|
| |
|
|
A036353
|
|
Square pentagonal numbers.
|
|
9
|
|
|
|
0, 1, 9801, 94109401, 903638458801, 8676736387298001, 83314021887196947001, 799981229484128697805801, 7681419682192581869134354401, 73756990988431941623299373152801, 708214619789503821274338711878841001, 6800276705461824703444258688161258139001
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
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
|
|
|
LINKS
|
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).
|
|
|
FORMULA
|
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)
|
|
|
MATHEMATICA
|
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 *)
|
|
|
PROG
|
(PARI) for(n=0, 10^9, g=(n*(3*n-1)/2); if(issquare(g), print(g)))
(PARI) concat(0, Vec(x*(1+198*x+x^2)/((1-x)*(1-9602*x+x^2)) + O(x^20))) \\ Colin Barker, Jun 24 2015
|
|
|
CROSSREFS
|
Cf. A000326, A001078, A001079, A001110, A046172, A046173, A248205.
Sequence in context: A227489 A350918 A113937 * A174769 A031597 A031777
Adjacent sequences: A036350 A036351 A036352 * A036354 A036355 A036356
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
Jean-Francois Chariot (jeanfrancois.chariot(AT)afoc.alcatel.fr)
|
|
|
EXTENSIONS
|
More terms from Eric W. Weisstein
|
|
|
STATUS
|
approved
|
| |
|
|
