

A046172


Indices of pentagonal numbers which are also square.


8



1, 81, 7921, 776161, 76055841, 7452696241, 730288175761, 71560788528321, 7012226987599681, 687126683996240401, 67331402804643959601, 6597790348171111800481, 646516122717964312487521
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

if P_x=y^2 is a pentagonal number which is also a square, the least both pentagonal and square number which is greater as P_x, is P_(49*x+40*y8)=(60*x+49*y10)^2 (in fact P_(49*x+40*y8)(60*x+49*y10)^2=1.5*x^20.5*xy^2). [Richard Choulet, Apr 28 2009]
a(n)*(3*a(n)1)/2=m*m is equivalent to the Pell equation (6*a(n)1)^26*(2*m)^2=1 or x(n)^26*y(n)^2=1.  Paul Weisenhorn, May 15 2009
As n increases, this sequence is approximately geometric with common ratio r = lim(n > Infinity, a(n)/a(n1)) = (sqrt(2) + sqrt(3))^4 = 49 + 20 * sqrt(6).  Ant King, Nov 07 2011
Also numbers n such that the pentagonal number P(n) is equal to the sum of two consecutive triangular numbers.  Colin Barker, Dec 11 2014


LINKS

Colin Barker, Table of n, a(n) for n = 1..503
L. Euler, De solutione problematum diophanteorum per numeros integros, par. 21
W. Sierpinski, Sur les nombres pentagonaux, Bull. Soc. Roy. Sci. Liege 33 (1964) 513517.
Eric Weisstein's World of Mathematics, Pentagonal Square Number.
Index to sequences with linear recurrences with constant coefficients, signature (99,99,1).


FORMULA

a(n) = 98*a(n1)  a(n2)  16; g.f.: (118*x+x^2)/((1x)*(198*x+x^2)).  Warut Roonguthai Jan 05 2001
a(n+1) = 49*a(n)8+10*sqrt(8*(3a(n)^2a(n)) with a(1)=1. [Richard Choulet, Apr 28 2009]
a(n) = 1/6+((5+2*sqrt(6))^(2*n+1)/12)+((52*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(n1) b(n2), b(1)=1, b(2)=9, b(n)=((5+sqrt(24))^n(5sqrt(24))^n)/(2*sqrt(24)). [Sture Sjöstedt, Sep 21 2009]
From Ant King, Nov 07 2011: (Start)
a(n) = 99*a(n1)  99*a(n2) + a(n3).
a(n) = ceiling(1/12*(sqrt(3) + sqrt(2))^(4*n2)).
(End)


MATHEMATICA

LinearRecurrence[{99, 99, 1}, {1, 81, 7921}, 13] (* Ant King, Nov 07 2011 *)


CROSSREFS

Cf. A036353, A046173.
Cf. A000217, A000326, A251914.
Sequence in context: A205056 A186132 A206504 * A123847 A115443 A186527
Adjacent sequences: A046169 A046170 A046171 * A046173 A046174 A046175


KEYWORD

nonn,easy,changed


AUTHOR

Eric W. Weisstein


STATUS

approved



