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A036428 Square octagonal numbers. 4
1, 225, 43681, 8473921, 1643897025, 318907548961, 61866420601441, 12001766689130625, 2328280871270739841, 451674487259834398561, 87622522247536602581025, 16998317641534841066320321 (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

REFERENCES

S. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, Math. Mag. 84 (5) (2011) 339.

LINKS

Table of n, a(n) for n=1..12.

C. Gill, solution to question no. 8, Mathematical Miscellany, 1 (1836), pp. 220-225, at p. 223.

Eric Weisstein's World of Mathematics, Octagonal Square Number.

Index to sequences with 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*(3*a(n)^2+a(n))^0.5. - 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 [From 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 *)

CROSSREFS

Cf. A000567, A016754.

Cf. A006060, A006051, A028230, A046184.

Sequence in context: A051364 A192934 A061051 * A183822 A171109 A239478

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 October 24 03:37 EDT 2014. Contains 248491 sequences.