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A290968 a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) + a(n-5), with a(0)=a(1)=a(2)=1, a(3)=-1 and a(4)=1.

%I A290968 #21 Jul 11 2024 07:50:30
%S A290968 1,1,1,-1,1,1,5,5,9,11,21,33,57,89,145,231,377,609,989,1597,2585,4179,
%T A290968 6765,10945,17713,28657,46369,75023,121393,196417,317813,514229,
%U A290968 832041,1346267,2178309,3524577,5702889,9227465,14930353,24157815
%N A290968 a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) + a(n-5), with a(0)=a(1)=a(2)=1, a(3)=-1 and a(4)=1.
%C A290968 The array of successive differences begins:
%C A290968 1,   1,   1,  -1,   1,   1,   5,   5,   9,  11,  21,  33,  57, ...
%C A290968 0,   0,  -2,   2,   0,   4,   0,   4,   2,  10,  12,  24,  32, ...
%C A290968 0,  -2,   4,  -2,   4,  -4,   4,  -2,   8,   2,  12,   8,  24, ...
%C A290968 -2,  6,  -6,   6,  -8,   8,  -6,  10,  -6,  10,  -4,  16,   6, ...
%C A290968 8, -12,  12, -14,  16, -14,  16, -16,  16, -14,  20, -10,  24, ...
%C A290968 ...
%C A290968 First row is a(n) = 2*A141325(n) - A141325(n+1).
%C A290968 Main diagonal is A099430(n).
%C A290968 The first upper subdiagonal, 1, -2, -2, -8, -14, ..., has -3*A078008(n) as first differences.
%C A290968 The second upper subdiagonal is A000079(n) = 2^n.
%C A290968 a(n) is related to Fibonacci numbers a(n) = A000045(n-2) + period 6: repeat [2, 0, 1, -2, 0, -1].
%H A290968 G. C. Greubel, <a href="/A290968/b290968.txt">Table of n, a(n) for n = 0..1000</a>
%H A290968 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,1,1).
%F A290968 G.f.: (1-x^2-2*x^3+x^4)/((1+x)*(1-x+x^2)*(1-x-x^2)).
%F A290968 a(n) ~ phi^(n-2)/sqrt(5), where phi is the golden ratio.
%F A290968 a(n) = (1/2 + sqrt(5)/2)^n*(3*sqrt(5)/10-1/2) - (-1/2 + sqrt(5)/2)^n*(3*sqrt(5)/10 + 1/2)*(-1)^n + 2*sqrt(3)*sin(Pi*(n/3 + 1/3))/3 + (-1)^n. - _Eric Simon Jacob_, Jul 11 2024
%t A290968 LinearRecurrence[{1,1,-1,1,1}, {1,1,1,-1,1}, 40]
%o A290968 (PARI) my(x='x+O('x^40)); Vec((1-x^2-2*x^3+x^4)/((1+x^3)*(1-x-x^2))) \\ _G. C. Greubel_, Jun 11 2019
%o A290968 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x^2-2*x^3+x^4)/((1+x^3)*(1-x-x^2)) )); // _G. C. Greubel_, Jun 11 2019
%o A290968 (Sage) ((1-x^2-2*x^3+x^4)/((1+x^3)*(1-x-x^2))).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 11 2019
%Y A290968 Cf. A000045, A078008, A099430, A131531, A141325.
%K A290968 easy,sign
%O A290968 0,7
%A A290968 _Jean-François Alcover_ and _Paul Curtz_, Aug 16 2017