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A046192
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Octagonal hexagonal numbers.
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3
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1, 11781, 113123361, 1086210502741, 10429793134197921, 100146872588357936901, 961610260163619775927681, 9233381617944204500099658261, 88658929333889991446337142696641, 851303030230630079923524744073490821, 8174211607615580693535693146256516168801, 78488779005021775588699645666830324179338581
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OFFSET
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1,2
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COMMENTS
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As n increases, this sequence is approximately geometric with common ratio r = lim_{n->infinity} a(n)/a(n-1) = (sqrt(3) + sqrt(2))^8 = 4801 + 1960*sqrt(6). - Ant King, Dec 27 2011
Intersection of A000384 and A000567. - Michel Marcus, Jun 20 2015
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LINKS
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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 (9603,-9603,1).
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FORMULA
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From Ant King, Dec 27 2011: (Start)
G.f.: x*(1 + 2178*x + 21*x^2)/((1-x)*(1 - 9602*x + x^2)).
a(n) = 9603*a(n-1) - 9603*a(n-2) + a(n-3).
a(n) = 9602*a(n-1) - a(n-2) + 2200.
a(n) = 1/96*((3*sqrt(3) - sqrt(2))*(sqrt(3) + sqrt(2))^(8n-5)+ (3*sqrt(3) + sqrt(2))*(sqrt(3) - sqrt(2))^(8n-5) - 22).
a(n) = floor(1/96*(3*sqrt(3) - sqrt(2))*(sqrt(3) + sqrt(2))^(8n-5)).
(End)
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MAPLE
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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, ", (y^2-1)/8) 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, ", (y^2-1)/8) fi: od: # Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 19 2008
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MATHEMATICA
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LinearRecurrence[{9603, -9603, 1}, {1, 11781, 113123361}, 9] (* Ant King, Dec 27 2011 *)
CoefficientList[Series[(1 + 2178 x + 21 x^2) / ((1 - x) (1 - 9602 x + x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Aug 10 2017 *)
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PROG
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(MAGMA) I:=[1, 11781, 113123361]; [n le 3 select I[n] else 9603*Self(n-1)-9603*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Aug 10 2017
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CROSSREFS
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Cf. A046190, A046191.
Sequence in context: A235316 A336658 A321158 * A210151 A278193 A031868
Adjacent sequences: A046189 A046190 A046191 * A046193 A046194 A046195
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KEYWORD
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nonn,easy
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AUTHOR
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Eric W. Weisstein
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EXTENSIONS
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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
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STATUS
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approved
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