A046199 Indices of pentagonal numbers that are also heptagonal.

1, 54, 3337, 206830, 12820113, 794640166, 49254870169, 3053007310302, 189237198368545, 11729653291539478, 727049266877079081, 45065324893087363534, 2793323094104539460017, 173140966509588359157510, 10731946600500373728305593, 665207548264513582795789246

Offset

1,2

Comments

As n increases, this sequence is approximately geometric with common ratio r = lim_{n->infinity} a(n)/a(n-1) = (4+sqrt(15))^2 = 31 + 8*sqrt(15). - Ant King, Dec 15 2011

Links

Colin Barker, Table of n, a(n) for n = 1..558

Eric Weisstein's World of Mathematics, Heptagonal Pentagonal Number.

Index entries for linear recurrences with constant coefficients, signature (63,-63,1).

Formula

From Ant King, Dec 15 2011: (Start)

a(n) = 63*a(n-1) - 63*a(n-2) + a(n-3).

a(n) = 62*a(n-1) - a(n-2) - 10.

a(n) = (1/60)*((3*sqrt(15)-5)*(4+sqrt(15))^(2*n-1) - (3*sqrt(15)+5)*(4-sqrt(15))^(2*n-1)+10).

a(n) = ceiling((1/60)*(3*sqrt(15)-5)*(4+sqrt(15))^(2*n-1)).

G.f.: x*(1-9*x-2*x^2)/((1-x)*(1-62*x+x^2)).

(End)

Mathematica

LinearRecurrence[{63, -63, 1}, {1, 54, 3337}, 14] (* Ant King, Dec 15 2011 *)
CoefficientList[Series[x (1-9x-2x^2)/((1-x)(1-62x+x^2)), {x, 0, 20}], x] (* Harvey P. Dale, Jul 09 2021 *)

Prog

(PARI) Vec(x*(2*x^2+9*x-1)/((x-1)*(x^2-62*x+1)) + O(x^30)) \\ Colin Barker, Jun 23 2015

Crossrefs

Cf. A046198, A048900.

Sequence in context: A187304 A309422 A212705 * A245831 A368366 A291071

Adjacent sequences: A046196 A046197 A046198 * A046200 A046201 A046202

Keyword

nonn,easy

Author

Eric W. Weisstein

Status

approved