A226841 Minimum k such that (F(n) - 1) | sum_{i=n..n+k} F(i), where F(i) are Fibonacci numbers (A000045).

1, 1, 2, 9, 10, 11, 22, 16, 38, 17, 58, 81, 82, 55, 110, 64, 142, 69, 178, 217, 218, 131, 262, 144, 310, 153, 362, 417, 418, 239, 478, 256, 542, 269, 610, 681, 682, 379, 758, 400, 838, 417, 922, 1009, 1010, 551, 1102, 576, 1198, 597, 1298, 1401, 1402, 755, 1510

Offset

1,3

Comments

a(8*n) = 16*n^2, for n>0.

a(8*n + 2) = a(8*n + 1)/2-2, for n>=1.

a(8*n + 3) = 32*n^2 + 24*n + 2, for n>=0.

a(8*n + 5) = a(8*n + 4) + 1, for n>=0.

a(8*n + 7) = 2*a(8*n + 6), for n>=0.

Values of Sum_{i=n..n+k}{F(i)} / A000071(n) are listed in A226842.

Links

Paolo P. Lava, Table of n, a(n) for n = 1..200

Example

The sum of first 10 Fibonacci numbers is 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143. We need to add at least 17 consecutive Fibonacci numbers, starting from F(11)=89, in order to have 89 + 144 + 233 + 377 + 610 + 987 + 1597 + 2584 + 4181 + 6765 + 10946 + 17711 + 28657 + 46368 + 75025 + 121393 + 196418 = 514085 and 514085 / 143 = 3595.

Maple

with(numtheory); with(combinat); ListA226841:= proc(q)
local n, a, b, k, p; a:=0;
for n from 1 to q do a:=a+fibonacci(n); b:=fibonacci(n+1); k:=1;
  while not type(b/a, integer) do k:=k+1; b:=b+fibonacci(n+k); od; print(k); od; end: ListA226841(10^4);

Crossrefs

Cf. A000045, A000071.

Sequence in context: A281899 A037457 A037314 * A218560 A373261 A031443

Adjacent sequences: A226838 A226839 A226840 * A226842 A226843 A226844

Keyword

nonn

Author

Paolo P. Lava, Jun 19 2013

Status

approved