A275788 a(0) = 0, a(n+1) = 2*a(n) + (-1)^floor(n/3).

0, 1, 3, 7, 13, 25, 49, 99, 199, 399, 797, 1593, 3185, 6371, 12743, 25487, 50973, 101945, 203889, 407779, 815559, 1631119, 3262237, 6524473, 13048945, 26097891, 52195783, 104391567, 208783133, 417566265, 835132529, 1670265059, 3340530119, 6681060239

Offset: 0

Author

Paul Curtz, Aug 09 2016

Keywords

• nonn

Comments

a(n) and its successive differences:

0, 1, 3, 7, 13, 25, 49, ...

1, 2, 4, 6, 12, 24, 50, 100, ...

1, 2, 2, 6, 12, 26, 50, 100, 198, ...

1, 0, 4, 6, 14, 24, 50, 98, 200, 398, ...

-1, 4, 2, 8, 10, 26, 48, 102, 198, 400, 794, ...

5, -2, 6, 2, 16, 22, 54, 96, 202, 394, 800, 1590, ...

-7, 8, -4, 14, 6, 32, 42, 106, 192, 406, 790, 1600, 3178, ...

... .

Each row has the recurrence a(n) + a(n+3) = 7*2^n.

Main diagonal: 2*A001045(n).

Upper diagonals: A084214(n+1), 3*2^n, ... .

Subdiagonals: 2^n, A078008(n), A084214(n+1), -2^n, ... .

a(-n) = 0, 1/2, 3/4, 7/8, -1/16, -17/32, -49/64, 15/128, ... .

b(n), numerators of a(-n), and first differences:

0, 1, 3, 7, -1, -17, -49, 15, 143, 399, -113, -1137, ...

1, 2, 4, -8, -16, -32, 64, 128, 256, -512, -1024, ... = A000079(n)*A130151(n), not in the OEIS.

Examples

			a(1)=2*0+1=1, a(2)=2*1+1=3, a(2)=2*3+1=7, a(3)=2*7-1=13, a(4)=2*13-1=25, ... .
		

Links

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

Crossrefs

Cf. A000079, A001045, A002264, A005009, A007283, A078008, A084214, A113405, A130151, A274817.

Programs

Mathematica

CoefficientList[Series[x (1 + x + x^2)/((1 + x) (1 - 2 x) (1 - x + x^2)), {x, 0, 33}], x] (* Michael De Vlieger, Aug 11 2016 *)
LinearRecurrence[{2,0,-1,2}, {0, 1, 3, 7}, 25] (* G. C. Greubel, Aug 16 2016 *)

PARI

concat(0, Vec(x*(1+x+x^2)/((1+x)*(1-2*x)*(1-x+x^2)) + O(x^40))) \\ Colin Barker, Aug 10 2016

Formula

From Colin Barker, Aug 09 2016: (Start)

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

G.f.: x*(1 + x + x^2) / ((1+x)*(1-2*x)*(1-x+x^2)).

(End)

a(n+3) = 7*2^n - a(n), a(0)=0, a(1)=1, a(2)=3.

Extensions

More terms from Colin Barker, Aug 10 2016