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A294199 Number of partitions of n into powers of 2 such that 1 and 2 cannot both be parts of a particular partition, and 4 and 8 cannot both be parts of a particular partition, and 16 and 32, and so on. 1
1, 1, 2, 1, 3, 2, 4, 2, 6, 4, 8, 4, 9, 5, 10, 5, 13, 8, 16, 8, 18, 10, 20, 10, 24, 14, 28, 14, 30, 16, 32, 16, 38, 22, 44, 22, 48, 26, 52, 26, 60, 34, 68, 34, 72, 38, 76, 38, 85, 47, 94, 47, 99, 52, 104, 52, 114, 62, 124, 62, 129, 67, 134, 67, 147, 80, 160 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Bin Lan and James A. Sellers, Properties of a Restricted Binary Partition Function a la Andrews and Lewis, #A23 INTEGERS 15 (2015), p.2.
FORMULA
G.f.: Product_{k>=1} (1 - x^(2^(2*k-2) + 2^(2*k-1))) / ((1 - x^(2^(2*k-2))) * (1 - x^(2^(2*k-1)))).
G.f.: Product_{k>=1} (1 - x^(3*2^(2*k-2))) / (1 - x^(2^(k-1))).
For n>=1 a(2*n) = a(2*n-2) + a([n/2]).
For n>=1 a(2*n+1) = a(2*n) - a(2*n-1).
EXAMPLE
a(10) = 8 where the partitions are the following: 8+2, 8+1+1, 4+4+2, 4+2+2+2, 4+4+1+1, 4+1+1+1+1+1+1, 2+2+2+2+2, 1+1+1+1+1+1+1+1+1+1.
MATHEMATICA
nmax = 20; CoefficientList[Series[Product[(1-x^(3*2^(2*k-2)))/(1-x^(2^(k-1))), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[1] = 1; a[2] = 2; a[3] = 1; Flatten[{1, 1, 2, 1, Table[If[EvenQ[n], a[n] = a[n-2] + a[Floor[n/4]], a[n] = a[n-1] - a[n-2]], {n, 4, 100}]}]
CROSSREFS
Cf. A070047.
Sequence in context: A334677 A365876 A179080 * A078658 A307719 A351594
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Oct 24 2017
STATUS
approved

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Last modified July 23 21:46 EDT 2024. Contains 374575 sequences. (Running on oeis4.)