

A239342


Number of 1's in all compositions of n into odd parts.


3



0, 1, 2, 3, 6, 11, 20, 36, 64, 113, 198, 345, 598, 1032, 1774, 3039, 5190, 8839, 15016, 25452, 43052, 72685, 122502, 206133, 346346, 581136, 973850, 1630011, 2725254, 4551683, 7594748, 12660660, 21087448, 35094377, 58360134, 96979089, 161042110, 267248664
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OFFSET

0,3


COMMENTS

a(n+1) is the number of ways to tile a strip of length n+1 using white tiles of only odd lengths, with total length n, and one red square of length one.  Gregory L. Simay, Aug 14 2016
A029907, the number of compositions of n with exactly one even part, is equal to a(n+12) + a(n+14) + a(n+16) + ...  Gregory L. Simay, Aug 14 2016
Apart from the initial 0 and 1, this is the pINVERT transform of (1,0,1,0,1,0,...) for p(S) = (1  S)^2. See A291219.  Clark Kimberling, Sep 02 2017


REFERENCES

S. Heubach and T. Mansour, Combinatorics of Compositions and Words, Chapman and Hall, 2010, page 70.


LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..4771
Mengmeng Liu, Andrew Yezhou Wang, The Number of Designated Parts in Compositions with Restricted Parts, J. Int. Seq., Vol. 23 (2020), Article 20.1.8.


FORMULA

For n >= 4, a(n) = a(n1) + a(n2) + A000045(n2).
G.f.: x*(1  x^2)^2/(1  x  x^2)^2.


EXAMPLE

a(5) = 11 because in the compositions of 5 into odd parts there are a total of 11 1's: 5, 3+1+1, 1+3+1, 1+1+3, 1+1+1+1+1.
Let r represent the red square and 1,3,5 represent the possible odd lengths of the white squares for n=5. Then a(5+1) = a(6) = 20 because r combined with a tile of length 5 generates 2 compositions; r combined with 3,1,1 generates 12 compositions; and r combined with 1,1,1,1,1 generates 6 compositions. 2+12+6 = 20.  Gregory L. Simay, Aug 14 2016


MATHEMATICA

nn=30; CoefficientList[Series[x (1x^2)^2/(1xx^2)^2, {x, 0, nn}], x]
(* or *)
Table[Count[Flatten[Level[Map[Permutations, IntegerPartitions[n, n, Table[2k+1, {k, 0, n/2}]]], {2}]], 1], {n, 0, 30}]


CROSSREFS

Cf. A000045, A029907.
Sequence in context: A090167 A352500 A002985 * A093608 A320328 A079976
Adjacent sequences: A239339 A239340 A239341 * A239343 A239344 A239345


KEYWORD

nonn


AUTHOR

Geoffrey Critzer, Mar 16 2014


STATUS

approved



