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A046191
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Indices of hexagonal numbers which are also octagonal.
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3
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1, 77, 7521, 736957, 72214241, 7076258637, 693401132161, 67946234693117, 6658037598793281, 652419738447048397, 63930476330211949601, 6264534260622324012477, 613860427064657541273121
(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+22*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)-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)
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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+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
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MATHEMATICA
| LinearRecurrence[{99, -99, 1}, {1, 77, 7521}, 13] (* Ant King, Dec 27 2011 *)
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CROSSREFS
| Cf. A046190, A046192.
Sequence in context: A180837 A107787 A099638 * A115616 A120799 A196543
Adjacent sequences: A046188 A046189 A046190 * A046192 A046193 A046194
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KEYWORD
| nonn,easy
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com)
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EXTENSIONS
| 1 more term 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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