A046178 Indices of pentagonal numbers that are also hexagonal.

1, 165, 31977, 6203341, 1203416145, 233456528757, 45289363162681, 8785902997031325, 1704419892060914337, 330648673156820350021, 64144138172531086989705, 12443632156797874055652717, 2414000494280615035709637361, 468303652258282519053613995285

Offset

1,2

Comments

The reason we obtain the same Diophantine equation with various parameters is the following: the number that is written 361 in base 4*A046179(n)-2 is the square of 6*A046178(n)-1. That is, 361 in base 110770 is 3*110770^2 + 6*110770 + 1 = 36810643321, i.e., the square of 191861 if we consider the third terms of A046179 and A046178, which are 27693 and 31977, respectively. - Richard Choulet, Oct 03 2007

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)

Formula

From Warut Roonguthai Jan 08 2001: (Start)

a(n) = 194*a(n-1) - a(n-2) - 32.

G.f.: x*(1-30*x-3*x^2)/((1-x)*(1-194*x+x^2)). (End)

a(n+1) = 97*a(n) - 16 + 28*sqrt(12*a(n)^2 - 4*a(n) + 1). - Richard Choulet, Oct 09 2007

From Ant King, Dec 14 2011: (Start)

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

a(n) = (1/12)*((sqrt(3)-1)*(2+sqrt(3))^(4n-2) - (sqrt(3)+1)* (2-sqrt(3))^(4n-2) + 2).

a(n) = ceiling((1/12)*(sqrt(3)-1)*(2+sqrt(3))^(4n-2)).

(End)

Mathematica

LinearRecurrence[{195, -195, 1}, {1, 165, 31977}, 11] (* Ant King, Dec 14 2011 *)

Prog

(PARI) Vec(x*(3*x^2+30*x-1)/((x-1)*(x^2-194*x+1)) + O(x^20)) \\ Colin Barker, Jun 21 2015

Crossrefs

Cf. A046179, A046180.

Sequence in context: A140912 A132055 A273441 * A176018 A184287 A233466

Adjacent sequences: A046175 A046176 A046177 * A046179 A046180 A046181

Keyword

nonn,easy

Author

Eric W. Weisstein

Status

approved