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Self-inverse permutation of nonnegative integers.

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,

...

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).

Here, the minimal polynomial is required to have integer coefficients with no common divisors.

If two numbers have eventually periodic continued fraction expansions with the same periodic part, the discriminants of their respective minimal polynomials are the same.

a(n) = A087276(n) - n.

a(n) = A087275(n) + n.

Numbers such that the sequence of its binary digits change periodically with linearly increasing period depending of its position: after the first '1', k-th most significant bit changes with period 2*(k-1). For instance, the second most significant bit changes with period 2, third bit changes with period 4, fourth bit with period 6, and so on. For example on the first 15 terms we have:

a(n) Binary digits

1 1

2 1 0

6 1 1 0

8 1 0 0 0

28 1 1 1 0 0

40 1 0 1 0 0 0

104 1 1 0 1 0 0 0

144 1 0 0 1 0 0 0 0

496 1 1 1 1 1 0 0 0 0

672 1 0 1 0 1 0 0 0 0 0

1632 1 1 0 0 1 1 0 0 0 0 0

2240 1 0 0 0 1 1 0 0 0 0 0 0

7872 1 1 1 1 0 1 1 0 0 0 0 0 0

11648 1 0 1 1 0 1 1 0 0 0 0 0 0 0

27520 1 1 0 1 0 1 1 1 0 0 0 0 0 0 0

/ | | | | | \ etc.

MSB / | | | | Period 12

/ | | | \

Period 2 / | | Period 10

/ | \

Period 4 | Period 8

/

Period 6

Regarding the periodicity of the binary digits, this sequence is similar to A059893 where the periodicity of its binary digits are powers of two.

By truncating the least significant bits in such a way to leave only k most significant bits of a(n) with n>k-1, it is obtained a periodic sequence with period p given by p=Least common multiple (LCM) of {2,4,6,..,2k}. In general any subsequence obtained by a selection of a subset of its most significant bits including the most significant bit is periodic.

T(n,k) is the swapping of the positions of the bits in n according to the active bits in K, where K = A057716(k). The bit in the same position as the first active bit in K switches positions with the bit in the same position as the last active bit in K, the bit in the same position as the second active bit in K switches with the one in the same as the second to last position, and so on until all have swapped (without repeating).

Any sequence, f, of the form "reverse the order of the a-th, b-th, ... and z-th bits in n" can be expressed as f(n) = T(n,k), where A057716(k) = 2^a + 2^b + ... 2^z. As a result, this operation combines 1 or more bit-swapping operations, which could be useful for bit-manipulation in computer programming.