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%I A332829 #17 Apr 13 2022 07:40:21
%S A332829 1,1,1,1,3,4,6,9,15,23,36,55,87,136,212,329,515,802,1251,1949,3043,
%T A332829 4745,7401,11535,17994,28063,43766,68243,106433,165981,258854,403670,
%U A332829 629530,981750,1531055,2387660,3723569,5806905,9055889,14122638,22024291,34346886
%N A332829 Number of compositions of n such that the difference between adjacent parts is at least two.
%H A332829 Alois P. Heinz, <a href="/A332829/b332829.txt">Table of n, a(n) for n = 0..1750</a>
%F A332829 a(n) ~ c * d^n, where d = 1.55950091106966174000570854045613844480247532446123619115121795622156266..., c = 0.42021981384104890468461570042297109905705539874851026797544718780579866... - _Vaclav Kotesovec_, Feb 28 2020
%e A332829 a(4) = 3: 13, 31, 4.
%e A332829 a(5) = 4: 131, 14, 41, 5.
%e A332829 a(6) = 6: 141, 24, 42, 15, 51, 6.
%e A332829 a(7) = 9: 313, 142, 241, 151, 25, 52, 16, 61, 7.
%e A332829 a(8) = 15: 1313, 3131, 242, 314, 413, 152, 251, 35, 53, 161, 26, 62, 17, 71, 8.
%e A332829 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.
%p A332829 b:= proc(n, i) option remember; `if`(n=0, 1, add(`if`(abs
%p A332829       (i-j)<2, 0, b(n-j, `if`(n<2*j-1, -1, j))), j=1..n))
%p A332829     end:
%p A332829 a:= n-> b(n, -1):
%p A332829 seq(a(n), n=0..50);
%t A332829 b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[If[Abs[i - j] < 2, 0,
%t A332829      b[n - j, If[n < 2*j - 1, -1, j]]], {j, 1, n}]];
%t A332829 a[n_] := b[n, -1];
%t A332829 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Apr 13 2022, after _Alois P. Heinz_ *)
%Y A332829 Cf. A003242, A328222.
%K A332829 nonn
%O A332829 0,5
%A A332829 _Alois P. Heinz_, Feb 25 2020

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