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 A210456 Period of the sequence of the digital roots of Fibonacci n-step numbers. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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 A062374 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

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Last modified February 19 19:30 EST 2020. Contains 332047 sequences. (Running on oeis4.)