A123108 a(n) = a(n-1) + a(n-2) - a(n-3), for n > 3, with a(0)=1, a(1)=0, a(2)=1, a(3)=1.

1, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37

Offset: 0

Author

Philippe Deléham, Sep 28 2006

Keywords

• nonn
• easy

Comments

Diagonal sums of triangle A123110. - Philippe Deléham, Oct 08 2009

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).

Crossrefs

Cf. A065033, A110654, A123110.

Programs

Magma

I:=[0,1,1]; [1] cat [n le 3 select I[n] else Self(n-1) +Self(n-2) -Self(n-3): n in [1..31]]; // G. C. Greubel, Jul 21 2021

Mathematica

LinearRecurrence[{1,1,-1},{1,0,1,1},90] (* Harvey P. Dale, Aug 10 2020 *)

Sage

[(2*n - 1 + (-1)^n)/4 + bool(n==0) for n in (0..90)] # G. C. Greubel, Jul 21 2021

Formula

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

a(n) = A110654(n-1). - R. J. Mathar, Jun 18 2008

From G. C. Greubel, Jul 21 2021: (Start)

a(n) = (1/4)*(2*n - 1 + (-1)^n) + [n=0].

E.g.f.: (1/2)*(2 + x*cosh(x) + (x-1)*sinh(x)). (End)