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 A097895 Number of compositions of n with at least 1 odd and 1 even part. 1
 0, 0, 2, 3, 11, 20, 51, 99, 222, 441, 935, 1872, 3863, 7751, 15774, 31653, 63939, 128232, 257963, 517011, 1037630, 2078417, 4165647, 8340192, 16702191, 33428943, 66912446, 133891725, 267921227, 536022488, 1072395555, 2145272571, 4291442718, 8584166169 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (3,1,-8,2,4). FORMULA G.f.: x^3*(3*x-2)/((2*x-1)*(2*x^2-1)*(x^2+x-1)). - Vladeta Jovovic, Sep 03 2004 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 From Gregory L. Simay, May 27 2021: (Start) a(2*n) = 2^(2*n - 1) - 2^(n-1) - A000045(2*n). a(2*n+1) = 2^(2*n) - A000045(2*n + 1). (End) EXAMPLE n=4: 2+1+1, 1+2+1, 1+1+2. Total=3. MAPLE 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 # second Maple program b:= proc(n, o, e) option remember; `if`(n=0, `if`(o and e, 1, 0),       add(`if`(irem(i, 2)=1, b(n-i, true, e),                              b(n-i, o, true)), i=1..n))     end: a:= n-> b(n, false\$2): seq(a(n), n=1..50);  # Alois P. Heinz, Jun 11 2013 MATHEMATICA 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 *) CROSSREFS 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). Cf. A000045, A000041, A000009, A035363, A006477. Cf. A007179. Sequence in context: A086791 A291633 A004687 * A298351 A023182 A302310 Adjacent sequences:  A097892 A097893 A097894 * A097896 A097897 A097898 KEYWORD nonn AUTHOR Dubois Marcel (dubois.ml(AT)club-internet.fr), Sep 03 2004 EXTENSIONS More terms from Emeric Deutsch, Feb 15 2005 STATUS approved

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Last modified January 26 09:35 EST 2022. Contains 350598 sequences. (Running on oeis4.)