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A226841
Minimum k such that (F(n) - 1) | sum_{i=n..n+k} F(i), where F(i) are Fibonacci numbers (A000045).
2
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
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
Sequence in context: A281899 A037457 A037314 * A218560 A373261 A031443
KEYWORD
nonn
AUTHOR
Paolo P. Lava, Jun 19 2013
STATUS
approved