cp's OEIS Frontend

Row 1: A005578

Row 2: A160113

For a background discussion of dispersions, see A191426.

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Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The six sequences and dispersions are listed here:

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A191663=dispersion of A042948 (0 or 1 mod 4 and >1)

A054582=dispersion of A005843 (0 or 2 mod 4 and >1; evens)

A191664=dispersion of A014601 (0 or 3 mod 4 and >1)

A191665=dispersion of A042963 (1 or 2 mod 4 and >1)

A191448=dispersion of A005408 (1 or 3 mod 4 and >1, odds)

A191666=dispersion of A042964 (2 or 3 mod 4)

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EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):

A191663 has 1st col A042964, all else A042948

A054582 has 1st col A005408, all else A005843

A191664 has 1st col A042963, all else A014601

A191665 has 1st col A014601, all else A042963

A191448 has 1st col A005843, all else A005408

A191666 has 1st col A042948, all else A042964

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There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):

If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by

a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by

a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.

The sequence is a permutation of the odd natural numbers, since A(n,k) = 2*A054582(n-1,k-1) - 1 and A054582 is a permutation of the natural numbers.

For j a natural number, 2*j - 1 appears in row A001511(j) of A.

This is the square array A075300 with the first row omitted. - Peter Bala, Feb 07 2017

Every integer occurs in A191362 (infinitely many times).

Represent the array as {f(i,j): i>=1, j>=1}. Then for m>=0, (diagonal #m) is the sequence (f(i,i+m)), i>=1;

for m<0, (diagonal #m) is the sequence (f(i+m,i)), i>=1.

Lemma: The sequence is a permutation of A008586\{0} = {4*m : m = 1,2,...}.

Proof: Write A(n,k)/4 = A054582(n-1,k-1). The sequence A054582 is known to be a permutation of the natural numbers, and the result follows. QED

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).

This is +2 at the bit position of the odd part of n, that being the least significant 1-bit.

The least significant run of 1-bits changes from 0111..111 in n to 1000..001 in a(n).

Arrays A054582 and A135764 arrange terms into rows having the same number of trailing 0 bits. a(n) is the term to the right of n, i.e., next in its row.

The nonzero Fibbinary numbers (A003714) arranged in rows where each successive term is twice the preceding term; a (transposed) Fibbinary equivalent of A054582.

Write the first term in each row as Sum_{i in S} 2^i, where S is a set of nonnegative integers, then n = Sum_{i in S} F_i, where F_i is the i-th Fibonacci number, A000045(i).

More generally, if the terms are represented in binary, and the binary weighting of the digits (2^0, 2^1, 2^2, ...) is replaced with Fibonacci weighting (F_0, F_1, F_2, ...), we get the extended Wythoff array (A287870). If the weighting of the Zeckendorf representation is used (F_2, F_3, F_4, ...), we get the (unextended) Wythoff array (A035513).

Every primes appears exactly once in the array.