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

1, 1, 2, 3, 6, 9, 17, 28, 50, 85, 149, 257, 448, 775, 1347, 2336, 4057, 7038, 12219, 21204, 36807, 63880, 110878, 192442, 334020, 579739, 1006237, 1746482, 3031310, 5261324, 9131892, 15849876, 27510049, 47748159, 82874713, 143842547, 249662173, 433329337, 752113633, 1305415658, 2265761441

Offset: 0

Author

Vladeta Jovovic, May 22 2007

Keywords

• nonn

Examples

			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.
From _Joerg Arndt_, May 21 2013: (Start)
The a(6)=17 such compositions are
01:  [ 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 2 ]
03:  [ 1 1 2 1 1 ]
04:  [ 1 1 2 2 ]
05:  [ 1 1 3 1 ]
06:  [ 1 1 4 ]
07:  [ 2 1 1 1 1 ]
08:  [ 2 1 2 1 ]
09:  [ 2 1 3 ]
10:  [ 2 2 2 ]
11:  [ 3 1 1 1 ]
12:  [ 3 1 2 ]
13:  [ 3 3 ]
14:  [ 4 1 1 ]
15:  [ 4 2 ]
16:  [ 5 1 ]
17:  [ 6 ]
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000

Crossrefs

Cf. A129852.

Cf. A025048, A025049.

Programs

Maple

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
# second Maple program:
b:= proc(n, l, t) option remember; `if`(n=0, 1, add(
      b(n-j, j, not t), j=`if`(t, l..n, 1..min(n, l))))
    end:
a:= n-> b(n, 1, true):
seq(a(n), n=0..40);  # Alois P. Heinz, May 23 2023

Mathematica

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]]]}]];
a[n_] := b[n, 1, True];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 16 2023, after Alois P. Heinz *)

Extensions

More terms from R. J. Mathar, Oct 31 2007

More terms from Joerg Arndt, May 21 2013

a(0)=1 prepended by Alois P. Heinz, May 23 2023