1,2

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 37.

Colin Barker, Table of n, a(n) for n = 1..399

Joseph 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).

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

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

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

Limit_{n->oo} a(2*n+1)/a(2*n) = (1/2)*(2207+987*sqrt(5)).

Limit_{n->oo} a(2*n)/a(2*n-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:

Limit_{n->oo} a(2*n+1)/a(2*n) = (1/2)*(L(16)+F(16)*sqrt(5)),

Limit_{n->oo} a(2*n)/a(2*n-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)

From Hans J. H. Tuenter, Mar 02 2026: (Start)

The formula expressing a(n) in terms of the Lucas or Fibonacci numbers gives the asymptotics a(2*n+1)/a(2*n) ~ phi^16 = 2206.9995... and a(2*n)/a(2*n-1) ~ phi^8 = 46.9787..., where phi=(1+sqrt(5))/2, the golden ratio. Since phi^n = (L(n)+F(n)*sqrt(5))/2, the previous limit results follow.

a(n) = (L(12*n-8+4*(n mod 2))-7)/40 = 1+F(6*n+2*(n mod 2))*F(6*n-8+2*(n mod 2))/8.

sqrt(8*a(n)+1) = F(6*n-4+2*(n mod 2)). (End)

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

Intersection of A000217 and A000566.

Cf. A039835, A046193.

Sequence in context: A027580 A265980 A287053 * A172808 A243315 A172856

Adjacent sequences: A046191 A046192 A046193 * A046195 A046196 A046197

nonn,easy

approved