A253298 Digital root for the following sequences, F(4*n)/F(4); F(12*n)/F(12); F(20*n)/F(20), where the pattern increases by 8, ad infinitum, with the Fibonacci numbers F = A000045.

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

Offset: 1

Author

Peter M. Chema, Dec 30 2014

Keywords

• nonn
• easy

Comments

Cyclical and palindromic in two parts with periodicity 18: {1, 7, 3, 5, 5, 3, 7, 1, 9, 8, 2, 6, 4, 4, 6, 2, 8, 9}.

Digital root of the period is 9, its mean and median is 5, and its product is (9!)^2.

See A253368 for the initial motivation for this sequence.

From Peter M. Chema, Jul 04 2016: (Start)

A composite of three respective digital root sequences in alternation: a "halving sequence" of 1, 5, 7, 8, 4, 2, a "doubling sequence" of 7, 5, 1, 2, 4, 8, and a three-six-nine circuit of 3, 3, 9, 6, 6, 9.

Also the digital root of A000045(4n)/3 or A004187(n). In general terms, sequences defined by Fib(x*n)/ Fib(x) where x=(8*a-4) all share the same digital root (e.g., F(4*n)/F(4); F(12*n)/F(12); F(20*n)/F(20); F(28*n)/F(28); F(36*n)/F(36), etc.) (End)

Links

• Tom Barnett, Phi VBM Tori Array, YouTube video (see first two minutes).
• Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1).

Crossrefs

Cf. A010888, A253368, A070366 and A001370.

Programs

Mathematica

f[n_] := Mod[ Fibonacci[ 12n]/144, 9]; Array[f, 5*18] (* Robert G. Wilson v, Jan 23 2015 *)
LinearRecurrence[{1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1},{1, 7, 3, 5, 5, 3, 7, 1, 9, 8, 2, 6, 4, 4, 6, 2, 8},72] (* Ray Chandler, Aug 12 2015 *)

Formula

a(n) = A010888(A253368(n)).

G.f.: x*(1 + 7*x + 3*x^2 + 5*x^3 + 5*x^4 + 3*x^5 + 7*x^6 + x^7 + 9*x^8 + 8*x^9 + 2*x^10 + 6*x^11 + 4*x^12 + 4*x^13 + 6*x^14 + 2*x^15 + 8*x^16 + 9*x^17)/(1 - x^18). - Vincenzo Librandi, Mar 28 2016

Extensions

Edited. Numbers and name changed to fit A253368. Formula adapted. Cross reference added. - Wolfdieter Lang, Jan 28 2015

Name generalized by Peter M. Chema, Jul 04 2016