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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A033579 Four times pentagonal numbers: a(n) = 2*n*(3*n-1).

Original entry on oeis.org

0, 4, 20, 48, 88, 140, 204, 280, 368, 468, 580, 704, 840, 988, 1148, 1320, 1504, 1700, 1908, 2128, 2360, 2604, 2860, 3128, 3408, 3700, 4004, 4320, 4648, 4988, 5340, 5704, 6080, 6468, 6868, 7280, 7704, 8140, 8588, 9048, 9520, 10004, 10500, 11008, 11528, 12060
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Subsequence of A062717: A010052(6*a(n)+1) = 1. - Reinhard Zumkeller, Feb 21 2011
Sequence found by reading the line from 0, in the direction 0, 4, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011

Links

Crossrefs

Cf. A000326, A001318, A014642, A033579, A033580, A033581, A049450, A186423.

Programs

Formula

a(n) = 4*n*(3*n-1)/2 = 6*n^2 - 2*n = 4*A000326(n). - Omar E. Pol, Dec 11 2008
a(n) = 2*A049450(n). - Omar E. Pol, Dec 13 2008
a(n) = a(n-1) + 12*n - 8 for n > 0, a(0)=0. - Vincenzo Librandi, Aug 05 2010
a(n) = A014642(n)/2. - Omar E. Pol, Aug 19 2011
G.f.: x*(4+8*x)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 06 2012
a(n) = A191967(2*n). - Reinhard Zumkeller, Jul 07 2012
a(n) = A181617(n+1) - A181617(n). - J. M. Bergot, Jun 28 2013
a(n) = (A174371(n) - 1)/6. - Miquel Cerda, Jul 28 2016
From Ilya Gutkovskiy, Jul 28 2016: (Start)
E.g.f.: 2*x*(2 + 3*x)*exp(x).
a(n+1) = Sum_{k=0..n} A017569(k).
Sum_{i>0} 1/a(i) = (9*log(3) - sqrt(3)*Pi)/12 = 0.3705093754425278... (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(2*sqrt(3)) - log(2). - Amiram Eldar, Feb 20 2022

Extensions

More terms from Michel Marcus, Mar 04 2014

A016970 a(n) = (6*n + 5)^2.

Original entry on oeis.org

25, 121, 289, 529, 841, 1225, 1681, 2209, 2809, 3481, 4225, 5041, 5929, 6889, 7921, 9025, 10201, 11449, 12769, 14161, 15625, 17161, 18769, 20449, 22201, 24025, 25921, 27889, 29929, 32041, 34225, 36481, 38809, 41209, 43681, 46225, 48841, 51529
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The product of 4 successive terms of an arithmetic progression + square of the common difference is a square: a(n) = the square arising as the sum of first four terms of an arithmetic progression + n^2 where 1 is the first term and n is the common difference. a(1) = 25 = 1*2*3*4+1 a(2) = 121 = 1*3*5*7 +2^2 a(3) = 289 = 1*4*7*10 + 3^2, etc. - Amarnath Murthy, Mar 25 2004
If Y is a fixed 2-subset of a (6n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting Y. - Milan Janjic, Oct 21 2007
Sequence found by reading the line from 25 in the direction 25, 121,... in the square spiral whose vertices are the generalized 20-gonal numbers. - Omar E. Pol, Jul 28 2016

Links

Crossrefs

Cf. A016969 (6*n+5), A086731, A174371.

Programs

  • GAP
    List([0..40],n->(6*n+5)^2); # Muniru A Asiru, Dec 06 2018
  • Magma
    [(6*n+5)^2: n in [0..50]]; // Vincenzo Librandi, May 07 2011
  • Maple
    [(6*n+5)^2$n=0..40]; # Muniru A Asiru, Dec 06 2018
  • Mathematica
    Array[(6 # + 5)^2 &, 38, 0] (* or *)
CoefficientList[Series[(-25 - 46 x - x^2)/(x - 1)^3, {x, 0, 37}], x] (* Michael De Vlieger, Dec 06 2018 *)
CoefficientList[Series[E^x (25 + 96 x + 36 x^2), {x, 0, 50}], x]*Table[n!, {n, 0, 50}] (* Stefano Spezia, Dec 07 2018 *)
  • PARI
    a(n)=(6*n+5)^2 \\ Charles R Greathouse IV, Jul 28 2016
  • Sage
    s=((25+46*x+x^2)/(1-x)^3).series(x, 20); s.coefficients(x, sparse=False) # G. C. Greubel, Dec 07 2018

Formula

G.f.: (25 + 46*x + x^2) / (1-x)^3. - R. J. Mathar, Mar 10 2011
a(n) = 24 * A000326(n+1) + 1. - Jean-Bernard François, Oct 12 2014
a(n) = 6*A033579(n+1) + 1. - Miquel Cerda, Jul 28 2016
E.g.f.: exp(x)*(25 + 96*x + 36*x^2). - Stefano Spezia, Dec 07 2018
a(n) = A003215(3*n+2) + A002378(3*n+2). - Klaus Purath, Jun 09 2020
Sum_{n>=0} 1/a(n) = A086731. - Amiram Eldar, Nov 17 2020
Showing 1-2 of 2 results.

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

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