A046191 Indices of hexagonal numbers which are also octagonal.

1, 77, 7521, 736957, 72214241, 7076258637, 693401132161, 67946234693117, 6658037598793281, 652419738447048397, 63930476330211949601, 6264534260622324012477, 613860427064657541273121

Offset

1,2

Comments

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

Links

Herman Jamke, Table of n, a(n) for n = 1..20

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

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

Formula

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)

Maple

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

Mathematica

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

Crossrefs

Cf. A046190, A046192.

Sequence in context: A210811 A107787 A099638 * A115616 A226256 A120799

Adjacent sequences: A046188 A046189 A046190 * A046192 A046193 A046194

Keyword

nonn,easy

Author

Eric W. Weisstein

Extensions

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

Status

approved