A166444 a(0) = 0, a(1) = 1 and for n > 1, a(n) = sum of all previous terms.

0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 0

Author

Robert G. Wilson v, Oct 13 2009

Keywords

• nonn
• easy

Comments

Essentially a duplicate of A000079. - N. J. A. Sloane, Oct 15 2009

a(n) is the number of compositions of n into an odd number of parts.

Also 0 together with A011782. - Omar E. Pol, Oct 28 2013

Inverse INVERT transform of A001519. - R. J. Mathar, Dec 08 2022

Examples

			G.f. = x + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 16*x^6 + 32*x^7 + 64*x^8 + 128*x^9 + ...
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..3317
• Index entries for linear recurrences with constant coefficients, signature (2).

Crossrefs

Cf. A000045, A000079, A000213, A000288, A000322, A000383, A001519.

Cf. A011782, A034008, A060455, A123526, A127193, A127194, A127624.

Cf. A131577, A163551.

Programs

Magma

[n le 1 select n else 2^(n-2): n in [0..40]]; // G. C. Greubel, Jul 27 2024

Maple

a:= n-> `if`(n<2, n, 2^(n-2)):
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 02 2021

Mathematica

a[0] = 0; a[1] = 1; a[n_] := a[n] = Plus @@ Array[a, n - 1]; Array[a, 35, 0]

SageMath

[(2^n +2*int(n==1) -int(n==0))/4 for n in range(41)] # G. C. Greubel, Jul 27 2024

Formula

a(n) = A000079(n-1) for n > 0.

O.g.f.: x*(1 - x) / (1 - 2*x) = x / (1 - x / (1 - x)).

a(n) = (1-n) * a(n-1) + 2 * Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011

E.g.f.: (exp(2*x) + 2*x - 1)/4. - Stefano Spezia, Aug 07 2022