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A079501 Number of compositions of the integer n with strictly smallest part in the first position.

%I A079501 #30 Jan 03 2024 06:49:26
%S A079501 1,1,2,2,4,5,8,12,19,28,45,70,110,173,275,436,695,1107,1769,2831,4537,
%T A079501 7276,11683,18774,30194,48592,78247,126062,203192,327645,528518,
%U A079501 852815,1376491,2222294,3588628,5796196,9363458,15128631,24447014,39510108
%N A079501 Number of compositions of the integer n with strictly smallest part in the first position.
%C A079501 Also number of compositions of n such that the first part is divisible by the number of parts . [_Vladeta Jovovic_, Dec 02 2009]
%D A079501 Arnold Knopfmacher and Neville Robbins, Compositions with parts constrained by the leading summand, Ars Combin. 76 (2005), 287-295.
%H A079501 Alois P. Heinz, <a href="/A079501/b079501.txt">Table of n, a(n) for n = 1..1000</a>
%F A079501 G.f.: Sum_{k>=1} (1-z)*z^k/(1-z-z^(k+1)).
%F A079501 G.f.: Sum_{k>=1} z^(2*k-1)/((1-z^k)*(1-z)^(k-1)), cf. A105039. - _Vladeta Jovovic_, Apr 05 2005
%F A079501 a(n) ~ 1/sqrt(5) * ((1+sqrt(5))/2)^(n-2). - _Vaclav Kotesovec_, May 01 2014
%F A079501 G.f.: Sum_{n>=1} q^n/(1-Sum_{k>=n+1} q^k). - _Joerg Arndt_, Jan 03 2024
%e A079501 The a(9)=19 such compositions of 9 are
%e A079501 [ 1]  [ 1 2 2 2 2 ]
%e A079501 [ 2]  [ 1 2 2 4 ]
%e A079501 [ 3]  [ 1 2 3 3 ]
%e A079501 [ 4]  [ 1 2 4 2 ]
%e A079501 [ 5]  [ 1 2 6 ]
%e A079501 [ 6]  [ 1 3 2 3 ]
%e A079501 [ 7]  [ 1 3 3 2 ]
%e A079501 [ 8]  [ 1 3 5 ]
%e A079501 [ 9]  [ 1 4 2 2 ]
%e A079501 [10]  [ 1 4 4 ]
%e A079501 [11]  [ 1 5 3 ]
%e A079501 [12]  [ 1 6 2 ]
%e A079501 [13]  [ 1 8 ]
%e A079501 [14]  [ 2 3 4 ]
%e A079501 [15]  [ 2 4 3 ]
%e A079501 [16]  [ 2 7 ]
%e A079501 [17]  [ 3 6 ]
%e A079501 [18]  [ 4 5 ]
%e A079501 [19]  [ 9 ]
%e A079501 - _Joerg Arndt_, Jan 01 2013
%p A079501 b:= proc(n, s) option remember; `if`(n=0, 1, add(
%p A079501       `if`(n-j>0 and n-j<=s, 0, b(n-j, s)), j=s+1..n))
%p A079501     end:
%p A079501 a:= n-> 1 +add(b(n-j, j), j=1..n/2):
%p A079501 seq(a(n), n=1..60);  # _Alois P. Heinz_, Apr 29 2014
%t A079501 b[n_, s_] := b[n, s] = If[n == 0, 1, Sum[ If[n - j > 0 && n - j <= s, 0, b[n - j, s]], {j, s + 1, n}]]; a[n_] := 1 + Sum[b[n - j, j], {j, 1, n/2}]; Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Jan 16 2015, after _Alois P. Heinz_ *)
%Y A079501 Cf. A168655, A168656, A168657. [From _Vladeta Jovovic_, Dec 02 2009]
%K A079501 nonn
%O A079501 1,3
%A A079501 _Arnold Knopfmacher_, Jan 21 2003
%E A079501 More terms from _Benoit Cloitre_, Jan 21 2003