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A046173
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Indices of square numbers which are also pentagonal.
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5
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1, 99, 9701, 950599, 93149001, 9127651499, 894416697901, 87643708742799, 8588189040096401, 841554882220704499, 82463790268588944501, 8080609891439495856599, 791817305570802005002201, 77590015336047156994359099, 7603029685627050583442189501
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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
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LINKS
| 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 to sequences with linear recurrences with constant coefficients, signature (98,-1).
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FORMULA
| a(n) = 98*a(n-1) - a(n-2); g.f.: (1+x)/(1-98*x+x^2) - Warut Roonguthai (warut822(AT)yahoo.com) Jan 05 2001
a(1-n)=-a(n). - 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 [From Richard Choulet (richardchoulet(AT)yahoo.fr), Apr 29 2009]
a(n+1)=49*a(n)+10*sqrt(24*a(n)^2+1) for n>=0 with a(0)=1 [From Richard Choulet (richardchoulet(AT)yahoo.fr), 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*(ln(2)-ln(3)) [From Francesco Daddi (francesco.daddi(AT)libero.it), Aug 01 2011]
a(n) = floor(1/24 * sqrt(6) * (sqrt(2) + sqrt(3))^(4n-2)). - Ant King, Nov 07 2011
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MATHEMATICA
| CoefficientList[Series[(1 + x)/(1 - 98*x + x^2), {x, 0, 30}], x] (* T. D. Noe, Aug 01 2011 *)
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PROG
| (PARI) {a(n)=subst(poltchebi(n)-poltchebi(n-1), 'x, 49)/48} /* Michael Somos Sep 05 2006 */
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CROSSREFS
| Cf. A036353, A046172.
Sequence in context: A069363 A163051 A093233 * A171415 A098609 A195623
Adjacent sequences: A046170 A046171 A046172 * A046174 A046175 A046176
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KEYWORD
| nonn,easy
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com)
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