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A046190
Indices of octagonal numbers which are also hexagonal numbers.
4
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
Eric Weisstein's World of Mathematics, Octagonal Hexagonal number.
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
Sequence in context: A364745 A069407 A342586 * A296782 A292782 A251011
KEYWORD
nonn,easy
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