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A166444
a(0) = 0, a(1) = 1 and for n > 1, a(n) = sum of all previous terms.
23
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
0,4
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
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
EXAMPLE
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 + ...
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]
PROG
(Magma) [n le 1 select n else 2^(n-2): n in [0..40]]; // G. C. Greubel, Jul 27 2024
(SageMath) [(2^n +2*int(n==1) -int(n==0))/4 for n in range(41)] # G. C. Greubel, Jul 27 2024
KEYWORD
nonn,easy
AUTHOR
Robert G. Wilson v, Oct 13 2009
STATUS
approved