A226956 a(0)=a(1)=1, a(n+2) = a(n+1) + a(n) - A128834(n).

1, 1, 2, 2, 3, 5, 9, 15, 24, 38, 61, 99, 161, 261, 422, 682, 1103, 1785, 2889, 4675, 7564, 12238, 19801, 32039, 51841, 83881, 135722, 219602, 355323, 574925, 930249, 1505175, 2435424, 3940598, 6376021, 10316619, 16692641, 27009261, 43701902, 70711162, 114413063, 185124225, 299537289

Offset: 0

Author

Paul Curtz, Jun 24 2013

Keywords

• nonn
• easy

Comments

a(n) and differences:

1, 1, 2, 2, 3, 5, 9, 15, 24, 38, ... a(n)

0, 1, 0, 1, 2, 4, 6, 9, 14, 23, 38, ... b(n)

1, -1, 1, 1, 2, 2, 3, 5, 9, 15, 24, 38, ... a(n-2)

-2, 2, 0, 1, 0, 1, 2, 4, 6, 9, 14, 23, 38, ... b(n-2)

4, -2, 1,-1, 1, 1, 2, 2, 3, 5, 9, ... a(n-4)

-6, 3,-2, 2, 0, 1, 0, 1, 2, 4, 6, 9, ... b(n-4)

9, -5, 4,-2, 1,-1, 1, 1, 2, 2, 3, 5, 9, ... a(n-6)

-14, 9,-6, 3,-2, 2, 0, 1 0, 1, 2, ... b(n-6)

23,-15, 9,-5, 4,-2, 1, -1, 1, 1, 2, 2, ... a(n-8)

a(n)-b(n+1) = period 6: repeat 0, 1, 1, 0, -1, -1 = A128834(n).

Diagonals with the same number give 1, 2, 9, 38, ... = A001077(n).

Second column: the (n+2)-th term is identical to a(n+1) signed.

a(n+1) is identical to its twice shifted inverse binomial transform signed.

a(n+1)/a(n) tends to A001622 (the golden ratio) as n -> infinity.

Examples

			a(0) = a(1) = 1.
a(2) = a(3) = 2.
a(4) = 2*a(3) - a(2) + a(0) = 4-2+1 = 3.
a(5) = 6-2+1 = 5.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1)

Crossrefs

Cf. Diagonals in A024490.

Programs

Magma

I:=[1, 1, 2, 2]; [n le 4 select I[n] else 2*Self(n-1) - Self(n-2) + Self(n-4): n in [1..30]]; // G. C. Greubel, Jan 15 2018

Mathematica

a[n_] := (LucasL[n] + {0, 1, 1, 0, -1, -1}[[Mod[n, 6] + 1]])/2; Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Jun 28 2013, after R. J. Mathar *)
LinearRecurrence[{2,-1,0,1}, {1,1,2,2}, 30] (* G. C. Greubel, Jan 15 2018 *)

PARI

x='x+O('x^30); Vec((x-1)*(1+x^2)/((x^2+x-1)*(x^2-x+1))) \\ G. C. Greubel, Jan 15 2018

Formula

a(n+6) - a(n-6) = 20*A000045(n).

a(n) = 2*a(n-1) - a(n-2) + a(n-4).

a(n) = 3*a(n-3) + 5*a(n-6) + a(n-9) (plus many similar by telescoping the fundamental recurrence).

a(n+3) - a(n-3) = 2*A000032(n).

G.f.: (x-1)*(1+x^2) / ( (x^2+x-1)*(x^2-x+1) ). - R. J. Mathar, Jun 26 2013

2*a(n) = A000032(n) + A010892(n-1). - R. J. Mathar, Jun 26 2013

a(n+5) = a(n+4) + a(n+2) + A108014(n).

a(2n+1) + A226447(2n+2) = 2*A182895(n).

a(n) - a(n-2) = 0,2,1,1,1,3,6,... = abs(A111734(n-2)).