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A036428 Square octagonal numbers. 4
1, 225, 43681, 8473921, 1643897025, 318907548961, 61866420601441, 12001766689130625, 2328280871270739841, 451674487259834398561, 87622522247536602581025, 16998317641534841066320321, 3297585999935511630263561281, 639714685669847721430064568225 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also, numbers simultaneously octagonal and centered octagonal. - Steven Schlicker (schlicks(AT)gvsu.edu), Apr 24 2007

As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = ( 2 + sqrt(3))^4 = 97 + 56 * sqrt(3). - Ant King, Nov 15 2011

Each m-th octagonal number (or m-th term of A000567) is also a square number, when q=sqrt(m), q is integer and q=A079935(n) or n-th term of A079935. - Sergey Pavlov, Oct 18 2015

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 (schlicks(AT)gvsu.edu), 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)).

a(n) = -(1/6)+(7/12)*{[97-56*sqrt(3)]^n+[97+56*sqrt(3)]^n}-(1/3)*sqrt(3)*{[97-56*sqrt(3)]^n -[97+56*sqrt(3)]^n}, with n>=0. - Paolo P. Lava, Nov 25 2008

From Ant King, Nov 15 2011: (Start)

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)

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

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

Sequence in context: A192934 A264194 A061051 * A183822 A260864 A265420

Adjacent sequences:  A036425 A036426 A036427 * A036429 A036430 A036431

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

More terms from Eric W. Weisstein

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

STATUS

approved

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Last modified December 7 04:57 EST 2016. Contains 278841 sequences.