A345254 Dispersion of A004754, a rectangular array T(n,k) read by downward antidiagonals.

1, 2, 3, 4, 5, 6, 8, 9, 10, 7, 16, 17, 18, 11, 12, 32, 33, 34, 19, 20, 13, 64, 65, 66, 35, 36, 21, 14, 128, 129, 130, 67, 68, 37, 22, 15, 256, 257, 258, 131, 132, 69, 38, 23, 24, 512, 513, 514, 259, 260, 133, 70, 39, 40, 25, 1024, 1025, 1026, 515, 516, 261, 134

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 A000079 or powers of two = 1, 2, 4, 8, 16, ....

Except for the leftmost element "1" of the top row, rows of T(n,k) indexed n = 0, 1, 2, ..., consist entirely of even numbers (A005843) for n even and entirely of odd numbers (A005408) for n odd.

The left column (k = 1) of {T(n,k)} comprises a "1" for the top row (n = 0) and A004755(n) = n + 2^(floor(log_2(n)) + 1), for rows n = 1, 2, 3, ....

For rows indexed n = 0, 1, 2, ..., and columns indexed k = 1, 2, 3, ..., T(n,k) is given by T(0,k) = L^(k - 1)(1) and T(n,k) = L^(k - 1) R(n) for n = 1, 2, 3, ..., the image of n under a composition of branching functions L(n) = A004754(n) = n + 2^floor(log_2(n)) and R(n) = A004755(n) = n + 2^(floor(log_2(n)) + 1) (cf. generating tree A059893 and 2nd linked reference).

(Duality with array A054582): Consider A059893 and A000027 as labeled binary trees arranging the positive integers. In latter tree, node labels equal node positions, thus following their natural order. Rows of {T(n,k)} are the labels along maximal straight paths that always branch left in the former tree, while rows of (transposed) array A054582 are the labels along maximal straight paths that always branch left in the latter tree.

Column k of {T(n,k)} comprises the (sorted) labels in the k-th right clade of latter tree, while column k of (transposed) A054582 comprises the (sorted) labels in the k-th right clade of the former tree. This makes the arrays {T(n,k)} and (transposed) A054582 "blade-duals," blade being a contraction of branch-clade ('right clades' explained under tree A345253 and in 2nd link).

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

1,

2, 3,

4, 5, 6, 7,

8, 9, 10, 11, 12, 13, 14, 15

16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,

...

Analogous to A345252, its right-justified tableau of the positive integers in cohorts with lengths the Fibonacci numbers replaced by the above left-justified tableau with powers of two as lengths of the cohorts.

(Mirror duality): A "mirror dual" I-D array or "inverse I-D array" (see Kimberling, 1st linked reference) is obtained by substituting the left-justified tableau by a right-justified tableau and following the identical procedure, or equivalently by mirroring the tree A059893 cited above, i.e., taking maximal straight paths that always branch right in the tree A059893. With two types of duality then, {T(n,k)} forms a quartet of I-D arrays together with its mirror dual, its blade dual (transposed) A054582 and mirror dual A191448 of the latter.

(Para-sequences): Sequences of row and column indices (see Conway-Sloane correspondence under A019586, citing Kimberling). For rows indexed n = 0, 1, 2, ..., and columns indexed k = 1, 2, 3, ..., the row index n of positive integer T(n,k) is A053645(T) and the column index k of positive integer T(n,k) is A065120(T).

Examples

			Northwest corner of {T(n,k)}:
       k=1   k=2    k=3     k=4      k=5       k=6
  n=0:   1,    2,     4,      8,      16,       32, ...
  n=1:   3,    5,     9,     17,      33,       65, ...
  n=2:   6,   10,    18,     34,      66,      130, ...
  n=3:   7,   11,    19,     35,      67,      131, ...
  n=4:  12,   20,    36,     68,     132,      260, ...
  ...
Northwest corner of {T(n,k)} in base-2:
        k=1  k=2    k=3     k=4      k=5       k=6
  n=0:  1,   10,    100,    1000,    10000,    100000, ...
  n=1:  11,  101,   1001,   10001,   100001,   1000001, ...
  n=2:  110, 1010,  10010,  100010,  1000010,  10000010, ...
  n=3:  111, 1011,  10011,  100010,  1000011,  10000011, ...
  n=4:  1100,10100, 100100, 1000100, 10000100, 100000100, ...
  ...
		

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, 2021.
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A000027, A004754, A053645, A005408, A005843, A019586, A054582, A059893, A065120, A139706, A139708, A191448, A345252, A345253.

Rows n=0..1: A000079, A048578.

Columns k=1..2: A004755, A004757.

Programs

Mathematica

(*Simplified Formula*)
MatrixForm[Prepend[Table[n + 2^(Floor[Log[2, n]] + k), {n, 1, 4}, {k, 1, 6}], Table[2^(k - 1), {k, 1, 6}]]]
(*Branching Formula*)
MatrixForm[Prepend[Table[NestList[Function[# + 2^(Floor[Log[2, #]])], n + 2^(Floor[Log[2, n]] + 1), 5], {n, 1, 4}], NestList[Function[# + 2^(Floor[Log[2, #]])], 1, 5]]]

PARI

T(n, k) = if (n==0, 2^(k-1), n + 2^(log(n)\log(2) + k));
matrix(7, 7, n, k, n--; T(n, k)) \\ Michel Marcus, Jul 30 2021

Formula

T(0,k) = 2^(k - 1) and T(n,k) = n + 2^(floor(log_2(n)) + k) for n >= 1.

T(0,k) = L^(k - 1)(1) and T(n,k) = L^(k - 1) R(n) for n = 1, 2, 3, ..., where L(n) = A004754(n) = n + 2^floor(log_2(n)) and R(n) = A004755(n) = n + 2^(floor(log_2(n)) + 1).

Let b(n) = A054582(n-1). Then for all n >= 1, a(n) = A139706(b(n)) and b(n) = A139708(a(n)).