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%I #57 Jan 31 2024 08:04:54
%S 2,-1,2,1,1,3,3,5,4,5,8,11,9,13,19,27,22,31,46,65,53,75,111,157,128,
%T 181,268,379,309,437,647,915,746,1055,1562,2209,1801,2547,3771,5333,
%U 4348,6149,9104,12875,10497,14845,21979,31083,25342,35839,53062,75041,61181,86523
%N a(n) = 2*a(n-4) + a(n-8) for n >= 8.
%C Previous name was: a(2n) = a(2n - 4) + a(2n - 3) and a(2n + 1) = 2*a(2n - 4) + a(2n - 3), with a(0) = 2, a(1) = -1, a(2) = 2, a(3) = 1. Alternatively, interleave denominators (A266504) and numerators (A266505) of convergents to sqrt(2).
%C a(2n) gives all x in N | 2*x^2 - 7(-1)^x = y^2. a(2n+1) gives associated y values.
%H G. C. Greubel, <a href="/A266506/b266506.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,2,0,0,0,1).
%F From _Chai Wah Wu_, Sep 17 2016: (Start)
%F a(n) = 2*a(n-4) + a(n-8) for n > 7.
%F G.f.: (-3*x^7 + x^6 - 5*x^5 + 3*x^4 - x^3 - 2*x^2 + x - 2)/(x^8 + 2*x^4 - 1).
%F (End)
%t CoefficientList[Series[(-3*x^7 + x^6 - 5*x^5 + 3*x^4 - x^3 - 2*x^2 + x - 2)/(x^8 + 2*x^4 - 1), {x, 0, 50}], x] (* _G. C. Greubel_, Jul 27 2018 *)
%o (PARI) x='x+O('x^50); Vec((-3*x^7+x^6-5*x^5+3*x^4-x^3-2*x^2+x-2)/(x^8 + 2*x^4-1)) \\ _G. C. Greubel_, Jul 27 2018
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((-3*x^7+x^6-5*x^5+3*x^4-x^3-2*x^2+x-2)/(x^8+2*x^4-1))); // _G. C. Greubel_, Jul 27 2018
%Y Cf. A266504, A266505.
%K sign,easy,less
%O 0,1
%A _Raphie Frank_, Dec 30 2015
%E Edited, new name using given formula, _Joerg Arndt_, Jan 31 2024
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