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%I #28 Aug 28 2015 11:55:08
%S 1,55,121771,5720653,12625478965,593128762435,1309034909945503,
%T 61496776341083161,135723357520344181225,6376108764003055554511,
%U 14072069153115290487843091,661087708807868029661744485,1459020273797576190840203197981,68542895818241264287385936157403
%N Heptagonal triangular numbers.
%C From _Ant King_, Oct 18 2011: (Start)
%C lim(n->Infinity, u(2n+1)/u(2n)) = 1/2(2207+987*sqrt(5)),
%C lim(n->Infinity, u(2n)/u(2n-1)) = 1/2(47+21*sqrt(5)).
%C (End)
%C From _Raphie Frank_, Nov 30 2012: (Start)
%C Where L_n is a Lucas number and F_n is Fibonacci number:
%C lim(n->Infinity, u(2n+1)/u(2n)) = 1/2(L_16+F_16*sqrt(5)),
%C lim(n->Infinity, u(2n)/u(2n-1)) = 1/2(L_8+F_8*sqrt(5)),
%C 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).
%C (End)
%H Colin Barker, <a href="/A046194/b046194.txt">Table of n, a(n) for n = 1..399</a>
%H J. C. Su, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Su/su.html">On some properties of two simultaneous polygonal sequences</a>, JIS 10 (2007) 07.10.4, example 4.4
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeptagonalTriangularNumber.html">Heptagonal Triangular Number</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,103682,-103682,-1,1).
%F 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
%F From _Ant King_, Oct 18 2011: (Start)
%F a(n) = a(n-1)+103682a(n-2)-103682a(n-3)-a(n-4)+a(n-5).
%F 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).
%F a(n) = floor(1/80*(3-sqrt(5)*(-1)^n)*(2+sqrt(5))^(4n-2)).
%F 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)).
%F (End)
%t LinearRecurrence[{1,103682,-103682,-1,1},{1,55,121771,5720653,12625478965},12] (* _Ant King_, Oct 18 2011 *)
%o (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
%o (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
%Y Cf. A039835, A046193.
%K nonn,easy
%O 1,2
%A _Eric W. Weisstein_
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