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%I #29 May 29 2021 10:24:17
%S 0,0,2,3,11,20,51,99,222,441,935,1872,3863,7751,15774,31653,63939,
%T 128232,257963,517011,1037630,2078417,4165647,8340192,16702191,
%U 33428943,66912446,133891725,267921227,536022488,1072395555,2145272571,4291442718,8584166169
%N Number of compositions of n with at least 1 odd and 1 even part.
%H Alois P. Heinz, <a href="/A097895/b097895.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-8,2,4).
%F G.f.: x^3*(3*x-2)/((2*x-1)*(2*x^2-1)*(x^2+x-1)). - _Vladeta Jovovic_, Sep 03 2004
%F a(n) = 3*a(n-1) + a(n-2) - 8*a(n-3) + 2*a(n-4) + 4*a(n-5) for n > 5. - _Jinyuan Wang_, Mar 10 2020
%F From _Gregory L. Simay_, May 27 2021: (Start)
%F a(2*n) = 2^(2*n - 1) - 2^(n-1) - A000045(2*n).
%F a(2*n+1) = 2^(2*n) - A000045(2*n + 1). (End)
%e n=4: 2+1+1, 1+2+1, 1+1+2. Total=3.
%p G:=x^3*(3*x-2)/((2*x-1)*(2*x^2-1)*(x^2+x-1)): Gser:=series(G,x=0,37): seq(coeff(Gser,x^n),n=1..35); # _Emeric Deutsch_, Feb 15 2005
%p # second Maple program
%p b:= proc(n, o, e) option remember; `if`(n=0, `if`(o and e, 1, 0),
%p add(`if`(irem(i, 2)=1, b(n-i, true, e),
%p b(n-i, o, true)), i=1..n))
%p end:
%p a:= n-> b(n, false$2):
%p seq(a(n), n=1..50); # _Alois P. Heinz_, Jun 11 2013
%t e=(1-x^2)/(1-2x^2); o=(1-x^2)/(1-x-x^2); nn=30; Drop[CoefficientList[Series[(1-x)/(1-2x)-(o+e), {x,0,nn}], x], 1] (* _Geoffrey Critzer_, Jan 18 2012 *)
%Y Cf. A000041 (partitions), A006477 (partitions of n with at least 1 odd and 1 even part), A000009 (partitions into odd parts), A035363 (partitions into even parts); A000079 (compositions). Compositions into odd parts give Fibonacci numbers (A000045), into even parts gives 0, 1, 0, 2, 0, 4, 0, 8, 0, 16, 0, 32, 0, 64, ... (essentially A000079).
%Y Cf. A000045, A000041, A000009, A035363, A006477.
%Y Cf. A007179.
%K nonn
%O 1,3
%A Dubois Marcel (dubois.ml(AT)club-internet.fr), Sep 03 2004
%E More terms from _Emeric Deutsch_, Feb 15 2005
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