A046194 Heptagonal triangular numbers.

1, 55, 121771, 5720653, 12625478965, 593128762435, 1309034909945503, 61496776341083161, 135723357520344181225, 6376108764003055554511, 14072069153115290487843091, 661087708807868029661744485, 1459020273797576190840203197981, 68542895818241264287385936157403

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• easy

Comments

From Ant King, Oct 18 2011: (Start)

lim(n->oo, u(2n+1)/u(2n)) = 1/2(2207+987*sqrt(5)),

lim(n->oo, u(2n)/u(2n-1)) = 1/2(47+21*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->oo, u(2n+1)/u(2n)) = 1/2(L_16+F_16*sqrt(5)),

lim(n->oo, u(2n)/u(2n-1)) = 1/2(L_8+F_8*sqrt(5)),

a(n) = L_1*a(n-1) + L_24*a(n-2) - L_24*a(n-3)- L_1*a(n-4) + L_1*a(n-5). (End)

Links

• Colin Barker, Table of n, a(n) for n = 1..399
• J. C. Su, On some properties of two simultaneous polygonal sequences, JIS 10 (2007) 07.10.4, example 4.4
• Eric Weisstein's World of Mathematics, Heptagonal Triangular Number
• Index entries for linear recurrences with constant coefficients, signature (1,103682,-103682,-1,1).

Crossrefs

Cf. A039835, A046193.

Programs

Mathematica

LinearRecurrence[{1,103682,-103682,-1,1},{1,55,121771,5720653,12625478965},12] (* Ant King, Oct 18 2011 *)

PARI

a(n)=((3-sqrt(5)*(-1)^n)*(2+sqrt(5))^(4*n-2)+(3+sqrt(5)*(-1)^n)*(2-sqrt(5))^(4*n-2)-14)\/80 \\ Charles R Greathouse IV, Oct 18 2011

PARI

Vec(-x*(x^4+54*x^3+18034*x^2+54*x+1)/((x-1)*(x^2-322*x+1)*(x^2+322*x+1)) + O(x^20)) \\ Colin Barker, Jun 23 2015

Formula

The two bisections satisfy the same recurrence relation: a(n+2)=103682*a(n+1)-a(n)+18144 or a(n+1)=51841*a(n)+9072+2898*(320*a(n)^2+112*a(n)+9)^0.5. The g.f. satisfies f(z)=(z+55*z^2+18088*z^3+18088*z^4+55*z^5+z^6)/((1-z^2)*(1-103682*z^2+z^4))=1*z+55*z^2+121771*z^3+... - Richard Choulet, Sep 20 2007

From Ant King, Oct 18 2011: (Start)

a(n) = a(n-1)+103682a(n-2)-103682a(n-3)-a(n-4)+a(n-5).

a(n) = 1/80*((3-sqrt(5)*(-1)^n)*(2+sqrt(5))^(4n-2)+(3+sqrt(5)*(-1)^n)*(2-sqrt(5))^(4n-2)-14).

a(n) = floor(1/80*(3-sqrt(5)*(-1)^n)*(2+sqrt(5))^(4n-2)).

G.f.: x(1+54*x+18034*x^2+54*x^3+x^4)/((1-x)(1-322*x+x^2)(1+322*x+x^2)). (End)