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A046194 Heptagonal triangular numbers. 4
1, 55, 121771, 5720653, 12625478965, 593128762435, 1309034909945503, 61496776341083161, 135723357520344181225, 6376108764003055554511, 14072069153115290487843091, 661087708807868029661744485, 1459020273797576190840203197981, 68542895818241264287385936157403 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

From Ant King, Oct 18 2011: (Start)

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

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

lim(n->Infinity, 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).

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)

MATHEMATICA

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

PROG

(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

CROSSREFS

Cf. A039835, A046193.

Sequence in context: A196428 A231907 A027580 * A172808 A243315 A172856

Adjacent sequences:  A046191 A046192 A046193 * A046195 A046196 A046197

KEYWORD

nonn,easy,changed

AUTHOR

Eric W. Weisstein

STATUS

approved

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Last modified September 4 23:40 EDT 2015. Contains 261339 sequences.