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A046192
Octagonal hexagonal numbers.
3
1, 11781, 113123361, 1086210502741, 10429793134197921, 100146872588357936901, 961610260163619775927681, 9233381617944204500099658261, 88658929333889991446337142696641, 851303030230630079923524744073490821, 8174211607615580693535693146256516168801, 78488779005021775588699645666830324179338581
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))^8 = 4801 + 1960*sqrt(6). - Ant King, Dec 27 2011
Intersection of A000384 and A000567. - Michel Marcus, Jun 20 2015
LINKS
Eric Weisstein's World of Mathematics, Octagonal Hexagonal Number.
FORMULA
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)
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, ", (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
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Sequence in context: A235316 A336658 A321158 * A210151 A278193 A031868
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