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.

0, 0, 0, 0, 1, 1, 2, 2, 4, 6, 11, 17, 28, 44, 72, 116, 189, 305, 494, 798, 1292, 2090, 3383, 5473, 8856, 14328, 23184, 37512, 60697, 98209, 158906, 257114, 416020, 673134, 1089155, 1762289, 2851444, 4613732, 7465176, 12078908

Offset: 0

Author

Jean-François Alcover and Paul Curtz, Sep 28 2017

Keywords

• nonn
• easy

Comments

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.

Successive differences begin:

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

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

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

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

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

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

...

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

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

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

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

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,1).

Crossrefs

Cf. A000975, A001045, A005578, A026644.

Programs

Mathematica

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

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

Formula

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