A222407 -id:A222407 - cp's OEIS frontend

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

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

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

			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).

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

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.

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);

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}]

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

A280426 Digital roots of tetranacci numbers A000078.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 8, 6, 2, 2, 9, 1, 5, 8, 5, 1, 1, 6, 4, 3, 5, 9, 3, 2, 1, 6, 3, 3, 4, 7, 8, 4, 5, 6, 5, 2, 9, 4, 2, 8, 5, 1, 7, 3, 7, 9, 8, 9, 6, 5, 1, 3, 6, 6, 7, 4, 5, 4, 2, 6, 8, 2, 9, 7, 8, 8, 5, 1, 4, 9, 1, 6, 2, 9, 9, 8, 1, 9, 9, 9, 1, 1, 2, 4, 8, 6, 2

Offset: 0

Peter M. Chema, Jan 02 2017

From a(3) onwards, periodic with period length 78, twice the periodicity of the digital roots of tribonacci numbers A222407 (also the 12th triangular number).

Cf. A000078, A010888, A222407.

a(n) = A010888(A000078(n)).

cp's OEIS Frontend

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

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