A327710
Number of compositions of n into distinct parts such that the difference between any two parts is at least two.
Original entry on oeis.org
1, 1, 1, 1, 3, 3, 5, 5, 7, 13, 15, 21, 29, 35, 43, 55, 87, 99, 137, 173, 235, 277, 363, 429, 545, 755, 895, 1135, 1443, 1827, 2285, 2837, 3463, 4285, 5199, 6309, 8237, 9755, 12091, 14743, 18351, 22251, 27833, 33125, 40819, 49045, 59691, 70869, 86033, 106163
Offset: 0
a(9) = 13: 135, 153, 315, 351, 513, 531, 36, 63, 27, 72, 18, 81, 9.
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b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 0):
seq(a(n), n=0..50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
end:
a:= n-> add(k!*b(n-k^2, k), k=0..floor(sqrt(n))):
seq(a(n), n=0..50);
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b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
b[n, i - 1] + b[n - i, Min[n - i, i]]]];
a[n_] := Sum[k!*b[n - k^2, k], {k, 0, Floor[Sqrt[n]]}];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 12 2021, after Alois P. Heinz *)
A332829
Number of compositions of n such that the difference between adjacent parts is at least two.
Original entry on oeis.org
1, 1, 1, 1, 3, 4, 6, 9, 15, 23, 36, 55, 87, 136, 212, 329, 515, 802, 1251, 1949, 3043, 4745, 7401, 11535, 17994, 28063, 43766, 68243, 106433, 165981, 258854, 403670, 629530, 981750, 1531055, 2387660, 3723569, 5806905, 9055889, 14122638, 22024291, 34346886
Offset: 0
a(4) = 3: 13, 31, 4.
a(5) = 4: 131, 14, 41, 5.
a(6) = 6: 141, 24, 42, 15, 51, 6.
a(7) = 9: 313, 142, 241, 151, 25, 52, 16, 61, 7.
a(8) = 15: 1313, 3131, 242, 314, 413, 152, 251, 35, 53, 161, 26, 62, 17, 71, 8.
a(9) = 23: 13131, 1314, 1413, 3141, 4131, 414, 252, 135, 153, 315, 351, 513, 531, 162, 261, 36, 63, 171, 27, 72, 18, 81, 9.
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b:= proc(n, i) option remember; `if`(n=0, 1, add(`if`(abs
(i-j)<2, 0, b(n-j, `if`(n<2*j-1, -1, j))), j=1..n))
end:
a:= n-> b(n, -1):
seq(a(n), n=0..50);
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b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[If[Abs[i - j] < 2, 0,
b[n - j, If[n < 2*j - 1, -1, j]]], {j, 1, n}]];
a[n_] := b[n, -1];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 13 2022, after Alois P. Heinz *)
Showing 1-2 of 2 results.
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