A254051 Square array A by downward antidiagonals: A(n,k) = (3 + 3^n*(2*floor(3*k/2) - 1))/6, n,k >= 1; read as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

1, 3, 2, 4, 8, 5, 6, 11, 23, 14, 7, 17, 32, 68, 41, 9, 20, 50, 95, 203, 122, 10, 26, 59, 149, 284, 608, 365, 12, 29, 77, 176, 446, 851, 1823, 1094, 13, 35, 86, 230, 527, 1337, 2552, 5468, 3281, 15, 38, 104, 257, 689, 1580, 4010, 7655, 16403, 9842, 16, 44, 113, 311, 770, 2066, 4739, 12029, 22964, 49208, 29525, 18, 47

Offset

1,2

Comments

This is transposed dispersion of (3n-1), starting from its complement A032766 as the first row of square array A(row,col). Please see the transposed array A191450 for references and background discussion about dispersions.

For any odd number x = A135765(row,col), the result after one combined Collatz step (3x+1)/2 -> x (A165355) is found in this array at A(row+1,col).

Links

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array

Index entries for sequences that are permutations of the natural numbers

Index entries for sequences related to 3x+1 (or Collatz) problem

Formula

In A(n,k)-formulas below, n is the row, and k the column index, both starting from 1:

A(n,k) = (3 + ( A000244(n) * (2*A032766(k) - 1) )) / 6. - Antti Karttunen after L. Edson Jeffery's direct formula for A191450, Jan 24 2015

A(n,k) = A048673(A254053(n,k)). [Alternative formula.]

A(n,k) = (1/2) * (1 + A003961((2^(n-1)) * A254050(k))). [The above expands to this.]

A(n,k) = (1/2) * (1 + (A000244(n-1) * A007310(k))). [Which further reduces to this, equivalent to L. Edson Jeffery's original formula above.]

A(1,k) = A032766(k) and for n > 1: A(n,k) = (3 * A254051(n-1,k)) - 1. [The definition of transposed dispersion of (3n-1).]

A(n,k) = (1+A135765(n,k))/2, or when expressed one-dimensionally, a(n) = (1+A135765(n))/2.

A(n+1,k) = A165355(A135765(n,k)).

As a composition of related permutations. All sequences interpreted as one-dimensional:

a(n) = A048673(A254053(n)). [Proved above.]

a(n) = A191450(A038722(n)). [Transpose of array A191450.]

Example

The top left corner of the array:
   1,   3,   4,   6,   7,   9,  10,  12,   13,   15,   16,   18,   19,   21
   2,   8,  11,  17,  20,  26,  29,  35,   38,   44,   47,   53,   56,   62
   5,  23,  32,  50,  59,  77,  86, 104,  113,  131,  140,  158,  167,  185
  14,  68,  95, 149, 176, 230, 257, 311,  338,  392,  419,  473,  500,  554
  41, 203, 284, 446, 527, 689, 770, 932, 1013, 1175, 1256, 1418, 1499, 1661
...

Prog

(Scheme, several versions)
(define (A254051 n) (A254051bi (A002260 n) (A004736 n)))
(define (A254051bi row col) (/ (+ 3 (* (A000244 row) (- (* 2 (A032766 col)) 1))) 6))
(define (A254051 n) (A191450biv2 (A004736 n) (A002260 n))) ;; As transpose of A191450
(define (A254051bi row col) (/ (+ 1 (A003961 (* (A000079 (- row 1)) (+ -1 (* 2 (A249745 col)))))) 2))
(define (A254051bi row col) (/ (+ 1 (A003961 (* (A000079 (- row 1)) (A254050 col)))) 2))
(define (A254051bi row col) (/ (+ 1 (* (A000244 (- row 1)) (A007310 col))) 2))

Crossrefs

Inverse: A254052.

Transpose: A191450.

Row 1: A032766.

Cf. A007051, A057198, A199109, A199113 (columns 1-4).

Cf. A254046 (row index of n in this array, see also A253786), A253887 (column index).

Array A135765(n,k) = 2*A(n,k) - 1.

Other related arrays: A254055, A254101, A254102.

Cf. also A000079, A000244, A003961, A007310, A032766, A002260, A004736, A038722, A165355, A254050.

Related permutations: A048673, A254053, A183209, A249745, A254103, A254104.

Sequence in context: A317704 A201422 A231330 * A082228 A114650 A170949

Adjacent sequences: A254048 A254049 A254050 * A254052 A254053 A254054

Keyword

nonn,tabl

Author

L. Edson Jeffery & Antti Karttunen, Jan 24 2015

Status

approved