A226447 Expansion of (1-x+x^3)/(1-x^2+2*x^3-x^4).

1, -1, 1, -2, 4, -5, 9, -15, 23, -38, 62, -99, 161, -261, 421, -682, 1104, -1785, 2889, -4675, 7563, -12238, 19802, -32039, 51841, -83881, 135721, -219602, 355324, -574925, 930249, -1505175, 2435423, -3940598, 6376022, -10316619, 16692641, -27009261, 43701901, -70711162, 114413064, -185124225

Offset: 0

Author

Paul Curtz, Jun 28 2013

Keywords

• sign
• easy

Comments

a(n) and its differences:

. 1, -1, 1, -2, 4, -5, 9, -15, 23, -38, ...

. -2, 2, -3, 6, -9, 14, -24, 38, -61, 100, ...

. 4, -5, 9, -15, 23, -38, 62, -99, 161, -261, ...

. -9, 14, -24, 38, -61, 100, -161, 260, -422, 682, ...

. 23, -38, 62, -99, 161, -261, 421, -682, 1104, -1785, ...

. -61, 100, -161, 260, -422, 682, -1103, 1786, -2889, 4674, ...

. 161, -261, 421, -682, 1104, -1785, 2889, -4675, 7563, -12238, ...

The third row is the first shifted .

The main diagonal is A001077(n). The fourth is -A001077(n+1). By "shifted" antidiagonals there are one 1, two 2's (-2 of the first row and 2), generally A001651(n) (-1)^n *A001077(n).

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

Links

• Index entries for linear recurrences with constant coefficients, signature (0,1,-2,1).

Programs

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+x^3)/(1-x^2+2*x^3-x^4))); // Bruno Berselli, Jul 04 2013

Mathematica

a[0] = 1; a[1] = -1; a[n_] := a[n] = a[n-2] - a[n-1] - {-1, 0, 1, 1, 0, -1}[[Mod[n+1, 6] + 1]]; Table[a[n], {n, 0, 41}] (* Jean-François Alcover, Jul 04 2013 *)

Formula

a(0)=1, a(1)=-1; for n>1, a(n) = a(n-2) - a(n-1) + A010892(n+2).

a(n) = a(n-2) -2*a(n-3) +a(n-4).

a(n) = A226956(-n).

a(n+1) = A039834(n) - (-1)^n*A094686(n).

a(n+6) - a(n) = 2*(-1)^n* A000032(n+3).

a(2n+1) = -A226956(2n+1).

G.f. ( -1+x-x^3 ) / ( (x^2-x-1)*(1-x+x^2) ). - R. J. Mathar, Jun 29 2013

2*a(n) = A010892(n+2)+A061084(n+1). - R. J. Mathar, Jun 29 2013