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.

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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

A116138 a(n) = 3^n * n*(n + 1).

Original entry on oeis.org

0, 6, 54, 324, 1620, 7290, 30618, 122472, 472392, 1771470, 6495390, 23383404, 82904796, 290166786, 1004423490, 3443737680, 11708708112, 39516889878, 132497807238, 441659357460, 1464449448420, 4832683179786, 15878816162154
Offset: 0

Views

Author

Mohammad K. Azarian, Apr 08 2007

Keywords

Links

Crossrefs

Cf. A007758, A036289, A128796, A027472.

Programs

Formula

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 6*x/(1-3*x)^3.
a(n) = 6 * A027472(n+2). (End)
a(n) = 9*a(n-1) -27*a(n-2) +27*a(n-3). - Vincenzo Librandi, Feb 28 2013
E.g.f.: 3*x*(2 + 3*x)*exp(3*x). - G. C. Greubel, May 10 2019
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = 1 - 2*log(3/2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(4/3) - 1. (End)

A128988 a(n) = (n^3 - n^2)*5^n.

Original entry on oeis.org

0, 100, 2250, 30000, 312500, 2812500, 22968750, 175000000, 1265625000, 8789062500, 59082031250, 386718750000, 2475585937500, 15551757812500, 96130371093750, 585937500000000, 3527832031250000, 21011352539062500
Offset: 1

Views

Author

Mohammad K. Azarian, Apr 30 2007

Keywords

Links

Crossrefs

Cf. A128796; A036289.

Programs

  • Magma
    [(n^3-n^2)*5^n: n in [1..30]]; // Vincenzo Librandi, Oct 26 2011
  • Mathematica
    Table[(n^3-n^2)5^n,{n,20}] (* or *) LinearRecurrence[{20,-150,500,-625},{0,100,2250,30000},20]

Formula

a(1)=0, a(2)=100, a(3)=2250, a(4)=30000, a(n)=20*a(n-1)- 150*a(n-2)+ 500*a(n-3)- 625*a(n-4). - Harvey P. Dale, Oct 25 2011
G.f.: (50*(5*x^2+2*x))/(5*x-1)^4. - Harvey P. Dale, Oct 25 2011

Extensions

Offset corrected by Mohammad K. Azarian, Nov 19 2008

A116144 a(n) = 4^n * n*(n+1).

Original entry on oeis.org

0, 8, 96, 768, 5120, 30720, 172032, 917504, 4718592, 23592960, 115343360, 553648128, 2617245696, 12213813248, 56371445760, 257698037760, 1168231104512, 5257039970304, 23502061043712, 104453604638720, 461794883665920
Offset: 0

Views

Author

Mohammad K. Azarian, Apr 08 2007

Keywords

Links

Crossrefs

Cf. A007758, A036289, A038845, A128796.

Programs

Formula

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 8*x/(1-4*x)^3.
a(n) = 8*A038845(n-1). (End)
a(n) = 12*a(n-1) -48*a(n-2) +64*a(n-3). - Vincenzo Librandi, Feb 28 2013
E.g.f.: 8*x*(1 + 2*x)*exp(4*x). - G. C. Greubel, May 10 2019
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = 1 - 3*log(4/3).
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*log(5/4) - 1. (End)

A116156 a(n) = 5^n * n*(n + 1).

Original entry on oeis.org

0, 10, 150, 1500, 12500, 93750, 656250, 4375000, 28125000, 175781250, 1074218750, 6445312500, 38085937500, 222167968750, 1281738281250, 7324218750000, 41503906250000, 233459472656250, 1304626464843750, 7247924804687500
Offset: 0

Views

Author

Mohammad K. Azarian, Apr 08 2007

Keywords

Links

Crossrefs

Cf. A007758, A036289, A084902, A128796.

Programs

  • GAP
    List([0..30], n-> 5^n*n*(n+1)); # G. C. Greubel, May 10 2019
  • Magma
    [(n^2+n)*5^n: n in [0..30]]; // Vincenzo Librandi, Feb 28 2013
  • Magma
    I:=[0,10,150]; [n le 3 select I[n] else 15*Self(n-1)-75*Self(n-2)+125*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Feb 28 2013
  • Mathematica
    Table[(n^2 + n) 5^n, {n, 0, 30}] (* or *) CoefficientList[Series[10 x/(1 - 5 x)^3, {x, 0, 30}], x](* Vincenzo Librandi, Feb 28 2013 *)
  • PARI
    a(n)=(n^2+n)*5^n \\ Charles R Greathouse IV, Feb 28 2013
  • Sage
    [5^n*n*(n+1) for n in (0..30)] # G. C. Greubel, May 10 2019

Formula

G.f.: 10*x/(1-5*x)^3. - Vincenzo Librandi, Feb 28 2013
a(n) = 15*a(n-1) -75*a(n-2) +125*a(n-3). - Vincenzo Librandi, Feb 28 2013
a(n) = 10*A084902(n). - Bruno Berselli, Feb 28 2013
E.g.f.: 5*x*(2 + 5*x)*exp(5*x). - G. C. Greubel, May 10 2019
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = 1 - 4*log(5/4).
Sum_{n>=1} (-1)^(n+1)/a(n) = 6*log(6/5) - 1. (End)

A116164 a(n) = 6^n * n*(n+1).

Original entry on oeis.org

0, 12, 216, 2592, 25920, 233280, 1959552, 15676416, 120932352, 906992640, 6651279360, 47889211392, 339578044416, 2377046310912, 16456474460160, 112844396298240, 767341894828032, 5179557790089216, 34733505180598272
Offset: 0

Views

Author

Mohammad K. Azarian, Apr 08 2007

Keywords

Links

Crossrefs

Cf. A007758, A036289, A081136, A128796.

Programs

  • GAP
    List([0..30], n-> 6^n*n*(n+1) ); # G. C. Greubel, May 10 2019
  • Magma
    [(n^2+n)*6^n: n in [0..30]]; // Vincenzo Librandi, Feb 28 2013
  • Magma
    I:=[0,12,216]; [n le 3 select I[n] else 18*Self(n-1)-108*Self(n-2)+216*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Feb 28 2013
  • Mathematica
    Table[(n^2 + n) 6^n, {n, 0, 30}] (* or *) CoefficientList[Series[12 x/(1 - 6 x)^3, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 28 2013 *)
  • PARI
    a(n)=(n^2+n)*6^n \\ Charles R Greathouse IV, Feb 28 2013
  • Sage
    [6^n*n*(n+1) for n in (0..30)] # G. C. Greubel, May 10 2019

Formula

G.f.: 12*x/(1-6*x)^3. - Vincenzo Librandi, Feb 28 2013
a(n) = 18*a(n-1) - 108*a(n-2) + 216*a(n-3). - Vincenzo Librandi, Feb 28 2013
a(n) = 12*A081136(n+1). - Bruno Berselli, Feb 28 2013
E.g.f.: 12*x*(1 + 3*x)*exp(6*x). - G. C. Greubel, May 10 2019
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = 1 - 5*log(6/5).
Sum_{n>=1} (-1)^(n+1)/a(n) = 7*log(7/6) - 1. (End)

A116165 a(n) = 7^n * n*(n+1).

Original entry on oeis.org

0, 14, 294, 4116, 48020, 504210, 4941258, 46118408, 415065672, 3631824630, 31072277390, 261007130076, 2159240803356, 17633799894074, 142426845298290, 1139414762386320, 9039357114931472, 71184937280085342, 556917450485373558
Offset: 0

Views

Author

Mohammad K. Azarian, Apr 08 2007

Keywords

Links

Crossrefs

Cf. A007758, A036289, A027474, A128796.

Programs

Formula

G.f.: 14*x/(1-7*x)^3. - Vincenzo Librandi, Feb 28 2013
a(n) = 21*a(n-1) - 147*a(n-2) + 343*a(n-3). - Vincenzo Librandi, Feb 28 2013
a(n+1) = 14*A027474(n+2). - Bruno Berselli, Feb 28 2013
E.g.f.: 7*x*(2 + 7*x)*exp(7*x). - G. C. Greubel, May 11 2019
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = 1 - 6*log(7/6).
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(8/7) - 1. (End)

A116166 a(n) = 8^n * n*(n+1).

Original entry on oeis.org

0, 16, 384, 6144, 81920, 983040, 11010048, 117440512, 1207959552, 12079595520, 118111600640, 1133871366144, 10720238370816, 100055558127616, 923589767331840, 8444249301319680, 76561193665298432, 689050742987685888
Offset: 0

Views

Author

Mohammad K. Azarian, Apr 08 2007

Keywords

Links

Crossrefs

Cf. A007758, A036289, A081138, A128796.

Programs

Formula

G.f.: 16*x/(1-8*x)^3. - Vincenzo Librandi, Feb 28 2013
a(n) = 24*a(n-1) - 192*a(n-2) + 512*a(n-3). - Vincenzo Librandi, Feb 28 2013
a(n) = 16*A081138(n+1). - Bruno Berselli, Feb 28 2013
E.g.f.: 16*x*(1 + 4*x)*exp(8*x). - G. C. Greubel, May 11 2019
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = 1 - 7*log(8/7).
Sum_{n>=1} (-1)^(n+1)/a(n) = 9*log(9/8) - 1. (End)

A116176 a(n) = 9^n * n*(n+1).

Original entry on oeis.org

0, 18, 486, 8748, 131220, 1771470, 22320522, 267846264, 3099363912, 34867844010, 383546284110, 4142299868388, 44059007691036, 462619580755878, 4804126415541810, 49413871702715760, 504021491367700752
Offset: 0

Views

Author

Mohammad K. Azarian, Apr 08 2007

Keywords

Links

Crossrefs

Cf. A007758, A036289, A081139, A128796.

Programs

Formula

G.f.: 18*x/(1-9*x)^3. - Vincenzo Librandi, Feb 28 2013
a(n) = 27*a(n-1) - 243*a(n-2) + 729*a(n-3). - Vincenzo Librandi, Feb 28 2013
a(n) = 18*A081139(n+1). - Bruno Berselli, Mar 01 2013
E.g.f.: 9*x*(2 + 9*x)*exp(9*x). - G. C. Greubel, May 11 2019
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = 1 - 8*log(9/8).
Sum_{n>=1} (-1)^(n+1)/a(n) = 10*log(10/9) - 1. (End)

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
Previous Showing 11-20 of 40 results. Next

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