A046179 Indices of hexagonal numbers that are also pentagonal.

1, 143, 27693, 5372251, 1042188953, 202179284583, 39221739020101, 7608815190614963, 1476070925240282673, 286350150681424223551, 55550453161271059086173, 10776501563135904038493963, 2090585752795204112408742601, 405562859540706461903257570583

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• easy

Comments

As n increases, this sequence is approximately geometric with common ratio r = lim_{n->infinity} a(n)/a(n-1) = (2+sqrt(3))^4 = 97 + 56*sqrt(3). - Ant King, Dec 14 2011

Links

• Colin Barker, Table of n, a(n) for n = 1..438
• Eric Weisstein's World of Mathematics, Hexagonal Pentagonal Number.
• Index entries for linear recurrences with constant coefficients, signature (195,-195,1).

Crossrefs

Cf. A046178, A046180, A276915.

Programs

Mathematica

LinearRecurrence[{195, -195, 1}, {1, 143, 27693}, 11] (* Ant King, Dec 14 2011 *)

PARI

Vec(-x*(3*x^2-52*x+1)/((x-1)*(x^2-194*x+1)) + O(x^20)) \\ Colin Barker, Jun 21 2015

Formula

From Warut Roonguthai, Jan 08 2001: (Start)

a(n) = 194*a(n-1) - a(n-2) - 48.

G.f.: x*(1-52*x+3*x^2)/((1-x)*(1-194*x+x^2)). (End)

From Ant King, Dec 14 2011: (Start)

a(n) = 195*a(n-1) - 195*a(n-2) + a(n-3).

a(n) = (1/24)*sqrt(3)*((sqrt(3)-1)*(2+sqrt(3))^(4n-2)+(sqrt(3)+1)* (2-sqrt(3))^(4n-2)+2*sqrt(3)).

a(n) = ceiling((1/24)*sqrt(3)*(sqrt(3)-1)*(2+sqrt(3))^(4n-2)).

(End)

a(n) = A276915(2n-1). - Daniel Poveda Parrilla, Dec 03 2016