A202565 Indices of decagonal numbers which are also pentagonal.

1, 56, 5451, 534106, 52336901, 5128482156, 502538914351, 49243685124206, 4825378603257801, 472837859434140256, 46333284845942487251, 4540189077042929610306, 444892196265361159322701, 43594895044928350684014356, 4271854822206713005874084151

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*(1-43*x+6*x^2) / ((1-x)*(1-98*x+x^2)).

a(n) = 98*a(n-1)-a(n-2)-36.

a(n) = 99*a(n-1)-99*a(n-2)+a(n-3).

a(n) = 1/48*(5*sqrt(3)*((sqrt(3)+sqrt(2))^(4n-3)+(sqrt(3)-sqrt(2))^(4n-3))+18).

a(n) = ceiling(5/48*sqrt(3)*(sqrt(3)+sqrt(2))^(4n-3)).

Example

The second decagonal number that is also pentagonal is A001107(56) = 12376. Hence a(2)=56.

Mathematica

LinearRecurrence[{99, -99, 1}, {1, 56, 5451}, 15]

Crossrefs

Cf. A202563, A202564, A000326, A001107.

Sequence in context: A278324 A009600 A278777 * A213693 A180372 A255960

Adjacent sequences: A202562 A202563 A202564 * A202566 A202567 A202568

Keyword

nonn,easy

Author

Ant King, Dec 22 2011

Status

approved