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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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)

A127960 a(n) = n^2*3^n.

Original entry on oeis.org

0, 3, 36, 243, 1296, 6075, 26244, 107163, 419904, 1594323, 5904900, 21434787, 76527504, 269440587, 937461924, 3228504075, 11019960576, 37321507107, 125524238436, 419576389587, 1394713760400, 4613015762523, 15188432850756
Offset: 0

Views

Author

Mohammad K. Azarian, Apr 07 2007

Keywords

Links

Crossrefs

Cf. A036289, A007758, A069996.

Programs

  • Magma
    [n^2*3^n: n in [0..30]]; // Vincenzo Librandi, Feb 07 2013
  • Mathematica
    LinearRecurrence[{9,-27,27}, {0,3,36}, 25] (* or *) CoefficientList[ Series[3*x(1 + 3*x)/(1 - 3*x)^3, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 07 2013 *)
  • PARI
    for(n=0,30, print1(n^2*3^n, ", ")) \\ G. C. Greubel, May 04 2018

Formula

G.f.: 3*x*(1 + 3*x)/(1 - 3*x)^3. - Vincenzo Librandi, Feb 07 2013
a(n) = 9*a(n-1) - 27*a(n-2) + 27*a(n-3). - Vincenzo Librandi, Feb 07 2013
a(n) = 3*A069996(n) for n>0. - Bruno Berselli, Feb 07 2013
E.g.f.: (9*x^2 + 3*x)*exp(3*x). - G. C. Greubel, May 04 2018

A128784 a(n) = n^2*5^n.

Original entry on oeis.org

0, 5, 100, 1125, 10000, 78125, 562500, 3828125, 25000000, 158203125, 976562500, 5908203125, 35156250000, 206298828125, 1196289062500, 6866455078125, 39062500000000, 220489501953125, 1235961914062500, 6885528564453125, 38146972656250000, 210285186767578125
Offset: 0

Views

Author

Mohammad K. Azarian, Apr 07 2007

Keywords

Links

Crossrefs

Cf. A036289, A007758.

Programs

Formula

From Harvey P. Dale, Dec 30 2011: (Start)
a(0)=0, a(1)=5, a(2)=100, a(n) = 15*a(n-1)-75*a(n-2)+125*a(n-3).
G.f.: -5*(5*x^2+x)/(5*x-1)^3. (End)

A128785 a(n) = n^2*6^n.

Original entry on oeis.org

0, 6, 144, 1944, 20736, 194400, 1679616, 13716864, 107495424, 816293376, 6046617600, 43898443776, 313456656384, 2207257288704, 15359376162816, 105791621529600, 722204136308736, 4891804579528704, 32905425960566784
Offset: 0

Views

Author

Mohammad K. Azarian, Apr 07 2007

Keywords

Links

Crossrefs

Cf. A036289; A007758.

Programs

  • Magma
    [n^2*6^n: n in [0..20]]; // Vincenzo Librandi, Oct 04 2011
  • Mathematica
    Table[n^2 6^n,{n,0,30}] (* or *) LinearRecurrence[{18,-108,216},{0,6,144},30] (* Harvey P. Dale, Oct 03 2011 *)

Formula

From R. J. Mathar, Jul 25 2009: (Start)
a(n) = 18*a(n-1) - 108*a(n-2) + 216*a(n-3).
G.f.: -6*x*(1+6*x)/(6*x-1)^3. (End)

A128786 n^2*7^n.

Original entry on oeis.org

0, 7, 196, 3087, 38416, 420175, 4235364, 40353607, 368947264, 3268642167, 28247524900, 239256535903, 1993145356944, 16374242758783, 132931722278404, 1068201339737175, 8507630225817856, 67230218542302823, 527606005722985476
Offset: 0

Views

Author

Mohammad K. Azarian, Apr 07 2007

Keywords

Links

Crossrefs

Cf. A036289; A007758.

Programs

Formula

G.f.: 7*x*(1 + 7*x)/((1 - 7*x)^3). - Vincenzo Librandi, Feb 06 2013
a(n) = 21*a(n-1) - 147*a(n-2) + 343*a(n-3). - Vincenzo Librandi, Feb 06 2013
Previous Showing 21-30 of 49 results. Next

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

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