A345252 2-1-Fibonacci cohort array, a rectangular array T(n,k) read by downward antidiagonals.

1, 2, 3, 4, 6, 5, 7, 11, 10, 8, 12, 19, 18, 16, 9, 20, 32, 31, 29, 17, 13, 33, 53, 52, 50, 30, 26, 14, 54, 87, 86, 84, 51, 47, 27, 15, 88, 142, 141, 139, 85, 81, 48, 28, 21, 143, 231, 230, 228, 140, 136, 82, 49, 42, 22, 232, 375, 374, 372, 229, 225, 137, 83, 76

Offset: 1

Author

J. Parker Shectman, Jun 12 2021

Keywords

• nonn
• tabl
• easy

Comments

As a sequence, {a(n)} permutes the positive integers. As an array, {T(n,k)} is an interspersion-dispersion or I-D array (refer to Kimberling, 1st linked reference).

The top row of {T(n,k)} is A000071 or one less than the Fibonacci numbers = 1, 2, 4, 7, 12, ....

The top row of {T(n,k)} alternates lower and upper Wythoff numbers, A000201 and A001950, respectively, while all subsequent rows consist entirely of lower Wythoff numbers or entirely of upper Wythoff numbers (cf. generating tree A345253).

The left column (k = 1) of {T(n,k)} is n + F(Finv(n) + 1), for rows indexed n = 0, 1, 2, ..., where F(n) = A000045(n) are the Fibonacci numbers and Finv(n) = A130233(n) is the 'lower' Fibonacci inverse.

For rows indexed n = 0, 1, 2, ..., and columns indexed k = 1, 2, 3, ..., T(n,k) is given by T(n,k) = L^(k - 1) R(n), the image of n under a composition of branching functions L(n) = n + F(Finv(n)) and R(n) = n + F(Finv(n) + 1) (cf. generating tree A345253).

Write the positive integers in natural order (A000027) as a right-justified "tetrangle" or "irregular triangle" tableau with F(t) (Fibonacci number) entries on each row t, for t = 1, 2, 3, .... Then, columns of the tableau equal rows of {T(n,k)} (2nd linked reference):

1,

2,

3, 4,

5, 6, 7,

8, 9, 10, 11, 12,

13, 14, 15, 16, 17, 18, 19, 20,

...

(Duality with array A194030): With the right-justified tableau substituted by a left-justified tableau, the above procedure yields the array A194030, making {T(n,k)} and A194030 "cohort-dual" arrays, where "cohort" t is the F(t)-sized block (F(t + 1), ..., F(t + 2) - 1) of successive positive integers on row t of both tableaux (2nd link under references, which calls A194030 the "1-2-Fibonacci cohort array", by analogy).

Analogous to A345254, its left-justified tableau of the positive integers in cohorts with lengths powers of two replaced by the above right-justified tableau with the Fibonacci numbers as lengths of the cohorts.

(Duality with array A083047): Consider the labeled binary trees A345253(n) = A232560(A059893(n)) and A232560(n) = A345253(A059893(n)). Rows of array {T(n,k)} are the labels along maximal straight paths that always branch left in A345253, while rows of array A083047 are the labels along maximal straight paths that always branch left in A232560.

Column k of {T(n,k)} comprises the (sorted) labels in the k-th right clade of binary tree A232560, while column k of A083047 comprises the (sorted) labels in the k-th right clades of binary tree A345253. This makes the arrays {T(n,k)} and A083047 "blade-duals", blade being a contraction of branch-clade (cf. A345253 and 2nd linked reference).

(Mirror duality): A "mirror dual" I-D array or "inverse I-D array" (see Kimberling in 1st linked reference) is obtained by mirroring the tree A345253 cited above, i.e., taking maximal straight paths that always branch right in A345253. With three types of duality then, {T(n,k)} forms an octet of I-D arrays together with its cohort dual A194030 and mirror duals of these two, its blade dual A083047 and cohort dual A035513 of the latter, and their respective mirror duals, A132817 and A191436.

(Para-sequences): Sequences of row and column indices (see Conway-Sloane correspondence under A019586, citing Kimberling). For rows indexed n = 0, 1, 2, ..., the row index n of positive integer T(n,k) follows a right-justified tableau with F(t) entries on each row t, for t = 1, 2, 3, ..., in which an entry on row t equals the entry immediately above it on row t-1, if such exists, or otherwise equals the minimum positive integer excluded from the tableau so far:

0,

0,

1, 0,

2, 1, 0,

3, 4, 2, 1, 0,

5, 6, 7, 3, 4, 2, 1, 0,

...

For columns indexed k = 1, 2, 3, ..., the column index k of positive integer T(n,k) follows a right-justified tableau with F(t) entries on each row t, for t = 1, 2, 3, ..., in which an entry x+1 on row t equals one plus the entry x immediately above it on row t-1, if such exists, or otherwise equals one:

1,

2,

1, 3,

1, 2, 4,

1, 1, 2, 3, 5,

1, 1, 1, 2, 2, 3, 4, 6,

...

Examples

			Northwest corner of {T(n,k)}:
       k=1 k=2 k=3 k=4 k=5  k=6 ...
  n=0:   1,  2,  4,  7, 12,  20, ...
  n=1:   3,  6, 11, 19, 32,  53, ...
  n=2:   5, 10, 18, 31, 52,  86, ...
  n=3:   8, 16, 29, 50, 84, 139, ...
  n=4:   9, 17, 30, 51, 85, 140, ...
  ...
Northwest corner of {T(n,k)} in maximal Fibonacci expansion (see link):
       k=1             k=2                  k=3                       ...
  n=0: F(1),           F(1)+F(2),           F(1)+F(2)+F(3),           ...
  n=1: F(1)+F(3),      F(1)+F(3)+F(4),      F(1)+F(3)+F(4)+F(5),      ...
  n=2: F(1)+F(2)+F(4), F(1)+F(2)+F(4)+F(5), F(1)+F(2)+F(4)+F(5)+F(6), ...
  ...
Northwest corner of {T(n,k)} as "Fibonacci gaps," or differences between successive indices in maximal Fibonacci expansion above, (see link):
       k=1   k=2    k=3     k=4      k=5       k=6  ...
  n=0:   *,    1,    11,    111,    1111,    11111, ...
  n=1:   2,   21,   211,   2111,   21111,   211111, ...
  n=2:  12,  121,  1211,  12111,  121111,  1211111, ...
  n=3:  22,  221,  2211,  22111,  221111,  2211111, ...
  n=4: 122, 1221, 12211, 122111, 1221111, 12211111, ...
  ...
		

Links

• Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
• Parker Shectman, A Quilt after Fibonacci, Part 2 of 3: Cohorts, Free Monoids, and Numeration

Crossrefs

Cf. A000027, A000045, A000071, A000201, A001950, A035513, A059893, A083047, A130233, A132817, A191436, A194030, A232560, A345253, A345254.

Programs

Mathematica

(* Define A000045 *)
F[n_] := Fibonacci[n]
(* Defined A130233 *)
Finv[n_] := Floor[Log[GoldenRatio, Sqrt[5]n + 1]]
(* Simplified Formula *)
MatrixForm[Table[n + F[Finv[n] + k + 2] - F[Finv[n] + 2], {n, 0, 4}, {k, 1, 6}]]
(* Branching Formula *)
MatrixForm[Table[NestList[Function[# + F[Finv[#]]], n + F[Finv[n] + 1], 5], {n, 0, 4}]]

Formula

T(n,k) = n + F(Finv(n) + k + 2) - F(Finv(n) + 2), for rows indexed n = 0, 1, 2, ... and columns indexed k = 1, 2, 3, ..., where F(n) = A000045(n) and Finv(n) = A130233.

T(n,k) = L^(k - 1) R(n), where L(n) = n + F(Finv(n)) and R(n) = n + F(Finv(n) + 1).