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Linear recurrence with signature (1,1,-1,1,1), where the first terms are powers of 2 (1,2,4,8,16).
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%I #9 Oct 29 2017 13:31:39

%S 1,2,4,8,16,23,37,56,94,152,250,401,649,1046,1696,2744,4444,7187,

%T 11629,18812,30442,49256,79702,128957,208657,337610,546268,883880,

%U 1430152,2314031,3744181,6058208,9802390,15860600,25662994,41523593,67186585,108710174,175896760,284606936

%N Linear recurrence with signature (1,1,-1,1,1), where the first terms are powers of 2 (1,2,4,8,16).

%C The interest of this sequence mainly lies in the peculiarities of its array of successive differences, which begins:

%C 1, 2, 4, 8, 16, 23, 37, 56, 94, ...

%C 1, 2, 4, 8, 7, 14, 19, 38, 58, ...

%C 1, 2, 4, -1, 7, 5, 19, 20, 40, ...

%C 1, 2, -5, 8, -2, 14, 1, 20, 13, ...

%C 1, -7, 13, -10, 16, -13, 19, -7, 31, ...

%C -8, 20, -23, 26, -29, 32, -26, 38, -23, ...

%C 28, -43, 49, -55, 61, -58, 64, -61, 67, ...

%C The main diagonal is A000079 (powers of 2).

%C The first upper subdiagonal is A254076.

%C The second upper subdiagonal (4, 8, 7, 14, 19, 38, ...) is not in the OEIS.

%C The third upper subdiagonal is A185346 (2^n-9).

%C Every row, once computed mod 9, is 6-periodic, repeating (1, 2, 4, 8, 7, 5) (A153130).

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,1,1).

%F G.f.: (1+x+x^2+3*x^3+5*x^4) / (1-x-x^2+x^3-x^4-x^5).

%F a(n) = (9/2)*fibonacci(n) + (-1)^n - sqrt(3)*sin(n*Pi/3).

%F a(n) ~ (9/2)*fibonacci(n).

%t LinearRecurrence[{1, 1, -1, 1, 1}, {1, 2, 4, 8, 16}, 40]

%Y Cf. A000045, A000079, A153130, A185346, A254076.

%K nonn,easy

%O 0,2

%A _Jean-François Alcover_ and _Paul Curtz_, Oct 29 2017