A320933 a(n) = 2^n - floor((n+3)/2).

0, 0, 2, 5, 13, 28, 60, 123, 251, 506, 1018, 2041, 4089, 8184, 16376, 32759, 65527, 131062, 262134, 524277, 1048565, 2097140, 4194292, 8388595, 16777203, 33554418, 67108850, 134217713, 268435441, 536870896

Offset: 0

Author

Paul Curtz, Oct 28 2018

Keywords

• nonn
• easy

Comments

The sequence 0, 0, a(n) is an autosequence of the second kind. The difference table is:

0, 0, 0, 0, 2, 5, 13, ...

0, 0, 0, 2, 3, 8, 15, ...

0, 0, 2, 1, 5, 7, 17, ...

0, 2, -1, 4, 2, 10, 14, ...

2, -3, 5, -2, 8, 4, 20, ...

-5, 8, -7, 10, -4, 16, 8, ...

13, -15, 17, -14, 20, -8, 32, ...

etc.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• OEIS Wiki, Autosequence
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

Crossrefs

Cf. A000079, A004526, A011377, A014551, A060182, A166920.

Programs

GAP

List([0..40],n->2^n-Int((n+3)/2)); # Muniru A Asiru, Oct 28 2018

Magma

[((-1)^n+2^(n+2)-2*n-5)/4: n in [0..40]]; // G. C. Greubel, Jun 04 2019

Maple

seq(2^n-floor((n+3)/2),n=0..40); # Muniru A Asiru, Oct 28 2018

Mathematica

a[n_]:=2^n - Floor[(n+3)/2]; Array[a, 40, 0] (* or *) CoefficientList[ Series[x^2*(2-x)/((1-x)^2*(1-x-2*x^2)), {x, 0, 40}], x] (* Stefano Spezia, Oct 28 2018 *)

PARI

concat([0,0], Vec(x^2*(2-x)/((1-x)^2*(1+x)*(1-2*x)) + O(x^40))) \\ Colin Barker, Oct 28 2018

Sage

[((-1)^n+2^(n+2)-2*n-5)/4 for n in (0..40)] # G. C. Greubel, Jun 04 2019

Formula

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

a(n+1) = a(n) + A166920(n).

a(n+4) - a(n) = 13, 28, 58, 118, ... = 15*2^n - 2 = A060182(n+2).

With b(n) = 0, 0, 0, A011377(n) = 0, 0, 0, 1, 3, 8, 18, ..., then a(n) = 2*b(n+1) - b(n).

a(n+2) - 2*a(n+1) + a(n) = A014551(n).

G.f.: x^2*(2 - x)/((1-x)^2*(1 - x - 2*x^2)). - Stefano Spezia, Oct 28 2018

a(n) = ((-1)^n + 2^(n+2) - 2*n - 5) / 4. - Colin Barker, Oct 28 2018

Extensions

Three terms corrected by Colin Barker, Oct 28 2018