A238422 - OEIS

%I #17 Nov 06 2014 09:53:55

%S 1,1,2,2,5,7,15,23,43,70,128,214,383,651,1149,1971,3457,5961,10412,

%T 18011,31384,54384,94639,164163,285454,495452,861129,1495126,2597970,

%U 4511573,7838280,13613289,23649355,41076088,71354998,123939602,215294730,373962643,649597906,1128352145

%N Number of compositions of n where no consecutive parts differ by 1.

%H Joerg Arndt and Alois P. Heinz, <a href="/A238422/b238422.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) ~ c * d^n, where c = 0.501153706040308227351395770679776260606990346633815... and d = 1.737029107886986816124470304294547513896522086125645631179... - _Vaclav Kotesovec_, Feb 26 2014

%e The a(6) = 15 such compositions are:

%e 01:  [ 1 1 1 1 1 1 ]

%e 02:  [ 1 1 1 3 ]

%e 03:  [ 1 1 3 1 ]

%e 04:  [ 1 1 4 ]

%e 05:  [ 1 3 1 1 ]

%e 06:  [ 1 4 1 ]

%e 07:  [ 1 5 ]

%e 08:  [ 2 2 2 ]

%e 09:  [ 2 4 ]

%e 10:  [ 3 1 1 1 ]

%e 11:  [ 3 3 ]

%e 12:  [ 4 1 1 ]

%e 13:  [ 4 2 ]

%e 14:  [ 5 1 ]

%e 15:  [ 6 ]

%p # b(n, i): number of compositions of n where the leftmost part j

%p #          and i do not have distance 1

%p b:= proc(n, i) option remember; `if`(n=0, 1,

%p       add(`if`(abs(i-j)=1, 0, b(n-j, j)), j=1..n))

%p     end:

%p a:= n-> b(n, -1):

%p seq(a(n), n=0..50);

%t b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[If[Abs[i - j] == 1, 0, b[n - j, j]], {j, 1, n}]]; a[n_] := b[n, -1]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Nov 06 2014, after Maple *)

%Y Cf. A116931 (partitions where no consecutive parts differ by 1).

%K nonn

%O 0,3

%A _Joerg Arndt_ and _Alois P. Heinz_, Feb 26 2014