A282772 Starting from F(n), minimum number, greater than 1, of consecutive Fibonacci numbers whose average is an integer.

4, 2, 3, 12, 2, 13, 3, 2, 6, 5, 2, 12, 4, 2, 3, 12, 2, 24, 3, 2, 6, 24, 2, 12, 4, 2, 3, 12, 2, 5, 3, 2, 6, 13, 2, 12, 4, 2, 3, 5, 2, 24, 3, 2, 5, 24, 2, 12, 4, 2, 3, 12, 2, 24, 3, 2, 6, 24, 2, 5, 4, 2, 3, 12, 2, 24, 3, 2, 6, 5, 2, 12, 4, 2, 3, 12, 2, 24, 3, 2, 6

Offset

0,1

Comments

Entries are 2, 3, 4, 5, 6, 12, 13 and 24.

Periodic with period equal to 420.

Links

Paolo P. Lava, Table of n, a(n) for n = 0..1000

Formula

a(3*k + 1) = 2;

a(12*k + 2) = a(12*k + 6) = 3;

a(12*k) = 4;

a(30*k + 9) = a(30*k + 29) = a(60*k + 44) = 5;

a(60*k + 8) = a(60*k + 20) = a(60*k + 32) = a(60*k + 56) = 6;

a(60*k + 3) = a(60*k + 11) = a(60*k + 15) = a(60*k + 23) = a(60*k + 27) = a(60*k + 35) = a(60*k + 47) = a(60*k + 51) = 12;

a(420*k + 5) = a(420*k + 33) = a(420*k + 117) = a(420*k + 173) = a(420*k + 201) = a(420*k + 257) = a(420*k + 285) = a(420*k + 341) = 13;

a(420*k + x) = 24, with x = 17, 21, 41, 45, 53, 57, 65, 77, 81, 93, 101, 105, 113, 125, 137, 141, 153, 161, 165, 177, 185, 197, 213, 221, 225, 233, 237, 245, 261, 273, 281, 293, 297, 305, 317, 321, 333, 345, 353, 365, 377, 381, 393, 401, 405, 413, 417.

Example

a(0) = 4 because F(0) + F(1) + F(2) + F(3) = 0 + 1 + 1 + 2 = 4 and 4/4 = 1;
a(1) = 2 because F(1) + F(2) = 1 + 1 = 2 and 2/2 = 1;
a(2) = 3 because F(2) + F(3) + F(4) = 1 + 2 + 3 = 6 and 6/3 = 2;
a(3) = 12 because F(3) + F(4) + ... + F(13) + F(14) =  2 + 3 + ... + 233 + 377 = 984 and 984/12 = 82.

Maple

with(combinat): P:=proc(q) local a, k, n; for k from 0 to q do a:=fibonacci(k); for n from 1 to q do a:=a+fibonacci(k+n);
if type(a/(n+1), integer) then print(n+1); break; fi; od; od; end: P(10^3);

Mathematica

Table[k = 1; While[! IntegerQ@ Mean@ Take[#, n ;; n + k], k++]; k + 1, {n, Length@ # - 24}] &@ Fibonacci@ Range[0, 419] (* Michael De Vlieger, Mar 06 2017 *)

Crossrefs

Cf. A000045, A101907, A111035, A254141.

Sequence in context: A026216 A292411 A069547 * A227041 A341055 A098046

Adjacent sequences: A282769 A282770 A282771 * A282773 A282774 A282775

Keyword

nonn,easy

Author

Paolo P. Lava, Mar 03 2017

Status

approved