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

%I A210456 #69 Feb 16 2025 08:33:17
%S A210456 1,24,39,78,312,2184,1092,240,273,26232,11553,9840,177144,14348904,
%T A210456 21523359,10315734,48417720,16120104,15706236,5036466318,258149112,
%U A210456 1162261464,141214768239,421900912158,8857200,2184,2271,28578504864,21938847432216,148698308091840
%N A210456 Period of the sequence of the digital roots of Fibonacci n-step numbers.
%C A210456 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.
%C A210456 Theorem: a(n) <= 9^n.
%C A210456 Conjecture: All entries >1 are divisible by 3.
%C A210456 Additional terms are a(242)=177144, a(243)=177879.
%C A210456 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
%C A210456 The modulus-9 Pisano periods of Fibonacci numbers, k-th order sequences. - _Hans Havermann_, Feb 10 2013
%C A210456 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
%H A210456 Hiroaki Yamanouchi, <a href="/A210456/b210456.txt">Table of n, a(n) for n = 1..100</a>
%H A210456 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>
%H A210456 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PisanoPeriod.html">Pisano Period</a>
%e A210456 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).
%p A210456 A210456:=proc(q,i)
%p A210456 local d,k,n,v;
%p A210456 v:=array(1..q);
%p A210456 for d from 1 to i do
%p A210456   for n from 1 to d do v[n]:=0; od; v[d+1]:=1;
%p A210456   for n from d+2 to q do v[n]:=1+((add(v[k],k=n-d-1..n-1)-1) mod 9);
%p A210456     if add(v[k],k=n-d+1..n)=9*d and v[n-d]=1 then print(n-d); break;
%p A210456 fi; od; od; end:
%p A210456 A210456 (100000000,100);
%t A210456 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}]
%Y A210456 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.
%Y A210456 Cf. also A010888.
%K A210456 nonn,base
%O A210456 1,2
%A A210456 _Paolo P. Lava_, _Robert G. Wilson v_, Jan 21 2013
%E A210456 a(23) from _Hans Havermann_, Jan 30 2013
%E A210456 a(24) from _Hans Havermann_, Feb 18 2013
%E A210456 a(28) from _Robert G. Wilson v_, Feb 21 2013
%E A210456 a(29)-a(30) from _Hiroaki Yamanouchi_, May 04 2015