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 A262445 Number of exact 3-colored partitions such that no adjacent parts have the same color. 3
 0, 0, 0, 6, 24, 72, 186, 438, 990, 2142, 4560, 9492, 19620, 40068, 81534, 164892, 332808, 669528, 1345554, 2699448, 5412636, 10843038, 21714972, 43467342, 86995428, 174069306, 348265164, 696694692, 1393652298, 2787646380, 5575837836, 11152384044, 22305891948, 44613248352, 89228806704, 178460625402, 356925987924 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS a(1) = a(2) = 0 because we need to use exactly three colors, which means the number of parts should be greater than two. All terms are multiples of 6. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 Ran Pan, A note on enumerating colored integer partitions, arXiv:1509.06107 [math.CO], 2015. Ran Pan, Exercise S, Project P. FORMULA G.f.: 3/2*Product_{k>=1} (1/(1-2*x^k)) - 6*Product_{k>=1} (1/(1-x^k)) + 3/(1-x) + 3/2. a(n) = A262444(n) - 6*A000041(n) + 3, for n >= 1. a(n) = 6 * A262495(n,3). - Alois P. Heinz, Sep 24 2015 EXAMPLE a(3)=6 because there are three partitions of 3 and there are no ways to color [3] or [2,1] but there are six ways to color [1,1,1]. MAPLE b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))     end: a:= n-> `if`(n=0, 0, b(n\$2, 2)/2*3-6*b(n\$2, 1)+3): seq(a(n), n=0..40);  # Alois P. Heinz, Sep 23 2015 MATHEMATICA b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k*b[n - i, i, k]]]]; a[n_] := If[n == 0, 0, b[n, n, 2]/2*3 - 6*b[n, n, 1] + 3]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *) CROSSREFS Cf. A000041, A262444, A139582, A262495, A320545. Sequence in context: A234271 A006528 A052749 * A090574 A294842 A290132 Adjacent sequences:  A262442 A262443 A262444 * A262446 A262447 A262448 KEYWORD nonn AUTHOR Ran Pan, Sep 23 2015 STATUS approved

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Last modified October 22 00:52 EDT 2019. Contains 328315 sequences. (Running on oeis4.)