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A025047 Wiggly sums: number of sums adding to n in which terms alternately increase and decrease or vice versa. 6
1, 1, 1, 3, 4, 7, 12, 19, 29, 48, 75, 118, 186, 293, 460, 725, 1139, 1789, 2814, 4422, 6949, 10924, 17168, 26979, 42404, 66644, 104737, 164610, 258707, 406588, 639009, 1004287, 1578363, 2480606, 3898599, 6127152, 9629623, 15134213, 23785388 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

I.e., compositions with alternating increases and decreases, starting with either an increase or a decrease. - Franklin T. Adams-Watters, May 27 2010

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..500

FORMULA

a(n) = A025048(n) + A025049(n) - 1 = sum_k[A059881(n, k)] = sum_k[S(n, k) + T(n, k)] - 1 where if n>k>0 S(n, k) = sum_j[T(n - k, j)] over j>k and T(n, k) = sum_j[S(n - k, j)] over k>j (note reversal) and if n>0 S(n, n) = T(n, n) = 1; S(n, k) = A059882(n, k), T(n, k) = A059883(n, k). - Henry Bottomley, Feb 05 2001

a(n) ~ c * d^n, where d = 1.571630806607064114100138865739690782401305155950789062725..., c = 0.82222360450823867604750473815253345888526601460811483897... . - Vaclav Kotesovec, Sep 12 2014

EXAMPLE

From Joerg Arndt, Dec 28 2012: (Start)

There are a(7)=19 such compositions of 7:

[ 1] +  [ 1 2 1 2 1 ]

[ 2] +  [ 1 2 1 3 ]

[ 3] +  [ 1 3 1 2 ]

[ 4] +  [ 1 4 2 ]

[ 5] +  [ 1 5 1 ]

[ 6] +  [ 1 6 ]

[ 7] -  [ 2 1 3 1 ]

[ 8] -  [ 2 1 4 ]

[ 9] +  [ 2 3 2 ]

[10] +  [ 2 4 1 ]

[11] +  [ 2 5 ]

[12] -  [ 3 1 2 1 ]

[13] -  [ 3 1 3 ]

[14] +  [ 3 4 ]

[15] -  [ 4 1 2 ]

[16] -  [ 4 3 ]

[17] -  [ 5 2 ]

[18] -  [ 6 1 ]

[19] 0  [ 7 ]

For A025048(7)-1=10 of these the first two parts are increasing (marked by '+'),

and for A025049(7)-1=8 the first two parts are decreasing (marked by '-').

The composition into one part is counted by both A025048 and A025049.

(End)

CROSSREFS

Cf. A003242. - Franklin T. Adams-Watters, May 27 2010

Sequence in context: A310007 A158237 A117950 * A050342 A293642 A214286

Adjacent sequences:  A025044 A025045 A025046 * A025048 A025049 A025050

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

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Last modified November 21 11:01 EST 2018. Contains 317447 sequences. (Running on oeis4.)