A046193 Indices of heptagonal numbers (A000566) which are also triangular numbers (A000217).

1, 5, 221, 1513, 71065, 487085, 22882613, 156839761, 7368130225, 50501915861, 2372515049741, 16261460067385, 763942477886281, 5236139639782013, 245987105364332645, 1686020702549740705, 79207083984837225313, 542893430081376724901, 25504435056012222218045

Offset

1,2

Comments

From Ant King, Oct 19 2011: (Start)

lim_{n->infinity} a(2n+1)/a(2n) = (1/2)*(47+21*sqrt(5)).

lim_{n->infinity} a(2n)/a(2n-1) = (1/2)*(7+3*sqrt(5)).

(End)

From Raphie Frank, Nov 30 2012: (Start)

Where L_n is a Lucas number and F_n is Fibonacci number:

lim_{n->infinity} a(2n+1)/a(2n) = (1/2)*(L_8 + F_8*sqrt(5)),

lim_{n->infinity} a(2n)/a(2n-1) = (1/2)*(L_4 + F_4*sqrt(5)),

a(n) = L_1*a(n-1) + L_12*a(n-2) - L_12*a(n-3)- L_1*a(n-4) + L_1*a(n-5).

(End)

Values of n such that 2*n-1 and 10*n-1 are both perfect squares. - Colin Barker, Dec 03 2016

Links

Colin Barker, Table of n, a(n) for n = 1..798

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

Index entries for linear recurrences with constant coefficients, signature (1,322,-322,-1,1).

Formula

For n odd, a(n+2) = 322*a(n+1) - a(n) - 96; for n even, a(n+1) = 161*a(n) - 48 + 36*sqrt(20*a(n)^2 - 12*a(n)+1). - Richard Choulet, Sep 29 2007, Oct 09 2007

From Ant King, Oct 19 2011: (Start)

a(n) = 322*a(n-2) - a(n-4) - 96.

a(n) = a(n-1) + 322*a(n-2) - 322*a(n-3) - a(n-4) + a(n-5).

a(n) = (1/20)*((sqrt(5)-(-1)^n)*(sqrt(5)+2)^(2n-1) + (sqrt(5)+(-1)^n)*(sqrt(5)-2)^(2n-1)+6).

a(n) = ceiling((1/20)*(sqrt(5)-(-1)^n)*(2+sqrt(5))^(2n-1)).

G.f.: x*(1 + 4*x - 106*x^2 + 4*x^3 + x^4)/((1-x)*(1-18*x+x^2)*(1+18*x+x^2)).

(End)

Mathematica

LinearRecurrence[{1, 322, -322, -1, 1}, {1, 5, 221, 1513, 71065}, 17] (* Ant King, Oct 19 2011 *)
Select[Range@240000000, IntegerQ@Sqrt[2 # - 1] && IntegerQ@Sqrt[10 # - 1] &] (* Vincenzo Librandi, Dec 04 2016 *)

Prog

(PARI) Vec(-x*(x^4+4*x^3-106*x^2+4*x+1)/((x-1)*(x^2-18*x+1)*(x^2+18*x+1)) + O(x^50)) \\ Colin Barker, Jun 23 2015
(Magma) [n: n in [1..2*10^8] |IsSquare(2*n-1) and IsSquare(10*n-1)]; // Vincenzo Librandi, Dec 04 2016

Crossrefs

Cf. A039835, A046194.

Sequence in context: A290179 A050617 A066462 * A195633 A112999 A337965

Adjacent sequences: A046190 A046191 A046192 * A046194 A046195 A046196

Keyword

nonn,easy

Author

Eric W. Weisstein

Status

approved