A202564 Indices of pentagonal numbers which are also decagonal.

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).

Links

Table of n, a(n) for n=1..15.

Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

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

Cf. A202563, A202565, A000326, A001107.

Sequence in context: A022253 A172174 A060078 * A006244 A054216 A109627

Adjacent sequences: A202561 A202562 A202563 * A202565 A202566 A202567

Keyword

nonn,easy

Author

Ant King, Dec 22 2011

Status

approved