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%I #11 Jan 06 2023 20:46:30
%S 19,45,71,160,414,1084
%N a(n) is the first number that is the sum of 5 distinct positive Fibonacci numbers in exactly n ways (with a single type of 1).
%C a(n) is the least k such that A358005(k) = n.
%C Conjecture: there are no other terms.
%e a(1) = 19 because 19 = 1 + 2 + 3 + 5 + 8 and no smaller number is the sum of 5 distinct positive Fibonacci numbers in exactly 1 way.
%e a(2) = 45 because 45 = 1 + 2 + 3 + 5 + 34
%e = 1 + 2 + 8 + 13 + 21 and no smaller number is the sum of 5 distinct positive Fibonacci numbers in exactly 2 ways.
%e a(3) = 71 because 71 = 1 + 2 + 5 + 8 + 55
%e = 1 + 2 + 13 + 21 + 34
%e = 3 + 5 + 8 + 21 + 34 and no smaller number is the sum of 5 distinct positive Fibonacci numbers in exactly 3 ways.
%e a(4) = 160 because 160 = 1 + 2 + 5 + 8 + 144
%e = 1 + 2 + 13 + 55 + 89
%e = 3 + 5 + 8 + 55 + 89
%e = 3 + 13 + 21 + 34 + 89 and no smaller number is the sum of 5 distinct positive Fibonacci numbers in exactly 4 ways.
%e a(5) = 414 because 414 = 1 + 2 + 13 + 21 + 377
%e = 1 + 2 + 34 + 144 + 233
%e = 3 + 5 + 8 + 21 + 377
%e = 3 + 13 + 21 + 144 + 233
%e = 3 + 34 + 55 + 89 + 233 and no smaller number is the sum of 5 distinct positive Fibonacci numbers in exactly 5 ways.
%e a(6) = 1084 because 1084 = 1 + 2 + 5 + 89 + 987
%e = 3 + 5 + 34 + 55 + 987
%e = 3 + 5 + 89 + 377 + 610
%e = 8 + 13 + 21 + 55 + 987
%e = 8 + 34 + 55 + 377 + 610
%e = 8 + 89 + 144 + 233 + 610 and no smaller number is the sum of 5 distinct positive Fibonacci numbers in exactly 6 ways.
%p G:= mul(1+t*x^combinat:-fibonacci(k),k=2..17):
%p S:= coeff(expand(G),t,5):
%p V:= Vector(6):
%p for i from 19 to combinat:-fibonacci(18) do
%p v:= coeff(S,x,i);
%p if v > 0 and V[v] = 0 then V[v]:= i fi
%p od:
%p convert(V,list);
%Y Cf. A358005.
%K nonn,more
%O 1,1
%A _Robert Israel_, Jan 06 2023