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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 AUTHOR STATUS approved

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