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%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
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