A203136 Indices of decagonal numbers that are also hexagonal.

1, 20, 667, 22646, 769285, 26133032, 887753791, 30157495850, 1024467105097, 34801724077436, 1182234151527715, 40161159427864862, 1364297186395877581, 46345943178031972880, 1574397770866691200327, 53483178266289468838226, 1816853663282975249299345

Offset

1,2

Comments

As n increases, this sequence is approximately geometric with common ratio (1+sqrt(2))^4=17+12*sqrt(2).

Links

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

Index entries for linear recurrences with constant coefficients, signature (35, -35, 1).

Formula

G.f.: x(1-15*x+2*x^2) / ((1-x)*(1-34*x+x^2))

a(n) = 35*a(n-1)-35*a(n-2)+a(n-3)

a(n) = 34*a(n-1)-a(n-2)-12

a(n) = 1/16 *((3-sqrt(2))*(1+sqrt(2))^(4*n-2)+(3+sqrt(2))*(1-sqrt(2))^(4*n-2)+6)

a(n) = ceiling(1/16*(3-sqrt(2))*(1+sqrt(2))^(4*n-2))

Example

The second decagonal number which is also hexagonal is A001107(20) = 1540. Hence a(2) = 20.

Mathematica

LinearRecurrence[{35, -35, 1}, {1, 20, 667}, 17]

Crossrefs

Cf. A203134, A203135, A000384, A001107.

Sequence in context: A201724 A006410 A159874 * A034404 A179712 A012802

Adjacent sequences: A203133 A203134 A203135 * A203137 A203138 A203139

Keyword

nonn,easy

Author

Ant King, Dec 30 2011

Status

approved