A359558 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).

19, 45, 71, 160, 414, 1084

Offset

1,1

Comments

a(n) is the least k such that A358005(k) = n.

Conjecture: there are no other terms.

Links

Table of n, a(n) for n=1..6.

Example

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.
a(2) = 45 because 45 = 1 + 2 + 3 +  5 + 34
                     = 1 + 2 + 8 + 13 + 21 and no smaller number is the sum of 5 distinct positive Fibonacci numbers in exactly 2 ways.
a(3) = 71 because 71 = 1 + 2 +  5 +  8 + 55
                     = 1 + 2 + 13 + 21 + 34
                     = 3 + 5 +  8 + 21 + 34 and no smaller number is the sum of 5 distinct positive Fibonacci numbers in exactly 3 ways.
a(4) = 160 because 160 = 1 +  2 +  5 +  8 + 144
                       = 1 +  2 + 13 + 55 +  89
                       = 3 +  5 +  8 + 55 +  89
                       = 3 + 13 + 21 + 34 +  89 and no smaller number is the sum of 5 distinct positive Fibonacci numbers in exactly 4 ways.
a(5) = 414 because 414 = 1 +  2 + 13 +  21 + 377
                       = 1 +  2 + 34 + 144 + 233
                       = 3 +  5 +  8 +  21 + 377
                       = 3 + 13 + 21 + 144 + 233
                       = 3 + 34 + 55 +  89 + 233 and no smaller number is the sum of 5 distinct positive Fibonacci numbers in exactly 5 ways.
a(6) = 1084 because 1084 = 1 +  2 +   5 +  89 + 987
                         = 3 +  5 +  34 +  55 + 987
                         = 3 +  5 +  89 + 377 + 610
                         = 8 + 13 +  21 +  55 + 987
                         = 8 + 34 +  55 + 377 + 610
                         = 8 + 89 + 144 + 233 + 610 and no smaller number is the sum of 5 distinct positive Fibonacci numbers in exactly 6 ways.

Maple

G:= mul(1+t*x^combinat:-fibonacci(k), k=2..17):
S:= coeff(expand(G), t, 5):
V:= Vector(6):
for i from 19 to combinat:-fibonacci(18) do
 v:= coeff(S, x, i);
 if v > 0 and V[v] = 0 then V[v]:= i fi
od:
convert(V, list);

Crossrefs

Cf. A358005.

Sequence in context: A044477 A140680 A183632 * A245864 A183624 A115249

Adjacent sequences: A359555 A359556 A359557 * A359559 A359560 A359561

Keyword

nonn,more

Author

Robert Israel, Jan 06 2023

Status

approved