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A097964 Rectangular array read by rows (n > 0, 1 <= k <= 3): T(n,k) = floor(b(n,k)/2^((A002264(n) + 1)/3)), where b(n,k) = b(n-3,k) + 3*b (n-6,k) + 2*b(n-9,k), with initial values given in comments. 1
2, 5, 7, 3, 5, 8, 2, 3, 6, 5, 8, 13, 6, 11, 17, 4, 8, 12, 10, 17, 27, 12, 21, 34, 9, 15, 24, 20, 34, 54, 25, 42, 68, 18, 30, 49, 40, 68, 108, 50, 85, 136, 36, 61, 97, 80, 135, 216, 101, 170, 271, 72, 121, 194, 160, 270, 430, 201, 339, 541, 144, 242, 387 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

From Franck Maminirina Ramaharo, Nov 08 2018: (Start)

The initial values for b(n,k), 1 <= n <= 9, 1 <= k <= 3, are

n\k |  1  2  3

----+---------

  1 |  4  8 12

  2 |  5  8 13

  3 |  4  6 10

  4 | 10 16 26

  5 | 13 22 35

  6 |  9 16 25

  7 | 26 44 70

  8 | 32 54 86

  9 | 23 38 61. (End)

LINKS

Table of n, a(n) for n=1..63.

FORMULA

From Franck Maminirina Ramaharo, Nov 08 2018: (Start)

Let M and A denote the following 3 X 3 matrices:

      0, 2, 0

  M = 1, 1, 1

      1, 1, 0

and

      0, 1, 1

  A = 1, 1, 2

      1, 2, 3.

Then applying floor() to the entries in (h*M)^(n + 1)*A, where h = 1/(2^(1/3)), yields row 3*n - 2 to 3*n. (End)

EXAMPLE

Array begins:

   2,  5,  7;

   3,  5,  8;

   2,  3,  6;

   5,  8, 13;

   6, 11, 17;

   4,  8, 12;

  10, 17, 27;

  12, 21, 34;

   9, 15, 24;

  20, 34, 54;

  25, 42, 68;

  18, 30, 49;

   ... - Franck Maminirina Ramaharo, Nov 08 2018

MATHEMATICA

M = N[4^(1/3)*({{0, 1, 0}, {1, 1, 0}, {0, 0, 0}}/2 + {{0, 1, 0}, {0, 0, 1}, {1, 1, 0}}/2)];

A[n_] := M.A[n - 1]; A[0] := {{0, 1, 1}, {1, 1, 2}, {1, 2, 3}};

Table[Floor[M.A[n]], {n, 1, 12}]//Flatten

CROSSREFS

Cf. A097966.

Sequence in context: A286463 A286362 A309200 * A133133 A024710 A140264

Adjacent sequences:  A097961 A097962 A097963 * A097965 A097966 A097967

KEYWORD

nonn,tabf,less

AUTHOR

Roger L. Bagula, Sep 06 2004

EXTENSIONS

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Nov 08 2018

STATUS

approved

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Last modified May 7 18:29 EDT 2021. Contains 343652 sequences. (Running on oeis4.)