cp's OEIS Frontend

A277752 a(n) = Sum_{k=0..n} (-1)^k*floor(phi^k), where phi is the golden ratio (A001622).

%I A277752 #50 Feb 16 2025 08:33:37
%S A277752 1,0,2,-2,4,-7,10,-19,27,-49,73,-126,195,-326,516,-848,1358,-2213,
%T A277752 3564,-5785,9341,-15135,24467,-39612,64069,-103692,167750,-271454,
%U A277752 439192,-710659,1149838,-1860511,3010335,-4870861,7881181,-12752058,20633223,-33385298,54018504,-87403820,141422306
%N A277752 a(n) = Sum_{k=0..n} (-1)^k*floor(phi^k), where phi is the golden ratio (A001622).
%C A277752 Alternating sum of A014217.
%H A277752 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldenRatio.html">Golden Ratio</a>
%H A277752 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,-1,-2,1).
%F A277752 G.f.: (1 - x^2 - x^3)/((1 - x)^2*(1 + 2*x - x^3)).
%F A277752 a(n) = 3*a(n-2) - a(n-3) - 2*a(n-4) + a(n-5).
%F A277752 a(n) = Sum_{k=0..n} (-1)^k*floor(Fibonacci(2k+3)/Fibonacci(k+3)).
%F A277752 a(n) = Sum_{k=0..n} (-1)^k*(L(k) - (1 + (-1)^k)/2), where L(k) is the Lucas numbers beginning at 2 (A000032).
%F A277752 a(n) = 2^(-n-2)*(9*2^n - 2^(n+1)*n - (-2)^n - 2*(1 + sqrt(5))*(sqrt(5) - 1)^n + 2*(sqrt(5) - 1)*(-1-sqrt(5))^n).
%F A277752 a(n) ~ (-1)^n*phi^(n-1).
%F A277752 a(n) = (-1)^n*Lucas(n-1) - (1/4)*(2*n -9 +(-1)^n). - _G. C. Greubel_, Oct 31 2016
%t A277752 Accumulate[Table[(-1)^n Floor[GoldenRatio^n], {n, 0, 40}]]
%t A277752 LinearRecurrence[{0, 3, -1, -2, 1}, {1, 0, 2, -2, 4}, 41]
%Y A277752 Cf. A000032, A000045, A001622, A014217, A020956.
%K A277752 easy,sign
%O A277752 0,3
%A A277752 _Ilya Gutkovskiy_, Oct 31 2016