A226956 -id:A226956 - cp's OEIS frontend

A286350 a(n) = 2*a(n-1) - a(n-2) + a(n-4) for n>3, a(0)=0, a(1)=a(2)=2, a(3)=3.

0, 2, 2, 3, 4, 7, 12, 20, 32, 51, 82, 133, 216, 350, 566, 915, 1480, 2395, 3876, 6272, 10148, 16419, 26566, 42985, 69552, 112538, 182090, 294627, 476716, 771343, 1248060, 2019404, 3267464, 5286867, 8554330, 13841197, 22395528, 36236726, 58632254, 94868979

Offset: 0

Paul Curtz, May 08 2017

This is b(n) in A286311(n). As mentioned in A286311, the pair A286311(n) and, here a(n), are autosequences of the first kind.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1).

Cf. A022086, A128834, A226956 (same recurrence), A286311.

I:=[0,2,2,3]; [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

LinearRecurrence[{2, -1, 0, 1}, {0, 2, 2, 3}, 40] (* or *)
CoefficientList[Series[x (2 - 2 x + x^2)/((1 - x + x^2) (1 - x - x^2)), {x, 0, 39}], x] (* Michael De Vlieger, May 09 2017 *)

concat(0, Vec(x*(2 - 2*x + x^2) / ((1 - x + x^2)*(1 - x - x^2)) + O(x^60))) \\ Colin Barker, May 09 2017

a(n) = A286311(n) + A128834(n).
a(n) = A022086(n) - A286311(n).
a(n) = (A022086(n) + A128834(n))/2.
G.f.: x*(2 - 2*x + x^2) / ((1 - x + x^2)*(1 - x - x^2)). - Colin Barker, May 09 2017

More terms from Colin Barker, May 09 2017

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

Original entry on oeis.org

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

Paul Curtz, Jun 28 2013

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.

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

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

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 *)

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

cp's OEIS Frontend

A286350 a(n) = 2*a(n-1) - a(n-2) + a(n-4) for n>3, a(0)=0, a(1)=a(2)=2, a(3)=3.

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