cp's OEIS Frontend

See A194029.

Suppose that c(1), c(2), c(3), ... is an increasing sequence of positive integers with c(1) = 1, and that the sequence c(k+1) - c(k) is strictly increasing. The natural fractal sequence f of c is defined by:

If c(k) <= n < c(k+1), then f(n) = 1 + n - c(k).

This defines the present sequence a(n) = f(n) for c = A000045.

The natural interspersion of c is here introduced as the array given by T(n,k) =(position of k-th n in f). Note that c = (row 1 of T).

As a different example from the one considered here (c = A000045), let c = A000217 = (1, 3, 6, 10, 15, ...), the triangular numbers, so that f = (1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, ...) = A002260, and a northwest corner of T = A194029 is:

1 3 6 10 15 ...

2 4 7 11 16 ...

5 8 12 17 23 ...

9 13 18 24 31 ...

...

Since every number in the set N of positive integers occurs exactly once in this (and every) interspersion, a listing of the terms of T by antidiagonals comprises a permutation, p, of N; letting q denote the inverse of p, we thus have for each c a fractal sequence, an interspersion T, and two permutations of N:

c f T / p q

A000045 A194029 A194030 A194031

A000290 A071797 A194032 A194033

A000217 A002260 A066182 A066181

A028387 A074294 A194034 A194035

A028872 A071797 A194036 A194037

A034856 A002260 A194038 A194040

It appears that this is also a triangle read by rows in which row n lists the first A000045(n) positive integers, n >= 1 (see example). - Omar E. Pol, May 28 2012

This is true, because the sequence c = A000045 has the property that c(k+1) - c(k) = c(k-1), so the number of integers {1, 2, 3, ...} to be filled in from index n = c(k) to n = c(k+1)-1 is equal to c(k-1); see also the first EXAMPLE. - M. F. Hasler, Apr 23 2022

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,

...

Arises naturally from the Fibonacci sort algorithm (see links).

The restriction of this sequence to the first F(k) - 1 entries, where k >= 2 and F is the Fibonacci sequence (A000045), is a permutation.

The entire sequence is a permutation of the positive integers (every positive integer occurs exactly once).

Differs from A194030 starting at n=10 with a(10) = 12 whereas A194030(10) = 11. Despite their similar starts, the two sequences are unrelated.

The terms consist of 1 together with numbers that appear in row m of the Wythoff array (A035513) if m is in the sequence.

a(0) = 1, otherwise a(n) is the number whose Zeckendorf representation is "10" followed by the Zeckendorf representation of n.

If we define an extended Zeckendorf representation to be the Zeckendorf representation with "01" appended, then the numbers in the sequence are exactly those whose extended representation starts 101... . This extended representation is a valid Fibonacci base representation if we specify the rightmost digit to have weight F(0) = 0.

Equivalently, for positive integer m, find the largest k with F(k) <= m, where F(k) is the k-th Fibonacci number. m is in the sequence if and only if m >= F(k) + F(k-2).

Numbers given to rabbits on Rabbit 1's branch of the generation tree described in the A035513 examples.

Equivalently, take the positive integers in turn, placing runs of them alternatively into 2 sets, with run lengths from A053602/A051792 (self-interleaved Fibonacci sequence) as follows:

set A: 1 0 1 1 2 3 5 ...

set B: 1 1 2 3 5 8 ...

The sequence lists the numbers in set A.