1,2

As n increases, this sequence is approximately geometric with common ratio r = lim_{n->oo} a(n)/a(n-1) = (sqrt(3) + sqrt(2))^4 = 49 + 20*sqrt(6). - Ant King, Dec 27 2011

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 41.

Harvey P. Dale, Table of n, a(n) for n = 1..503

Eric Weisstein's World of Mathematics, Octagonal Hexagonal Number.

Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

G.f.: x*(-1+22*x+3*x^2) / ( (x-1)*(x^2-98*x+1) ). - R. J. Mathar, Dec 21 2011

From Ant King, Dec 27 2011: (Start)

a(n) = 98*a(n-1) - a(n-2) - 24.

a(n) = (1/24)*sqrt(3)*((1+sqrt(6))*(sqrt(3) + sqrt(2))^(4n-3) + (1-sqrt(6))*(sqrt(3) - sqrt(2))^(4n-3) + 2*sqrt(3)).

a(n) = ceiling((1/24)*sqrt(3)*(1+sqrt(6))*(sqrt(3) + sqrt(2))^(4n-3)). (End)

E.g.f.: exp(x)*(3 - exp(48*x)*(39*cosh(20*sqrt(6)*x) - 16*sqrt(6)*sinh(20*sqrt(6)*x)))/12 + 3. - Stefano Spezia, Oct 10 2025

a:=5+2*sqrt(6): b:=5-2*sqrt(6): s:=n->a^n+b^n: d:=n->sqrt(6)*(a^n-b^n):for n from 0 to 40 do x:=simplify(s(n)-1/4*d(n)): y:=simplify(1/3*d(n)-s(n)/2): if(type((1+x/2)/3, integer) and type((1+y)/4, integer)) then printf("%d, ", (1+y)/4) fi: x:=simplify(s(n+1)+1/4*d(n+1)): y:=simplify(1/3*d(n+1)+s(n+1)/2): if(type((1+x/2)/3, integer) and type((1+y)/4, integer)) then printf("%d, ", (1+y)/4) fi: od: # Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 19 2008

LinearRecurrence[{99, -99, 1}, {1, 77, 7521}, 13] (* Ant King, Dec 27 2011 *)

nxt[{a_, b_}]:={b, 98b-a-24}; NestList[nxt, {1, 77}, 19][[;; , 1]] (* Harvey P. Dale, Dec 20 2025 *)

Cf. A046190, A046192.

Sequence in context: A210811 A107787 A099638 * A115616 A226256 A120799

Adjacent sequences: A046188 A046189 A046190 * A046192 A046193 A046194

nonn,easy

1 more term from Larry Reeves (larryr(AT)acm.org), May 07 2001

One more term from Lior Manor, Feb 13 2002

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 19 2008

approved