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A293014 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.

%I A293014 #18 Oct 12 2017 03:20:55
%S A293014 0,0,0,0,1,1,2,2,4,6,11,17,28,44,72,116,189,305,494,798,1292,2090,
%T A293014 3383,5473,8856,14328,23184,37512,60697,98209,158906,257114,416020,
%U A293014 673134,1089155,1762289,2851444,4613732,7465176,12078908
%N A293014 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 A293014 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 A293014 Successive differences begin:
%C A293014 0,  0,  0,  0,   1,  1,   2,  2,   4,  6,  11, 17, 28, 44, ...
%C A293014 0,  0,  0,  1,   0,  1,   0,  2,   2,  5,   6, 11, 16, 28, ...
%C A293014 0,  0,  1, -1,   1, -1,   2,  0,   3,  1,   5,  5, 12, 16, ...
%C A293014 0,  1, -2,  2,  -2,  3,  -2,  3,  -2,  4,   0,  7,  4, 13, ...
%C A293014 1, -3,  4, -4,   5, -5,   5, -5,   6, -4,   7, -3,  9,  1, ...
%C A293014 -4, 7, -8,  9, -10, 10, -10, 11, -10, 11, -10, 12, -8, 15, ...
%C A293014 ...
%C A293014 The main diagonal d0 (0, 1, 2, 5, 10, 21, 42, 85, ...) (with initial zero dropped) consists of the Lichtenberg numbers A000975.
%C A293014 Likewise, the first upper subdiagonal d1 (0, -1, -2, -5, -10, -21, -42, -85, ...) is the negated Lichtenberg numbers (so is d3).
%C A293014 The second upper subdiagonal d2 (0, 1, 1, 3, 5, 11, 21, 43, 85, ...) is the Jacobsthal numbers.
%C A293014 Subdiagonal d4 (1, 1, 2, 3, 6, 11, 22, 43, 86, ...) is A005578.
%C A293014 Subdiagonal d5 (1, 0, 0, -2, -4, -10, -20, -42, -84, ...) is negated A026644 from the 4th term on.
%H A293014 G. C. Greubel, <a href="/A293014/b293014.txt">Table of n, a(n) for n = 0..1000</a>
%H A293014 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,1,1).
%F A293014 G.f.: x^4/(1 - x - x^2 + x^3 - x^4 - x^5).
%t A293014 LinearRecurrence[{1, 1, -1, 1, 1}, {0, 0, 0, 0, 1}, 40]
%o A293014 (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 A293014 Cf. A000975, A001045, A005578, A026644.
%K A293014 nonn,easy
%O A293014 0,7
%A A293014 _Jean-François Alcover_ and _Paul Curtz_, Sep 28 2017