A036428 Square octagonal numbers.

1, 225, 43681, 8473921, 1643897025, 318907548961, 61866420601441, 12001766689130625, 2328280871270739841, 451674487259834398561, 87622522247536602581025, 16998317641534841066320321, 3297585999935511630263561281, 639714685669847721430064568225

Offset: 1

Author

Jean-Francois Chariot (jean-francois.chariot(AT)afoc.alcatel.fr)

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000567, A006051, A006060, A016754, A028230, A046184.

Programs

Magma

[Floor(1/12*(2+Sqrt(3))^(4*n-2)): n in [1..20]]; // Vincenzo Librandi, Dec 04 2015

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 *)

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

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)).

From Ant 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

Extensions

More terms from Eric W. Weisstein

Edited by N. J. A. Sloane, Oct 02 2007