A039835 - OEIS

%I #39 Mar 09 2026 23:56:16

%S 1,10,493,3382,158905,1089154,51167077,350704366,16475640049,

%T 112925716858,5305104928861,36361730124070,1708227311453353,

%U 11708364174233842,550043889183050965,3770056902373173214,177112424089630957537,1213946614199987541226,57029650512971985276109

%N Indices of triangular numbers which are also heptagonal.

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

%H Colin Barker, <a href="/A039835/b039835.txt">Table of n, a(n) for n = 1..798</a>

%H Eric Weisstein's World of Mathematics, <a href="https://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 (1,322,-322,-1,1).

%F G.f.: (-2x^4-9x^3+161x^2+9x+1)/((1-x)*(1-18x+x^2)*(1+18x+x^2)).

%F a(n+2) = 322*a(n+1)-a(n)+160 a(n+1) = 161*a(n)+80+36*(20*a(n)^2+20*a(n)+9)^0.5 - _Richard Choulet_, Sep 29 2007

%F From _Ant King_, Oct 19 2011: (Start)

%F a(n) = a(n-1)+322a(n-2)-322a(n-3)-a(n-4)+a(n-5).

%F a(n) = 1/20*sqrt(5)*(( sqrt(5)-(-1)^n)*(2+ sqrt(5))^(2n-1)+( sqrt(5)+(-1)^n)*(2- sqrt(5))^(2n-1)-2* sqrt(5)).

%F a(n) = floor(1/20* sqrt(5)*(sqrt(5)-(-1)^n)*(2+ sqrt(5))^(2n-1)).

%F Limit_{n->oo} a(2*n+1)/a(2*n) = (1/2)*(47+21*sqrt(5)).

%F Limit_{n->oo} a(2*n)/a(2*n-1)= (1/2)*(7+3*sqrt(5)). (End)

%F From _Hans J. H. Tuenter_, Mar 03 2026: (Start)

%F Let F(n) and L(n) be the Fibonacci and Lucas numbers, and phi=(1+sqrt(5))/2, the golden ratio.

%F a(n) = (F(6*n-4+2*(n mod 2))-1)/2 = F(3*n-1+(n mod 2))*L(3*n-3+(n mod 2))/2.

%F a(2*n+1)/a(2*n) ~ phi^8 = 46.9787... and a(2*n)/a(2*n-1) ~ phi^4 = 6.8541..., so that the limit results by King follow, since phi^n = (L(n)+F(n)*sqrt(5))/2. (End)

%t LinearRecurrence[{1,322,-322,-1,1},{1,10,493,3382,158905},16] (* _Ant King_, Oct 19 2011 *)

%o (PARI) Vec((-2*x^4-9*x^3+161*x^2+9*x+1)/((1-x)*(1-18*x+x^2)*(1+18*x+x^2))+O(x^99))

%Y Cf. A046193, A046194, A000032, A000045.

%K nonn,easy

%O 1,2

%A _Eric W. Weisstein_