A286311 -id:A286311 - 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

A286390 a(n) = a(n-2) - 2*a(n-3) + a(n-4) for n>3, a(0)=0, a(1)=2, a(2)=-1, a(3)=3.

Original entry on oeis.org

0, 2, -1, 3, -5, 7, -12, 20, -31, 51, -83, 133, -216, 350, -565, 915, -1481, 2395, -3876, 6272, -10147, 16419, -26567, 42985, -69552, 112538, -182089, 294627, -476717, 771343, -1248060, 2019404, -3267463, 5286867, -8554331, 13841197, -22395528, 36236726

Offset: 0

Paul Curtz, May 08 2017

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

Cf. A134667, A286311.

LinearRecurrence[{0, 1, -2, 1}, {0, 2, -1, 3}, 38] (* or *)
CoefficientList[Series[x (2 - x + x^2)/((1 + x - x^2) (1 - x + x^2)), {x, 0, 37}], x] (* Michael De Vlieger, May 08 2017 *)

concat(0, Vec(x*(2 - x + x^2) / ((1 + x - x^2)*(1 - x + x^2)) + O(x^60))) \\ Colin Barker, May 08 2017

a(n) = A286311(-n).
b(n) = A286311(n) + (period 6 sequence : repeat [0, 1, 0, 0, 0, -1]) = 0, 2, 1, 3, 5, 7, 12, 20, 31, 51, 83, 133,  ... . See A134667(n).
a(n) = -(-1)^n*b(n).
G.f.: x*(2 - x + x^2) / ((1 + x - x^2)*(1 - x + x^2)). - Colin Barker, May 08 2017

Typo in data fixed and more terms added by Colin Barker, May 08 2017

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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