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A226447 Expansion of (1-x+x^3)/(1-x^2+2*x^3-x^4). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

Table of n, a(n) for n=0..41.

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

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

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-Fran├žois Alcover, Jul 04 2013 *)

PROG

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

CROSSREFS

Sequence in context: A120770 A266990 A255213 * A073151 A279786 A255515

Adjacent sequences:  A226444 A226445 A226446 * A226448 A226449 A226450

KEYWORD

sign,easy

AUTHOR

Paul Curtz, Jun 28 2013

STATUS

approved

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Last modified January 29 08:12 EST 2020. Contains 331337 sequences. (Running on oeis4.)