OFFSET
1,2
COMMENTS
Also, numbers simultaneously octagonal and centered octagonal. -Steven Schlicker,Apr 24 2007
LINKS
Colin Barker,Table of n, a(n) for n = 1..437
C. Gill, solution to question no. 8,Mathematical Miscellany,1 (1836), pp. 220-225, at p. 223.
S. C. Schlicker,Numbers Simultaneously Polygonal and Centered Polygonal,Mathematics Magazine, Vol. 84, No. 5, December 2011, pp. 339-350.
Eric Weisstein's World of Mathematics,Octagonal Square Number.
Index entries for linear recurrences with constant coefficients,signature (195,-195,1).
FORMULA
Let x(n) + y(n)*sqrt(48) = (8+sqrt(48))*(7+sqrt(48))^n, s(n) = (y(n)+1)/2; then a(n) = (1/2)*(2+8*(s(n)^2-s(n))). -Steven Schlicker,Apr 24 2007
a(n+2) = 194*a(n+1) - a(n) + 32 and also a(n+1) = 97*a(n) + 56*sqrt(3*a(n)^2 + a(n)). -Richard Choulet,Sep 26 2007
G.f.: x*(x^2+30x+1)/((1-x)*(1-194x+x^2)).
FromAnt King,Nov 15 2011: (Start)
lim_{n->oo} a(n)/a(n-1) = (2 + sqrt(3))^4 = 97 + 56*sqrt(3).
a(n) = (1/12) * ((2 + sqrt(3))^(4n-2) + (2 - sqrt(3))^(4n-2) - 2).
a(n) = floor((1/12) * (2 + sqrt(3))^(4n-2)).
a(n) = (1/12) * ((tan(5*Pi/12))^(4n-2) + (tan(Pi/12))^(4n-2) - 2).
a(n) = floor((1/12) * tan(5*Pi/12)^(4n-2)).
(End)
a(n) =A028230(n)^2. -Bernard Schott,Dec 23 2022
MAPLE
A036428:= proc(n)
option remember;
if n < 4 then
op(n, [1, 225, 43681]);
else
195*(procname(n-1)-procname(n-2))+procname(n-3);
end if;
end proc: #R. J. Mathar,Nov 11 2011
MATHEMATICA
LinearRecurrence[{195, -195, 1}, {1, 225, 43681}, 12] (*Ant King,Nov 15 2011 *)
PROG
(PARI) Vec(-x*(x^2+30*x+1)/((x-1)*(x^2-194*x+1)) + O(x^20)) \\Colin Barker,Jun 24 2015
(PARI) vector(15, n, floor((2+sqrt(3))^(4*n-2)/12)) \\Altug Alkan,Oct 19 2015
(Magma) [Floor(1/12*(2+Sqrt(3))^(4*n-2)): n in [1..20]]; //Vincenzo Librandi,Dec 04 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jean-Francois Chariot (jean-francois.chariot(AT)afoc.alcatel.fr)
EXTENSIONS
More terms fromEric W. Weisstein
Edited byN. J. A. Sloane,Oct 02 2007
STATUS
approved