cp's OEIS Frontend

A224960 Number of compositions [p(1), p(2), ..., p(k)] of n such that p(j) >= p(1) - 1.

%I A224960 #19 Jun 02 2025 08:31:34
%S A224960 1,1,2,4,7,14,26,52,101,200,396,787,1564,3117,6214,12398,24749,49427,
%T A224960 98740,197303,394323,788201,1575695,3150265,6298732,12594595,25184598,
%U A224960 50361842,100711888,201404839,402779246,805509560,1610940381,3221753990
%N A224960 Number of compositions [p(1), p(2), ..., p(k)] of n such that p(j) >= p(1) - 1.
%H A224960 Alois P. Heinz, <a href="/A224960/b224960.txt">Table of n, a(n) for n = 0..1000</a>
%F A224960 a(n) ~ 3 * 2^(n-3). - _Vaclav Kotesovec_, May 01 2014
%e A224960 The a(5) = 14 such compositions of 5 are
%e A224960 01:  [ 1 1 1 1 1 ]
%e A224960 02:  [ 1 1 1 2 ]
%e A224960 03:  [ 1 1 2 1 ]
%e A224960 04:  [ 1 1 3 ]
%e A224960 05:  [ 1 2 1 1 ]
%e A224960 06:  [ 1 2 2 ]
%e A224960 07:  [ 1 3 1 ]
%e A224960 08:  [ 1 4 ]
%e A224960 09:  [ 2 1 1 1 ]
%e A224960 10:  [ 2 1 2 ]
%e A224960 11:  [ 2 2 1 ]
%e A224960 12:  [ 2 3 ]
%e A224960 13:  [ 3 2 ]
%e A224960 14:  [ 5 ]
%e A224960 (the two forbidden compositions are [ 3 1 1 ] and [ 4 1 ]).
%p A224960 b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j,
%p A224960       `if`(i=0, max(1, j-1), i)), j=`if`(i=0, 1, i)..n))
%p A224960     end:
%p A224960 a:= n-> b(n, 0):
%p A224960 seq(a(n), n=0..50);  # _Alois P. Heinz_, May 02 2013
%t A224960 b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - j, If[i == 0, Max[1, j - 1], i]], {j, If[i == 0, 1, i], n}]];
%t A224960 a[n_] := b[n, 0];
%t A224960 a /@ Range[0, 50] (* _Jean-François Alcover_, Dec 20 2020, after _Alois P. Heinz_ *)
%Y A224960 Cf. A171682 (compositions such that p(j) >= p(1)).
%Y A224960 Cf. A079501 (compositions such that p(j) > p(1)).
%Y A224960 Cf. A048888 (compositions such that p(j) <= p(1) + 1).
%Y A224960 Cf. A007059 (compositions such that p(j) < p(1)).
%Y A224960 Cf. A079500 (compositions such that p(j) <= p(1)).
%K A224960 nonn
%O A224960 0,3
%A A224960 _Joerg Arndt_, Apr 21 2013