vincenzo Librandi, Feb 22 2013
E.g.f.: 4*x*(1 + 3*x + 2*x^2)*exp(2*x). - G. C. Greubel, Mar 04 2019

A121607 (n^3+n)*3^n.

Original entry on oeis.org

6, 90, 810, 5508, 31590, 161838, 765450, 3411720, 14526054, 59639490, 237731274, 924707340, 3523453830, 13191428502, 48642794730, 177008116752, 636661003590, 2266409860650, 7994034370026, 27964010896020, 97092998430246
Offset: 1

Views

Author

Mohammad K. Azarian, May 02 2007

Keywords

Links

Crossrefs

Cf. A128796, A128960, A128985, A129002.

Programs

  • Magma
    [(n^3 + n)*3^n: n in [1..30]]; // Vincenzo Librandi, Feb 22 2013
  • Mathematica
    Table[(n^3 + n)*3^n, {n, 30}] (* or *) CoefficientList[Series[6 (1 + 3 x + 9 x^2)/(1 - 3 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 22 2013 *)

Formula

G.f.: 6*x*(1 + 3*x + 9*x^2)/(1 - 3*x)^4. - Vincenzo Librandi, Feb 22 2013
Showing 1-10 of 15 results. Next

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