A248205 -id:A248205 - cp's OEIS frontend

A036353 Square pentagonal numbers.

0, 1, 9801, 94109401, 903638458801, 8676736387298001, 83314021887196947001, 799981229484128697805801, 7681419682192581869134354401, 73756990988431941623299373152801, 708214619789503821274338711878841001, 6800276705461824703444258688161258139001

Offset: 0

Jean-Francois Chariot (jeanfrancois.chariot(AT)afoc.alcatel.fr)

Lim_{n -> oo} a(n)/a(n-1) = (sqrt(2) + sqrt(3))^8 = 4801 + 1960*sqrt(6). - Ant King, Nov 06 2011

Pentagonal numbers (A000326) which are also centered octagonal numbers (A016754). - Colin Barker, Jan 11 2015

Colin Barker, Table of n, a(n) for n = 0..252
Muniru A. Asiru, All square chiliagonal numbers, International Journal of Mathematical Education in Science and Technology, Volume 47, 2016 - Issue 7.
Byungchan Kim, Eunmi Kim, and Jeremy Lovejoy, On weighted overpartitions related to some q-series in Ramanujan's lost notebook, Int'l J. Number Theory (2021). Also at Université de Paris (France, 2020).
Eric Weisstein's World of Mathematics, Pentagonal Square Number
Index entries for linear recurrences with constant coefficients, signature (9603,-9603,1).

Cf. A000326, A001078, A001079, A001110, A046172, A046173, A248205.

Table[Floor[1/96 ( Sqrt[2] + Sqrt[3] ) ^ ( 8*n - 4 ) ] , {n, 0, 9}] (* Ant King, Nov 06 2011 *)
LinearRecurrence[{9603,-9603,1},{0,1,9801,94109401},20] (* Harvey P. Dale, Apr 14 2019 *)

for(n=0,10^9,g=(n*(3*n-1)/2); if(issquare(g),print(g)))

concat(0, Vec(x*(1+198*x+x^2)/((1-x)*(1-9602*x+x^2)) + O(x^20))) \\ Colin Barker, Jun 24 2015

a(n) = 9602*a(n-1) - a(n-2) + 200; g.f.: x*(1+198*x+x^2)/((1-x)*(1-9602*x+x^2)). - Warut Roonguthai, Jan 05 2001
a(n+1) = 4801*a(n)+100+980*(24*a(n)^2+a(n))^(1/2). - Richard Choulet, Sep 21 2007
From Ant King, Nov 06 2011: (Start)
a(n) = floor(1/96*(sqrt(2) + sqrt(3))^(8*n-4)).
a(n) = 9603*a(n-1) - 9603*a(n-2) + a(n-3).
(End)

More terms from Eric W. Weisstein

A046172 Indices of pentagonal numbers (A000326) that are also squares (A000290).

Original entry on oeis.org

1, 81, 7921, 776161, 76055841, 7452696241, 730288175761, 71560788528321, 7012226987599681, 687126683996240401, 67331402804643959601, 6597790348171111800481, 646516122717964312487521, 63351982236012331511976561, 6207847743006490523861215441

Offset: 1

Eric W. Weisstein

if P_x = y^2 is a pentagonal number that is also a square, the least both pentagonal and square number that is greater as P_x, is P_(49*x + 40*y - 8) = (60*x + 49*y - 10)^2 (in fact, P_(49*x + 40*y - 8) - (60*x + 49*y - 10)^2 = (3/2)*x^2 - (1/2)*x - y^2). - Richard Choulet, Apr 28 2009
a(n)*(3*a(n)-1)/2 = m^2 is equivalent to the Pell equation (6*a(n)-1)^2 - 6*(2*m)^2 = 1 or x(n)^2 - 6*y(n)^2 = 1. - Paul Weisenhorn, May 15 2009
As n increases, this sequence is approximately geometric with common ratio r = lim_{n -> oo} a(n)/a(n-1) = (sqrt(2) + sqrt(3))^4 = 49 + 20*sqrt(6). - Ant King, Nov 07 2011
Numbers k such that the k-th pentagonal number is equal to the sum of two consecutive triangular numbers. - Colin Barker, Dec 11 2014
Indices of pentagonal numbers (A000326) that are also centered octagonal numbers (A016754). - Colin Barker, Jan 11 2015

Muniru A. Asiru, All square chiliagonal numbers, International Journal of Mathematical Education in Science and Technology, Volume 47, 2016 - Issue 7; http://dx.doi.org/10.1080/0020739X.2016.1164346

Colin Barker, Table of n, a(n) for n = 1..503
Leonhard Euler, De solutione problematum diophanteorum per numeros integros, section 21.
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Circular Segment, Forum Geometricorum, Vol. 18 (2018), 47-55.
W. Sierpiński, Sur les nombres pentagonaux, Bull. Soc. Roy. Sci. Liege 33 (1964) 513-517.
Eric Weisstein's World of Mathematics, Pentagonal Square Number.
Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Cf. A036353, A046173.

Cf. A000217, A000290, A000326, A251914, A248205.

LinearRecurrence[{99, -99, 1}, {1, 81, 7921}, 13] (* Ant King, Nov 07 2011 *)
Table[Round[(1 + x^(2*n+1))^2 / (12*x^(2*n+1)) /. x->5+2*Sqrt@6],{n,0,99}] (* Federico Provvedi, Apr 24 2023 *)

a(n) = 98*a(n-1) - a(n-2) - 16; g.f.: x*(1 - 18*x + x^2)/((1-x)*(1 - 98*x + x^2)). - Warut Roonguthai Jan 05 2001 - Corrected by Colin Barker, Jan 11 2015
a(n+1) = 49*a(n) - 8 + 10*sqrt(8*(3*a(n)^2 - a(n))) with a(1) = 1. - Richard Choulet, Apr 28 2009
a(n) = 1/6+((5 + 2*sqrt(6))^(2*n+1)/12) + ((5 - 2*sqrt(6))^(2*n+1)/12) for n >= 0. - Richard Choulet, Apr 29 2009
From Paul Weisenhorn, May 15 2009: (Start)
x(n+2) = 98*x(n+1) - x(n) with x(1)=5, x(2)=485;
y(n+2) = 98*y(n+1) - y(n) with y(n)=A046173(n)*2;
m(n+2) = 98*m(n+1) - m(n) with m(n)=A046173(n);
a(n) = A072256(n)^2.
(End)
a(n) = b(n)*b(n), b(n) = 10*b(n-1)- b(n-2), b(1)=1, b(2)=9, b(n)=((5 + sqrt(24))^n - (5 - sqrt(24))^n)/(2*sqrt(24)). - Sture Sjöstedt, Sep 21 2009
From Ant King, Nov 07 2011: (Start)
a(n) = 99*a(n-1) - 99*a(n-2) + a(n-3).
a(n) = ceiling((1/12)*(sqrt(3) + sqrt(2))^(4*n-2)).
(End)
a(n) = (1 + x^(2n+1))^2 / (12*x^(2*n+1)), with x = 5 + 2*sqrt(6). - Federico Provvedi, Apr 24 2023

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A036353 Square pentagonal numbers.

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