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A046190
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Indices of octagonal numbers which are also hexagonal numbers.
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4
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1, 63, 6141, 601723, 58962681, 5777740983, 566159653621, 55477868313843, 5436264935102961, 532698485771776303, 52199015340698974701, 5114970804902727744363, 501214939865126619972841
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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
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LINKS
| Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 19 2008, Table of n, a(n) for n = 1..20
Eric Weisstein's World of Mathematics, Octagonal Hexagonal number.
Index to sequences with linear recurrences with constant coefficients, signature (99,-99,1).
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FORMULA
| G.f. -x*(1-36*x+3*x^2) / ( (x-1)*(x^2-98*x+1) ). - R. J. Mathar, Dec 21 2011
Contribution 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)
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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
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MATHEMATICA
| LinearRecurrence[{99, -99, 1}, {1, 63, 6141}, 13] (* Ant King, Dec 27 2011 *)
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CROSSREFS
| Cf. A046191, A046192.
Sequence in context: A194484 A195620 A069407 * A093263 A069433 A178634
Adjacent sequences: A046187 A046188 A046189 * A046191 A046192 A046193
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KEYWORD
| nonn,easy
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com)
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001
One more term from Lior Manor (lior.manor(AT)gmail.com) Feb 13 2002
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 19 2008
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