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A025047
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Number of alternating compositions, i.e., compositions with alternating increases and decreases, starting with either an increase or a decrease.
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155
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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, 37381849, 58750468
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OFFSET
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0,4
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COMMENTS
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Original name: Wiggly sums: number of sums adding to n in which terms alternately increase and decrease or vice versa.
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LINKS
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FORMULA
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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
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EXAMPLE
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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 '-').
(End)
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MAPLE
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b:= proc(n, l, t) option remember; `if`(n=0, 1, add(
b(n-j, j, 1-t), j=`if`(t=1, 1..min(l-1, n), l+1..n)))
end:
a:= n-> 1+add(add(b(n-j, j, i), i=0..1), j=1..n-1):
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MATHEMATICA
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wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], wigQ]], {n, 0, 15}] (* Gus Wiseman, Jun 17 2021 *)
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PROG
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(PARI)
D(n, f)={my(M=matrix(n, n, j, k, k>=j), s=M[, n]); for(b=1, n, f=!f; M=matrix(n, n, j, k, if(k<j, if(f, if(k>1, M[j-k, k-1]), M[j-k, n]-M[j-k, k] ))); for(k=2, n, M[, k]+=M[, k-1]); s+=M[, n]); s~}
seq(n) = concat([1], D(n, 0) + D(n, 1) - vector(n, j, 1)) \\ Andrew Howroyd, Jan 31 2024
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CROSSREFS
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The version allowing pairs (x,x) is A344604.
A032020 counts strict compositions.
A106356 counts compositions by number of maximal anti-runs.
A114901 counts compositions where each part is adjacent to an equal part.
A274174 counts compositions with equal parts contiguous.
A345164 counts alternating permutations of prime indices.
A345165 counts partitions w/o alternating permutation, ranked by A345171.
A345170 counts partitions w/ alternating permutation, ranked by A345172.
Cf. A000070, A008965, A238279, A333755, A344606, A344614, A344653, A344740, A345163, A345166, A345169.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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