login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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*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). [Richard Choulet, Apr 28 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. - Paul Weisenhorn, May 15 2009

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

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) 513-517.

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

a(n+1) = 49*a(n)-8+10*sqrt(8*(3a(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)

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified December 20 09:43 EST 2014. Contains 252241 sequences.