A210456 Period of the sequence of the digital roots of Fibonacci n-step numbers.

1, 24, 39, 78, 312, 2184, 1092, 240, 273, 26232, 11553, 9840, 177144, 14348904, 21523359, 10315734, 48417720, 16120104, 15706236, 5036466318, 258149112, 1162261464, 141214768239, 421900912158, 8857200, 2184, 2271, 28578504864, 21938847432216, 148698308091840

Offset

1,2

Comments

More precisely, start with 0,0,...,0,1 (with n-1 0's and a single 1); thereafter the next term is the digital root (A010888) of the sum of the previous n terms. This is a periodic sequence and a(n) is the length of the period.

Theorem: a(n) <= 9^n.

Conjecture: All entries >1 are divisible by 3.

Additional terms are a(242)=177144, a(243)=177879.

More: a(728)=1594320, a(729)=1596513, a(2186)=14348904, a(2187)=14355471, a(6560)=129140160, a(6561)=129159849, a(19682)=1162261464, a(19683)=1162320519. - Hans Havermann, Jan 30 2013, Feb 08 2013

The modulus-9 Pisano periods of Fibonacci numbers, k-th order sequences. - Hans Havermann, Feb 10 2013

Conjecture: a(3^n-1)=3^(2*n+1)-3, a(3^n)=3^(2*n+1)+3^(n+1)+3 - Fred W. Helenius (fredh(AT)ix.netcom.com), posting to MathFun, Feb 21 2013

Links

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..100

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Eric Weisstein's World of Mathematics, Pisano Period

Example

Digital roots of Fibonacci numbers (A030132) are 0, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9, 1, 1, 2, 3,... Thus the period is 24 (1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9).

Maple

A210456:=proc(q, i)
local d, k, n, v;
v:=array(1..q);
for d from 1 to i do
  for n from 1 to d do v[n]:=0; od; v[d+1]:=1;
  for n from d+2 to q do v[n]:=1+((add(v[k], k=n-d-1..n-1)-1) mod 9);
    if add(v[k], k=n-d+1..n)=9*d and v[n-d]=1 then print(n-d); break;
fi; od; od; end:
A210456 (100000000, 100);

Mathematica

f[n_] := f[n] = Block[{s = PadLeft[{1}, n], c = 1}, s = t = Nest[g, s, n]; While[t = g[t]; s != t, c++]; c]; g[lst_List] := Rest@Append[lst, 1 + Mod[-1 + Plus @@ lst, 9]]; Do[ Print[{n, f[n] // Timing}], {n, 100}]

Crossrefs

Cf. Fibonacci numbers, k-th order sequences, A000045 (Fibonacci numbers, k=2), A030132 (digital root, k=2), A001175 (Pisano periods, k=2), A000073 (tribonacci numbers, k=3), A222407 (digital roots, k=3), A046738 (Pisano periods, k=3), A029898 (Pitoun's sequence), A187772, A220555.

Cf. also A010888.

Sequence in context: A015805 A164534 A047982 * A095158 A271422 A362148

Adjacent sequences: A210453 A210454 A210455 * A210457 A210458 A210459

Keyword

nonn,base

Author

Paolo P. Lava, Robert G. Wilson v, Jan 21 2013

Extensions

a(23) from Hans Havermann, Jan 30 2013

a(24) from Hans Havermann, Feb 18 2013

a(28) from Robert G. Wilson v, Feb 21 2013

a(29)-a(30) from Hiroaki Yamanouchi, May 04 2015

Status

approved