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A214259 Number of compositions of n where the difference between largest and smallest parts equals one.

%I A214259 #19 Jan 10 2019 20:14:44
%S A214259 0,0,2,3,9,11,25,35,60,96,157,241,401,637,1019,1639,2651,4258,6870,
%T A214259 11075,17891,28895,46678,75412,121915,197109,318724,515414,833590,
%U A214259 1348301,2181020,3528138,5707564,9233625,14938477,24168522,39102322,63264680,102358836
%N A214259 Number of compositions of n where the difference between largest and smallest parts equals one.
%H A214259 Alois P. Heinz, <a href="/A214259/b214259.txt">Table of n, a(n) for n = 1..4785</a>
%F A214259 a(n) = A072951(n) - A000005(n).
%F A214259 a(n) ~ phi^(n+1) / sqrt(5), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jan 07 2019
%e A214259 a(3) = 2: [2,1], [1,2].
%e A214259 a(4) = 3: [2,1,1], [1,2,1], [1,1,2].
%e A214259 a(5) = 9: [3,2], [2,3], [2,2,1], [2,1,2], [1,2,2], [2,1,1,1], [1,2,1,1], [1,1,2,1], [1,1,1,2].
%e A214259 a(6) = 11: [2,2,1,1], [2,1,2,1], [2,1,1,2], [1,2,2,1], [1,2,1,2], [1,1,2,2], [2,1,1,1,1], [1,2,1,1,1], [1,1,2,1,1], [1,1,1,2,1], [1,1,1,1,2].
%p A214259 with(numtheory):
%p A214259 a:= n-> add(binomial(t, n mod t), t=1..n) -tau(n):
%p A214259 seq(a(n), n=1..50);
%Y A214259 Column k=1 of A214258.
%Y A214259 Cf. A000005, A001622, A072951.
%K A214259 nonn
%O A214259 1,3
%A A214259 _Alois P. Heinz_, Jul 08 2012