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A097873 Reconstruction by Markov matrices of a Hofstadter like (1,4) recurrence. 0
0, 1, 1, 2, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 17, 18, 18, 19, 20, 20, 21, 22, 20, 22, 22, 21, 24, 24, 25, 25, 24, 27, 28, 24, 26, 31, 27, 30, 31, 27, 32, 30, 33, 31, 27, 38 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

(* using a Hofstadter-type (1,4) chaotic recurring sequence as a pattern*)

(* set of solved Markov matrices are used to reconstruct to sequential domain*)

(* using an average Matrix to bridge extremes on nonlinearity*)

(* the reconstruction is extended to 60 from 32 where the first "bad spot" happens*)

(* periodic "bad spots" that run for several matrices in a row exist*)

LINKS

Table of n, a(n) for n=0..60.

FORMULA

A[n_]:=If[(n<33||n>33)&&(n<42||n>45)&&(n<56||n>59), a[[n]].A[n-1], c[[32]].A[n-1]] A[0] = {{0, 1, 1, 2}, {1, 1, 2, 3}, {1, 2, 3, 4}, {2, 3, 4, 5}};

MATHEMATICA

Clear[B, A, M, Hofstadter, ave] digits=60 Hofstadter[n_Integer?Positive] :=Hofstadter[n] =Hofstadter[n - Hofstadter[n-1]] + Hofstadter[n - Hofstadter[n-4]] Hofstadter[1] = Hofstadter[2] =1; Hofstadter[3] =2; Hofstadter[0]= 0; Hofstadter[4]= 3; n0=4 (* pattern matrices of the chaotic sequence*) A[k_]=Table[Hofstadter[k+i+j-2], {i, 1, n0}, {j, 1, n0}] M=Array[f, {n0, n0}] m1=Flatten[M] (* linear Markov matrix solutions *) a=Table[Flatten[M/.Solve[A[n]-M.A[n-1]==0, m1], 1], {n, 1, digits}]; (* function for average matrix calculation*) ave[n_Integer?Positive] :=ave[n] = (ave[n-1]*(n-1)+a[[n]])/n ave[1]=a[[1]]; c=Table[ave[n], {n, 1, 32}]; c[[32]] (* Matrix reconstruction tensors skipping the "bad spots" by substitution of an average matrix*) B[n_]:=If[(n<33||n>33)&&(n<42||n>45)&&(n<56||n>59), a[[n]].B[n-1], c[[32]].B[n-1]] B[0] = {{0, 1, 1, 2}, {1, 1, 2, 3}, {1, 2, 3, 4}, {2, 3, 4, 5}}; (* output sequence of the reconstruction*) b=Flatten[Table[Floor[B[n][[1, 1]]], {n, 0, digits}]] ListPlot[b, PlotJoined->True, PlotRange->All]

CROSSREFS

Cf. A005185.

Sequence in context: A079411 A198454 A063882 * A005375 A138370 A125051

Adjacent sequences:  A097870 A097871 A097872 * A097874 A097875 A097876

KEYWORD

nonn,uned

AUTHOR

Roger L. Bagula, Sep 01 2004

STATUS

approved

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Last modified November 30 03:03 EST 2021. Contains 349416 sequences. (Running on oeis4.)