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A046173 Indices of square numbers that are also pentagonal. 8
1, 99, 9701, 950599, 93149001, 9127651499, 894416697901, 87643708742799, 8588189040096401, 841554882220704499, 82463790268588944501, 8080609891439495856599, 791817305570802005002201, 77590015336047156994359099, 7603029685627050583442189501 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

a(n)^2 is of the form (2*m-1)*(3*m-2), and the corresponding values of m are 1, 41, 3961, 388081, 38027921, 3726348121, 365144087881, ..., with closed form ((5-2*sqrt(6))^(2n-1)+(5+2*sqrt(6))^(2n-1)+14)/24 (for n>0). - Bruno Berselli, Dec 12 2013

REFERENCES

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

LINKS

Colin Barker, Table of n, a(n) for n = 1..503

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

Tanya Khovanova, Recursive Sequences

Eric Weisstein's World of Mathematics, Pentagonal Square Number

Index entries for linear recurrences with constant coefficients, signature (98,-1).

FORMULA

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

a(1-n) = -a(n) for all n in Z. - Michael Somos, Sep 05 2006

Define f[x,s] = s x + Sqrt[(s^2-1)x^2+1]; f[0,s]=0. a(n) = f[f[a(n-1),5],5]. - Marcos Carreira, Dec 27 2006

a(n) = ((12+5*sqrt(6))/24)*(5+2*sqrt(6))^(2*n)+((12-5*sqrt(6))/24)*(5-2*sqrt(6))^(2*n) for n>=0. - Richard Choulet, Apr 29 2009

a(n+1) = 49*a(n)+10*sqrt(24*a(n)^2+1) for n>=0 with a(0)=1. - Richard Choulet, Apr 29 2009

a(n) = b such that (-1)^n*Integral_{x=-Pi/2,Pi/2} (cos(2*n-1)*x)/(5-sin(x)) dx = c + b*(log(2)-log(3)). - Francesco Daddi, Aug 01 2011

a(n) = floor(1/24 * sqrt(6) * (sqrt(2) + sqrt(3))^(4n-2)). - Ant King, Nov 07 2011

EXAMPLE

G.f. = x + 99*x^2 + 9701*x^3 + 950599*x^4 + 93149001*x^5 + ...

MATHEMATICA

CoefficientList[Series[(1 + x)/(1 - 98*x + x^2), {x, 0, 30}], x] (* T. D. Noe, Aug 01 2011 *)

LinearRecurrence[{98, -1}, {1, 99}, 30] (* Harvey P. Dale, Jul 31 2017 *)

PROG

(PARI) {a(n) = subst( poltchebi(n) - poltchebi(n-1), 'x, 49) / 48}; /* Michael Somos, Sep 05 2006 */

(PARI) Vec(x*(x+1)/(x^2-98*x+1) + O(x^30)) \\ Colin Barker, Jun 23 2015

CROSSREFS

Cf. A036353, A046172.

Sequence in context: A163051 A093233 A213155 * A278620 A171415 A098609

Adjacent sequences:  A046170 A046171 A046172 * A046174 A046175 A046176

KEYWORD

nonn,easy

AUTHOR

Eric W. Weisstein

STATUS

approved

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Last modified August 20 22:26 EDT 2017. Contains 290837 sequences.