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%I #17 Sep 08 2022 08:46:17
%S 1,-1,4,-10,23,-59,138,-340,813,-1973,4752,-11486,27715,-66927,161558,
%T -390056,941657,-2273385,5488412,-13250226,31988847,-77227939,
%U 186444706,-450117372,1086679429,-2623476253,6333631912,-15290740102,36915112091,-89120964311,215157040686,-519435045712,1254027132081
%N a(n) = Sum_{k=0..n} (-1)^k*floor((1 + sqrt(2))^k).
%C Alternating sum of A080039.
%H Robert Israel, <a href="/A277789/b277789.txt">Table of n, a(n) for n = 0..2610</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SilverRatio.html">Silver Ratio</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (-1,4,0,-3,1).
%F O.g.f.: (1 - x^2 - 2*x^3)/((1 - x)^2*(1 + x)*(1 + 2*x - x^2)).
%F E.g.f.: ((-4*sqrt(2)*sinh(sqrt(2)*x) - 1)*exp(-x) + (5 - 2*x)*exp(x))/4.
%F a(n) = -a(n-1) + 4*a(n-2) - 3*a(n-4) + a(n-5).
%F a(n) = (2*sqrt(2)*(-1 - sqrt(2))^n - 2*sqrt(2)*(sqrt(2) - 1)^n - (-1)^n - 2*n + 5)/4.
%F a(n) ~ (-1)^n*s^(n+1)/(s + 1), where s is the silver ratio (A014176).
%p f:= gfun:-rectoproc({a(n) = -a(n-1) + 4*a(n-2) - 3*a(n-4) + a(n-5),seq(a(i)=[ 1, -1, 4, -10, 23][i+1],i=0..4)},a(n),remember):
%p map(f, [$0..40]); # _Robert Israel_, Oct 31 2016
%t Accumulate[Table[(-1)^n Floor[(1 + Sqrt[2])^n], {n, 0, 32}]]
%t LinearRecurrence[{-1, 4, 0, -3, 1}, {1, -1, 4, -10, 23}, 33]
%o (Magma) I:=[1,-1,4,-10,23]; [n le 5 select I[n] else -Self(n-1)+4*Self(n-2)-3*Self(n-4)+Self(n-5): n in [1..35]]; // _Vincenzo Librandi_, Nov 01 2016
%o (PARI) x='x+O('x^30); Vec((1-x^2-2*x^3)/((1-x)^2*(1+x)*(1+2*x-x^2))) \\ _G. C. Greubel_, Sep 30 2018
%Y Cf. A000129, A014176, A020962, A080039.
%K easy,sign
%O 0,3
%A _Ilya Gutkovskiy_, Oct 31 2016
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