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a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) for n > 4, where a(n)=0 for n < 4 and a(4) = 1.
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%I #18 Oct 12 2017 03:20:55

%S 0,0,0,0,1,1,2,2,4,6,11,17,28,44,72,116,189,305,494,798,1292,2090,

%T 3383,5473,8856,14328,23184,37512,60697,98209,158906,257114,416020,

%U 673134,1089155,1762289,2851444,4613732,7465176,12078908

%N a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) for n > 4, where a(n)=0 for n < 4 and a(4) = 1.

%C The interest of this sequence is mainly in the array of its successive differences, the diagonals of which are closely related to the Jacobsthal numbers A001045.

%C Successive differences begin:

%C 0, 0, 0, 0, 1, 1, 2, 2, 4, 6, 11, 17, 28, 44, ...

%C 0, 0, 0, 1, 0, 1, 0, 2, 2, 5, 6, 11, 16, 28, ...

%C 0, 0, 1, -1, 1, -1, 2, 0, 3, 1, 5, 5, 12, 16, ...

%C 0, 1, -2, 2, -2, 3, -2, 3, -2, 4, 0, 7, 4, 13, ...

%C 1, -3, 4, -4, 5, -5, 5, -5, 6, -4, 7, -3, 9, 1, ...

%C -4, 7, -8, 9, -10, 10, -10, 11, -10, 11, -10, 12, -8, 15, ...

%C ...

%C The main diagonal d0 (0, 1, 2, 5, 10, 21, 42, 85, ...) (with initial zero dropped) consists of the Lichtenberg numbers A000975.

%C Likewise, the first upper subdiagonal d1 (0, -1, -2, -5, -10, -21, -42, -85, ...) is the negated Lichtenberg numbers (so is d3).

%C The second upper subdiagonal d2 (0, 1, 1, 3, 5, 11, 21, 43, 85, ...) is the Jacobsthal numbers.

%C Subdiagonal d4 (1, 1, 2, 3, 6, 11, 22, 43, 86, ...) is A005578.

%C Subdiagonal d5 (1, 0, 0, -2, -4, -10, -20, -42, -84, ...) is negated A026644 from the 4th term on.

%H G. C. Greubel, <a href="/A293014/b293014.txt">Table of n, a(n) for n = 0..1000</a>

%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.: x^4/(1 - x - x^2 + x^3 - x^4 - x^5).

%t LinearRecurrence[{1, 1, -1, 1, 1}, {0, 0, 0, 0, 1}, 40]

%o (PARI) a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,1,-1,1,1]^n)[1,5] \\ _Charles R Greathouse IV_, Sep 28 2017

%Y Cf. A000975, A001045, A005578, A026644.

%K nonn,easy

%O 0,7

%A _Jean-François Alcover_ and _Paul Curtz_, Sep 28 2017