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A202564
Indices of pentagonal numbers which are also decagonal.
2
1, 91, 8901, 872191, 85465801, 8374776291, 820642610701, 80414601072391, 7879810262483601, 772140991122320491, 75661937319724924501, 7414097716341920280591, 726505914264188462573401, 71190165500174127411912691, 6975909713102800297904870301
OFFSET
1,2
COMMENTS
As n increases, this sequence is approximately geometric with common ratio r = lim(n->oo, a(n)/a(n-1)) = (sqrt(3)+sqrt(2))^4 = 49+20*sqrt(6).
FORMULA
G.f.: x*(1+x)*(1-9*x) / ((1-x)*(1-98*x+x^2)).
a(n) = 98*a(n-1)-a(n-2)-16.
a(n) = 99*a(n-1)-99a(n-2)+a(n-3).
a(n) = 1/24*(5*sqrt(2)*((sqrt(3)+sqrt(2))^(4n-3)-(sqrt(3)-sqrt(2))^(4n-3))+4).
a(n) = ceiling(5/24*sqrt(2)*(sqrt(3)+sqrt(2))^(4n-3)).
EXAMPLE
The second pentagonal number that is also decagonal is A000326(91) = 12376. Hence a(2)=91.
MATHEMATICA
LinearRecurrence[{99, -99, 1}, {1, 91, 8901}, 15]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ant King, Dec 22 2011
STATUS
approved