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A046172 Indices of pentagonal numbers which are also square. 6
1, 81, 7921, 776161, 76055841, 7452696241, 730288175761, 71560788528321, 7012226987599681, 687126683996240401, 67331402804643959601, 6597790348171111800481, 646516122717964312487521 (list; graph; refs; listen; history; 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*y-8)=(60*x+49*y-10)^2 (in fact P_(49*x+40*y-8)-(60*x+49*y-10)^2=1.5*x^2-0.5*x-y^2). [From Richard Choulet (richardchoulet(AT)yahoo.fr), Apr 28 2009]

Contribution from Weisenhorn Paul (paulweisenhorn(AT)online.de), May 15 2009: a(n)*(3*a(n)-1)/2=m*m 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

As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (sqrt(2) + sqrt(3))^4 = 49 + 20 * sqrt(6). - Ant King, Nov 07 2011

LINKS

L. Euler, De solutione problematum diophanteorum per numeros integros, par. 21

W. Sierpinski, Sur les nombres pentagonaux, Bull. Soc. Roy. Sci. Liege 33 (1964) 513-517.

Eric Weisstein's World of Mathematics, Pentagonal Square Number.

FORMULA

a(n) = 98*a(n-1) - a(n-2) - 16; g.f.: (1-18*x+x^2)/((1-x)*(1-98*x+x^2)) - Warut Roonguthai (warut822(AT)yahoo.com) Jan 05 2001

a(n+1)=49*a(n)-8+10*sqrt(8*(3a(n)^2-a(n)) with a(1)=1 [From Richard Choulet (richardchoulet(AT)yahoo.fr), 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 [From Richard Choulet (richardchoulet(AT)yahoo.fr), Apr 29 2009]

Contribution from Weisenhorn Paul (paulweisenhorn(AT)online.de), 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)) [From Sture Sjostedt (sture.sjostedt(AT)spray.se), 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)

MATHEMATICA

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

CROSSREFS

Cf. A036353, A046173.

Sequence in context: A205056 A186132 A206504 * A123847 A115443 A186527

Adjacent sequences:  A046169 A046170 A046171 * A046173 A046174 A046175

KEYWORD

nonn

AUTHOR

Eric Weisstein (eric(AT)weisstein.com)

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Last modified February 18 00:14 EST 2012. Contains 206085 sequences.