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A046192 Octagonal hexagonal numbers. 3
1, 11781, 113123361, 1086210502741, 10429793134197921, 100146872588357936901, 961610260163619775927681, 9233381617944204500099658261, 88658929333889991446337142696641, 851303030230630079923524744073490821, 8174211607615580693535693146256516168801, 78488779005021775588699645666830324179338581 (list; graph; refs; listen; history; text; internal format)
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

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).

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

Cf. A046190, A046191.

Sequence in context: A229411 A235316 A321158 * A210151 A278193 A031868

Adjacent sequences:  A046189 A046190 A046191 * A046193 A046194 A046195

KEYWORD

nonn,easy

AUTHOR

Eric W. Weisstein

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

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Last modified December 9 19:48 EST 2018. Contains 318023 sequences. (Running on oeis4.)