A203409 Indices of heptagonal numbers that are also decagonal.

1, 15, 1075, 21201, 1549717, 30571395, 2234690407, 44083929957, 3222422016745, 63568996426167, 4646730313455451, 91666448762602425, 6700581889580743165, 132182955546676270251, 9662234438045118188047, 190607730231858419099085, 13932935359079170846420177

Offset

1,2

Comments

As n increases, the ratios of consecutive terms settle into an approximate 2-cycle with a(n)/a(n-1) bounded above and below by 1/9*(329+104*sqrt(10)) and 1/9*(89+28*sqrt(10)) respectively.

Links

Table of n, a(n) for n=1..17.

Index entries for linear recurrences with constant coefficients, signature (1,1442,-1442,-1,1).

Formula

G.f.: x*(1+14*x-382*x^2-62*x^3-3*x^4) / ((1-x)*(1-38*x+x^2)*(1+38*x+x^2)).

a(n) = 1442*a(n-2)-a(n-4)-432.

a(n) = a(n-1)+1442*a(n-2)-1442*a(n-3)-a(n-4)+a(n-5).

a(n) = 1/40*(((-1)^n-sqrt(10))*(2-sqrt(10))*(3+sqrt(10))^(2*n-1)+((-1)^n+sqrt(10))*(2+sqrt(10))*(3-sqrt(10))^(2*n-1)+12).

a(n) = ceiling(1/40*((-1)^n-sqrt(10))*(2-sqrt(10))*(3+sqrt(10))^(2*n-1)).

Example

The second heptagonal number that is also decagonal is A000566(15)=540. Hence a(2)=15.

Mathematica

LinearRecurrence[{1, 1442, -1442, -1, 1}, {1, 15, 1075, 21201, 1549717}, 17]

Crossrefs

Cf. A203408, A203410, A001107, A000566.

Sequence in context: A370735 A233509 A206162 * A207980 A131313 A208346

Adjacent sequences: A203406 A203407 A203408 * A203410 A203411 A203412

Keyword

nonn,easy

Author

Ant King, Jan 02 2012

Status

approved