A203135 Indices of hexagonal numbers that are also decagonal

1, 28, 943, 32026, 1087933, 36957688, 1255473451, 42649139638, 1448815274233, 49217070184276, 1671931570991143, 56796456343514578, 1929407584108504501, 65543061403345638448, 2226534680129643202723, 75636636063004523254126, 2569419091462024147437553

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(1-7*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)-8

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

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

Example

The second hexagonal number which is also decagonal is A000384(28)=1540. Hence a(2) = 28.

Mathematica

LinearRecurrence[{35, -35, 1}, {1, 28, 943}, 17]

Crossrefs

Cf. A203134, A203136, A000384, A001107.

Sequence in context: A160263 A129461 A239336 * A097579 A091549 A034904

Adjacent sequences: A203132 A203133 A203134 * A203136 A203137 A203138

Keyword

nonn,easy

Author

Ant King, Dec 30 2011

Status

approved