A328881 a(n+3) = 2^n - a(n), a(0)=a(2)=1, a(1)=0 for n >= 0.

1, 0, 1, 0, 2, 3, 8, 14, 29, 56, 114, 227, 456, 910, 1821, 3640, 7282, 14563, 29128, 58254, 116509, 233016, 466034, 932067, 1864136, 3728270, 7456541, 14913080, 29826162, 59652323, 119304648, 238609294, 477218589, 954437176, 1908874354, 3817748707

Offset: 0

Author

Paul Curtz, Oct 29 2019

Keywords

• nonn
• easy

Comments

The array of a(n) and its repeated differences:

1, 0, 1, 0, 2, 3, 8, 14, ...

-1, 1, -1, 2, 1, 5, 6, 15, ...

2, -2, 3, -1, 4, 1, 9, 12, ...

-4, 5, -4, 5, -3, 8, 3, 19, ...

9, -9, 9, -8, 11, -5, 16, 5, ...

-18, 18, -17, 19, -16, 21, -11, 32, ...

36, -35, 36, -35, 37, -32, 43, -21, ...

-71, 71, -71, 72, -69, 75, -64, 85, ...

...

The recurrence is the same for every row.

From Jean-François Alcover, Nov 28 2019: (Start)

It appears that, when odd, a(n) is never a multiple of 5.

Main and 3rd upper diagonals of the difference array are A001045 (Jacobsthal numbers); first upper diagonal is negated A001045; second upper diagonal is A000079 (powers of 2); 4th upper diagonal is A062092.

(End)

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. A024495, A062092, A131714.

Cf. A015565, A092220, A113405.

Programs

Mathematica

a[0] = a[2] = 1; a[1] = 0; a[n_] := a[n] = 2^(n - 3) - a[n - 3]; Array[a, 36, 0] (* Amiram Eldar, Nov 06 2019 *)

PARI

Vec((1 - 2*x + x^2 - x^3) / ((1 + x)*(1 - 2*x)*(1 - x + x^2)) + O(x^40)) \\ Colin Barker, Oct 29 2019

Formula

a(n+1) - 2*a(n) = period 6: repeat [-2, 1, -2, 2, -1, 2].

a(n+12) - a(n) = 455*2^n.

From Colin Barker, Oct 29 2019: (Start)

G.f.: (1 - 2*x + x^2 - x^3) / ((1 + x)*(1 - 2*x)*(1 - x + x^2)).

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

(End)

a(n+2) - a(n) = A024495(n).

a(n+6) - a(n) = 7*2^n.

a(n+9) + a(n) = 57*2^n.

a(n) = A113405(n) + A092220(n+5).

9*a(n) = 2^n + 5*(-1)^n + 3*A010892(n). - R. J. Mathar, Nov 28 2019