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 A261981 Number T(n,k) of compositions of n such that k is the minimal distance between two identical parts; triangle T(n,k), n>=2, 1<=k<=floor((sqrt(8*n-7)-1)/2), read by rows. 6
 1, 1, 4, 1, 9, 2, 18, 3, 41, 8, 2, 89, 16, 4, 185, 34, 10, 388, 57, 10, 810, 113, 30, 6, 1670, 213, 52, 12, 3435, 396, 104, 28, 7040, 733, 176, 50, 14360, 1333, 278, 62, 29226, 2419, 512, 152, 24, 59347, 4400, 878, 246, 48, 120229, 7934, 1492, 458, 108 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,3 LINKS Alois P. Heinz, Rows n = 2..55, flattened FORMULA T(n,k) = A261960(n,k-1) - A261960(n,k). T((n+1)*(n+2)/2+1,n+1) = A000142(n) for n>=0. EXAMPLE T(5,1) = 9: 311, 113, 221, 122, 2111, 1211, 1121, 1112, 11111. T(5,2) = 2: 131, 212. T(7,2) = 8: 151, 313, 232, 3121, 1213, 2131, 1312, 12121. T(7,3) = 2: 1231, 1321. Triangle T(n,k) begins: n\k: 1 2 3 4 5 ---+--------------------------- 02 : 1; 03 : 1; 04 : 4, 1; 05 : 9, 2; 06 : 18, 3; 07 : 41, 8, 2; 08 : 89, 16, 4; 09 : 185, 34, 10; 10 : 388, 57, 10; 11 : 810, 113, 30, 6; 12 : 1670, 213, 52, 12; 13 : 3435, 396, 104, 28; 14 : 7040, 733, 176, 50; 15 : 14360, 1333, 278, 62; 16 : 29226, 2419, 512, 152, 24; MAPLE b:= proc(n, l) option remember; `if`(n=0, 1, add(`if`(j in l, 0, b(n-j, `if`(l=[], [], [subsop(1=NULL, l)[], j]))), j=1..n)) end: T:= (n, k)-> b(n, [0\$(k-1)])-b(n, [0\$k]): seq(seq(T(n, k), k=1..floor((sqrt(8*n-7)-1)/2)), n=2..20); MATHEMATICA b[n_, l_] := b[n, l] = If[n == 0, 1, Sum[If[MemberQ[l, j], 0, b[n-j, If[l == {}, {}, Append[Rest[l], j]]]], {j, 1, n}]]; A[n_, k_] := b[n, Array[0&, Min[n, k]]]; T[n_, k_] := A[n, k-1] - A[n, k]; Table[T[n, k], {n, 2, 20}, {k, 1, Floor[(Sqrt[8*n-7]-1)/2]}] // Flatten (* Jean-François Alcover, Apr 13 2017, after Alois P. Heinz *) CROSSREFS Columns k=1-2 give: A261983, A261984. Row sums give A261982. Cf. A000142, A261960, A262191. Sequence in context: A304526 A333352 A260253 * A153265 A331153 A331149 Adjacent sequences: A261978 A261979 A261980 * A261982 A261983 A261984 KEYWORD nonn,tabf AUTHOR Alois P. Heinz, Sep 07 2015 STATUS approved

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Last modified September 10 06:17 EDT 2024. Contains 375773 sequences. (Running on oeis4.)