A046190 Indices of octagonal numbers which are also hexagonal numbers.

1, 63, 6141, 601723, 58962681, 5777740983, 566159653621, 55477868313843, 5436264935102961, 532698485771776303, 52199015340698974701, 5114970804902727744363, 501214939865126619972841, 49113949135977506029594023, 4812665800385930464280241381, 471592134488685207993434061283

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 - 36*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) - 32.

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

a(n) = ceiling((1/24)*sqrt(2)*(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+x/2)/3) 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+x/2)/3) fi: od: # Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 19 2008

Mathematica

LinearRecurrence[{99, -99, 1}, {1, 63, 6141}, 13] (* Ant King, Dec 27 2011 *)
CoefficientList[Series[(1 - 36 x + 3 x^2) / ((1 - x) (x^2 -98 x + 1)), {x, 0, 33}], x] (* Vincenzo Librandi, Aug 10 2017 *)

Prog

(Magma) I:=[1, 63, 6141]; [n le 3 select I[n] else 99*Self(n-1)-99*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Aug 10 2017

Crossrefs

Cf. A046191, A046192.

Sequence in context: A364745 A069407 A342586 * A296782 A292782 A251011

Adjacent sequences: A046187 A046188 A046189 * A046191 A046192 A046193

Keyword

nonn,easy

Author

Eric W. Weisstein

Extensions

More terms 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