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A350240 Number of representations of n as a sum of distinct Fibonacci numbers where 1 can be included twice.

%I A350240 #15 Dec 22 2021 14:14:58
%S A350240 1,1,2,2,2,3,2,3,3,3,4,3,3,4,3,5,4,4,5,3,4,4,4,6,5,5,6,4,6,5,5,6,4,4,
%T A350240 5,4,7,6,6,8,5,7,6,6,8,6,6,7,5,8,6,6,7,4,5,5,5,8,7,7,9,6,9,8,8,10,7,7,
%U A350240 8,6,10,8,8,10,6,8,7,7,10,8,8,9,6
%N A350240 Number of representations of n as a sum of distinct Fibonacci numbers where 1 can be included twice.
%C A350240 A part of size 1 can be included twice in the partitions enumerated by this sequence, but there is only 1 way to include it once. The sequence A000119 only allows 1 to be included once and A000121 allows it to be included twice, but it in two different ways once.
%C A350240 Some connections with the upper Wythoff sequence (A001950):
%C A350240   a(n) = A000121(n) for n in A001950.
%C A350240   a(n) = A000119(n) for n-1 in A001950.
%C A350240   a(n) = A000121(n) - A000119(n) for n+1 in A001950.
%F A350240 G.f.: (1 + x + x^2)*Product_{k>=3} (1 + x^Fibonacci(k)). - _Andrew Howroyd_, Dec 21 2021
%e A350240 The a(10)=4 partitions are: 8+2 = 8+1+1 = 5+3+1+1 = 5+3+2.
%e A350240 The a(11)=3 partitions are: 8+3 = 8+2+1 = 5+3+2+1.
%e A350240 The a(12)=3 partitions are: 8+3+1 = 8+2+1+1 = 5+3+2+1+1.
%o A350240 (PARI) seq(n)=my(m=2); while(fibonacci(m)<n, m++); Vec((1 + x + x^2 + O(x*x^n))*prod(k=3, m, 1 + x^fibonacci(k) + O(x*x^n))) \\ _Andrew Howroyd_, Dec 21 2021
%Y A350240 Cf. A000045, A000119, A000121, A001950, A003107, A007000, A239002.
%K A350240 nonn
%O A350240 0,3
%A A350240 _Kung Yue Tong_, Dec 21 2021