A254076 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2, a(0)=-1, a(1)=-2, a(2)=-4.

-1, -2, -4, 1, 2, 13, 26, 61, 122, 253, 506, 1021, 2042, 4093, 8186, 16381, 32762, 65533, 131066, 262141, 524282, 1048573, 2097146, 4194301, 8388602, 16777213, 33554426, 67108861, 134217722, 268435453, 536870906, 1073741821, 2147483642, 4294967293

Offset: 0

Author

Paul Curtz, Jan 29 2015

Keywords

• sign
• easy

Comments

The main diagonal of the difference table is -A000079(n) = -2^n.

a(n) mod 9 is of period 6: repeat 8, 7, 5, 1, 2, 4.

a(n) + a(n+1) = -3, -6, -3, 3, 15, ...; all are multiples of 3.

Links

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

Crossrefs

Cf. A000079, A010704, A141725, A153130, A156067, A157823.

Programs

Mathematica

a[0] = -1; a[n_] := 2^(n-1) + 3*Mod[n, 2] - 6; Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Feb 04 2015 *)

PARI

Vec((9*x^3+x^2-1)/((x-1)*(x+1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Jan 30 2015

Formula

a(2n+1) = A141725(n-1), a(2n+2) = 2*a(2n+1).

a(n+1) = 2*a(n) + (period 2: repeat 0, 9), n>0.

a(n) = -A157823(n) - (period 2: repeat 6, 3).

a(n+1) = a(n) - A156067(n).

a(n+2) = a(n) + 3*2^(n-1), n>0.

a(n+4) = a(n) + 15*2^(n-1), n>0.

a(n+6) = a(n) + 63*2^(n-1), n>0.

a(n) = (2^n - 3*(-1)^n - 9)/2 for n>0. - Colin Barker, Jan 30 2015

G.f.: (9*x^3+x^2-1) / ((x-1)*(x+1)*(2*x-1)). - Colin Barker, Jan 30 2015