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A036428 Square octagonal numbers.

%I A036428 #74 Feb 16 2025 08:32:37
%S A036428 1,225,43681,8473921,1643897025,318907548961,61866420601441,
%T A036428 12001766689130625,2328280871270739841,451674487259834398561,
%U A036428 87622522247536602581025,16998317641534841066320321,3297585999935511630263561281,639714685669847721430064568225
%N A036428 Square octagonal numbers.
%C A036428 Also, numbers simultaneously octagonal and centered octagonal. - _Steven Schlicker_, Apr 24 2007
%H A036428 Colin Barker, <a href="/A036428/b036428.txt">Table of n, a(n) for n = 1..437</a>
%H A036428 C. Gill, solution to question no. 8, <a href="https://archive.org/details/mathematicalmis00unkngoog">Mathematical Miscellany</a>, 1 (1836), pp. 220-225, at p. 223.
%H A036428 S. C. Schlicker, <a href="http://www.jstor.org/stable/10.4169/math.mag.84.5.339">Numbers Simultaneously Polygonal and Centered Polygonal</a>, Mathematics Magazine, Vol. 84, No. 5, December 2011, pp. 339-350.
%H A036428 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OctagonalSquareNumber.html">Octagonal Square Number.</a>
%H A036428 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (195,-195,1).
%F A036428 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
%F A036428 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
%F A036428 G.f.: x*(x^2+30x+1)/((1-x)*(1-194x+x^2)).
%F A036428 From _Ant King_, Nov 15 2011: (Start)
%F A036428 lim_{n->oo} a(n)/a(n-1) = (2 + sqrt(3))^4 = 97 + 56*sqrt(3).
%F A036428 a(n) = (1/12) * ((2 + sqrt(3))^(4n-2) + (2 - sqrt(3))^(4n-2) - 2).
%F A036428 a(n) = floor((1/12) * (2 + sqrt(3))^(4n-2)).
%F A036428 a(n) = (1/12) * ((tan(5*Pi/12))^(4n-2) + (tan(Pi/12))^(4n-2) - 2).
%F A036428 a(n) = floor((1/12) * tan(5*Pi/12)^(4n-2)).
%F A036428 (End)
%F A036428 a(n) = A028230(n)^2. - _Bernard Schott_, Dec 23 2022
%p A036428 A036428 := proc(n)
%p A036428         option remember;
%p A036428         if n < 4 then
%p A036428                 op(n,[1,225,43681]) ;
%p A036428         else
%p A036428                 195*(procname(n-1)-procname(n-2))+procname(n-3) ;
%p A036428         end if;
%p A036428 end proc: # _R. J. Mathar_, Nov 11 2011
%t A036428 LinearRecurrence[{195,-195,1}, {1,225,43681}, 12] (* _Ant King_, Nov 15 2011 *)
%o A036428 (PARI) Vec(-x*(x^2+30*x+1)/((x-1)*(x^2-194*x+1)) + O(x^20)) \\ _Colin Barker_, Jun 24 2015
%o A036428 (PARI) vector(15, n, floor((2+sqrt(3))^(4*n-2)/12)) \\ _Altug Alkan_, Oct 19 2015
%o A036428 (Magma) [Floor(1/12*(2+Sqrt(3))^(4*n-2)): n in [1..20]]; // _Vincenzo Librandi_, Dec 04 2015
%Y A036428 Cf. A000567, A006051, A006060, A016754, A028230, A046184.
%K A036428 nonn,easy
%O A036428 1,2
%A A036428 Jean-Francois Chariot (jean-francois.chariot(AT)afoc.alcatel.fr)
%E A036428 More terms from _Eric W. Weisstein_
%E A036428 Edited by _N. J. A. Sloane_, Oct 02 2007