A254141 The average of a(n) consecutive Fibonacci numbers is never an integer.

8, 16, 21, 28, 32, 40, 52, 55, 56, 64, 65, 68, 69, 80, 84, 85, 87, 88, 92, 93, 99, 104, 105, 112, 117, 119, 128, 132, 133, 136, 140, 141, 145, 148, 152, 153, 155, 156, 160, 161, 164, 165, 171, 172, 176, 184, 187, 188, 196, 200, 203, 204, 205, 207, 208, 209, 212

Offset

1,1

Comments

Subset of A033949 and A175594 (essentially the same sequence).

Numbers of the form 2^k, with k>=3, appear to be part of the sequence.

The file "List of indexes and steps (k, x, y)" (see Links) for k = 1, 2, 3, 4, ... consecutive Fibonacci numbers gives the minimum index to start to calculate the average ( x ) and the step to add to get all the other averages ( y ).

e.g.: for k = 7 we have 7, 6, 8. This means that we must start from the 6th Fibonacci number to add 7 consecutive Fibonacci numbers and get an average that is integer. Fibonacci(6) + Fibonacci(7) + ... + Fibonacci(12) = 8 + 13 + 21 + 34 + 55 + 89 + 144 = 364 and 364 / 7 = 52.

Then 6 + 1*8 = 14, 6 + 2*8 = 22, 6 + 3*8 = 30, etc. are the other indexes:

Fibonacci(14) + Fibonacci (15) + ... + Fibonacci(20) = 377 + 610 + 987 + 1597 + 2584 + 4181 + 6765 = 17101 and 17101 / 7 = 2443;

Fibonacci(22) + Fibonacci(23) + ... + Fibonacci(28) = 17711 + 28657 + 46368 + 75025 + 121393 + 196418 + 317811 = 803383 and 803383 / 7 = 114769;

Fibonacci(30) + Fibonacci(31) + ... + Fibonacci(36) = 832040 + 1346269 + 2178309 + 3524578 + 5702887 + 9227465 + 14930352 = 37741900 and 37741900 / 7 = 5391700; etc.

In particular we note that:

x = 0 is A219612; x = 1 is A124456; x = 0 and y = k - 1 is A106535;

y = 1 is A141767; x = k - 1 and y = k + 1 is A000057;

x = y - 1 or y|k is A023172; y = k is A000351;

x = y - k + 1 appears to give only prime numbers: 3,11,19,31,59,71,79,131,179,191,239,251,271,311,359,379,419,431,439,479,491,499,571,599,631,659,719,739,751,839,971, etc.

Links

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

Paolo P. Lava, List of indexes and steps (k, x, y)

Maple

with(numtheory); with(combinat):P:=proc(q) local a, b, k, j, n, ok;
for j from 1 to q do b:=0; ok:=1;
for n from 0 to q do a:=add(fibonacci(n+k), k=0..j-1)/j;
if type(a, integer) then ok:=0; break; fi; od;
if ok=1 then print(j); fi; od; end: P(20000);

Crossrefs

Cf. A000045, A000057, A000071, A023172, A033949, A106535, A124456, A141767, A175594, A219612.

Sequence in context: A112679 A212021 A096912 * A188438 A282256 A037370

Adjacent sequences: A254138 A254139 A254140 * A254142 A254143 A254144

Keyword

nonn

Author

Paolo P. Lava, Jan 26 2015

Status

approved