A203410 Indices of decagonal numbers that are also heptagonal.

1, 12, 850, 16761, 1225159, 24168810, 1766677888, 34851406719, 2547548288797, 50255704319448, 3673562865766846, 72468690777236757, 5297275104887502595, 104499801845071083606, 7638667027684912974604, 150688641791901725322555, 11014952556646539621875833

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+11*x-604*x^2+49*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)-540.

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

a(n) = 1/80*((sqrt(10)-(-1)^n)*(5-sqrt(10))* (3+sqrt(10))^(2*n-1)-(sqrt(10)+(-1)^n)*(5+sqrt(10))*(3-sqrt(10))^(2*n-1)+30).

a(n) = ceiling(1/80*(sqrt(10)-(-1)^n)*(5-sqrt(10))*(3+sqrt(10))^(2*n-1))

Example

The second decagonal number that is also heptagonal is A001107(12)=540. Hence a(2)=12.

Mathematica

LinearRecurrence[{1, 1442, -1442, -1, 1}, {1, 12, 850, 16761, 1225159}, 17]

Crossrefs

Cf. A203408, A203409, A001107, A000566.

Sequence in context: A350412 A207817 A349468 * A275568 A271433 A140412

Adjacent sequences: A203407 A203408 A203409 * A203411 A203412 A203413

Keyword

nonn,easy

Author

Ant King, Jan 03 2012

Status

approved