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A129853 Nonascending wiggly sums: number of sums adding to n in which terms alternately do not increase and do not decrease.

%I A129853 #22 Aug 16 2023 08:22:10
%S A129853 1,1,2,3,6,9,17,28,50,85,149,257,448,775,1347,2336,4057,7038,12219,
%T A129853 21204,36807,63880,110878,192442,334020,579739,1006237,1746482,
%U A129853 3031310,5261324,9131892,15849876,27510049,47748159,82874713,143842547,249662173,433329337,752113633,1305415658,2265761441
%N A129853 Nonascending wiggly sums: number of sums adding to n in which terms alternately do not increase and do not decrease.
%H A129853 Alois P. Heinz, <a href="/A129853/b129853.txt">Table of n, a(n) for n = 0..2000</a>
%e A129853 The a(4)=6 sums that add to 4 are 4, 3+1, 2+2, 2+1+1, 1+1+2 and 1+1+1+1. The 2 = 2^(n-1)-a(n) sums 1+2+1 and 1+3 do not satisfy the criterion and do not count.
%e A129853 From _Joerg Arndt_, May 21 2013: (Start)
%e A129853 The a(6)=17 such compositions are
%e A129853 01:  [ 1 1 1 1 1 1 ]
%e A129853 02:  [ 1 1 1 1 2 ]
%e A129853 03:  [ 1 1 2 1 1 ]
%e A129853 04:  [ 1 1 2 2 ]
%e A129853 05:  [ 1 1 3 1 ]
%e A129853 06:  [ 1 1 4 ]
%e A129853 07:  [ 2 1 1 1 1 ]
%e A129853 08:  [ 2 1 2 1 ]
%e A129853 09:  [ 2 1 3 ]
%e A129853 10:  [ 2 2 2 ]
%e A129853 11:  [ 3 1 1 1 ]
%e A129853 12:  [ 3 1 2 ]
%e A129853 13:  [ 3 3 ]
%e A129853 14:  [ 4 1 1 ]
%e A129853 15:  [ 4 2 ]
%e A129853 16:  [ 5 1 ]
%e A129853 17:  [ 6 ]
%e A129853 (End)
%p A129853 A129853rec := proc(part,n) local asum,a,k ; asum := add(i,i=part) ; if asum > n then RETURN(0) ; elif asum = n then RETURN(1) ; else a := 0 ; if nops(part) mod 2 = 0 then for k from op(-1,part) to n-asum do a := a+A129853rec([op(part),k],n) ; od: else for k from 1 to min(op(-1,part),n-asum) do a := a+A129853rec([op(part),k],n) ; od: fi ; RETURN(a) ; fi ; end: A129853 := proc(n) local a,a1 ; a := 0 ; for a1 from 1 to n do a := a+A129853rec([a1],n) ; od: RETURN(a) ; end: seq(A129853(n),n=1..20) ; # _R. J. Mathar_, Oct 31 2007
%p A129853 # second Maple program:
%p A129853 b:= proc(n, l, t) option remember; `if`(n=0, 1, add(
%p A129853       b(n-j, j, not t), j=`if`(t, l..n, 1..min(n, l))))
%p A129853     end:
%p A129853 a:= n-> b(n, 1, true):
%p A129853 seq(a(n), n=0..40);  # _Alois P. Heinz_, May 23 2023
%t A129853 b[n_, l_, t_] := b[n, l, t] = If[n == 0, 1, Sum[b[n - j, j, !t], {j, If[t, Range[l, n], Range[Min[n, l]]]}]];
%t A129853 a[n_] := b[n, 1, True];
%t A129853 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Aug 16 2023, after _Alois P. Heinz_ *)
%Y A129853 Cf. A129852.
%Y A129853 Cf. A025048, A025049.
%K A129853 nonn
%O A129853 0,3
%A A129853 _Vladeta Jovovic_, May 22 2007
%E A129853 More terms from _R. J. Mathar_, Oct 31 2007
%E A129853 More terms from _Joerg Arndt_, May 21 2013
%E A129853 a(0)=1 prepended by _Alois P. Heinz_, May 23 2023