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A036428
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Square octagonal numbers.
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6
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1, 225, 43681, 8473921, 1643897025, 318907548961, 61866420601441, 12001766689130625, 2328280871270739841, 451674487259834398561, 87622522247536602581025, 16998317641534841066320321, 3297585999935511630263561281, 639714685669847721430064568225
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OFFSET
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1,2
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COMMENTS
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Also, numbers simultaneously octagonal and centered octagonal. - Steven Schlicker, Apr 24 2007
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LINKS
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FORMULA
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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)).
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)
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MAPLE
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option remember;
if n < 4 then
op(n, [1, 225, 43681]) ;
else
195*(procname(n-1)-procname(n-2))+procname(n-3) ;
end if;
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MATHEMATICA
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LinearRecurrence[{195, -195, 1}, {1, 225, 43681}, 12] (* Ant King, Nov 15 2011 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Jean-Francois Chariot (jean-francois.chariot(AT)afoc.alcatel.fr)
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EXTENSIONS
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STATUS
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approved
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