A046183 - OEIS

%I #39 Oct 09 2025 16:49:56

%S 1,21,11781,203841,113123361,1957283461,1086210502741,18793835590881,

%T 10429793134197921,180458407386358101,100146872588357936901,

%U 1732761608929974897121,961610260163619775927681,16637976788487211575799941,9233381617944204500099658261,159757851390292596620856138561

%N Octagonal triangular numbers.

%C From _Ant King_, Oct 31 2011: (Start)

%C Limit_{n->oo} a(2*n+1)/a(2*n) = (6937 + 2832*sqrt(6))/25.

%C Limit_{n->oo} a(2*n)/a(2*n-1) = (217 + 88*sqrt(6))/25. (End)

%C Intersection of A000217 and A000567. - _Michel Marcus_, Feb 07 2015

%D Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 39.

%H Colin Barker, <a href="/A046183/b046183.txt">Table of n, a(n) for n = 1..500</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OctagonalTriangularNumber.html">Octagonal Triangular Number.</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,9602,-9602,-1,1).

%F a(n+1) = 4801*a(n) + 1100 + 980*sqrt(24*a(n)^2+11*a(n)+1).

%F G.f.: -z*(z^4+20*z^3+2158*z^2+20*z+1) / ((z-1)*(z^2-98*z+1)*(z^2+98*z+1)). - _Richard Choulet_, Oct 03 2007, factored by _Colin Barker_, Feb 07 2015

%F From _Ant King_, Oct 31 2011: (Start)

%F a(n) = a(n-1) + 9602*a(n-2) - 9602*a(n-3) - a(n-4) + a(n-5).

%F a(n) = 9602*a(n-2) - a(n-4) + 2200.

%F a(n) = 1/96*((7-2*sqrt(6)*(-1)^n)*(sqrt(3)+sqrt(2))^(4*n-2)+(7+2*sqrt(6)*(-1)^n)*(sqrt(3)-sqrt(2))^(4*n-2)-22).

%F a(n) = floor(1/96*(7-2*sqrt(6)*(-1)^n)*(sqrt(3)+sqrt(2))^(4n-2)).

%F (End)

%t LinearRecurrence[{1,9602,-9602,-1,1}, {1,21,11781,203841,113123361}, 13] (* _Ant King_, Oct 31 2011 *)

%o (PARI) Vec(-z*(z^4+20*z^3+2158*z^2+20*z+1)/((z-1)*(z^2-98*z+1)*(z^2+98*z+1)) + O(z^36)) \\ _Joerg Arndt_, Feb 07 2015, factored by _Colin Barker_, Feb 07 2015

%Y Cf. A046181, A046182.

%K nonn,easy

%O 1,2

%A _Eric W. Weisstein_

%E More terms from _Richard Choulet_, Oct 03 2007