|
%I #62 Sep 03 2015 14:57:36
%S 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,
%T 284,608,365,12,29,77,176,446,851,1823,1094,13,35,86,230,527,1337,
%U 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
%N 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), ...
%C 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.
%C 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).
%H Antti Karttunen, <a href="/A254051/b254051.txt">Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%F In A(n,k)-formulas below, n is the row, and k the column index, both starting from 1:
%F 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
%F A(n,k) = A048673(A254053(n,k)). [Alternative formula.]
%F A(n,k) = (1/2) * (1 + A003961((2^(n-1)) * A254050(k))). [The above expands to this.]
%F 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.]
%F 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).]
%F A(n,k) = (1+A135765(n,k))/2, or when expressed one-dimensionally, a(n) = (1+A135765(n))/2.
%F A(n+1,k) = A165355(A135765(n,k)).
%F As a composition of related permutations. All sequences interpreted as one-dimensional:
%F a(n) = A048673(A254053(n)). [Proved above.]
%F a(n) = A191450(A038722(n)). [Transpose of array A191450.]
%e The top left corner of the array:
%e 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21
%e 2, 8, 11, 17, 20, 26, 29, 35, 38, 44, 47, 53, 56, 62
%e 5, 23, 32, 50, 59, 77, 86, 104, 113, 131, 140, 158, 167, 185
%e 14, 68, 95, 149, 176, 230, 257, 311, 338, 392, 419, 473, 500, 554
%e 41, 203, 284, 446, 527, 689, 770, 932, 1013, 1175, 1256, 1418, 1499, 1661
%e ...
%o (Scheme, several versions)
%o (define (A254051 n) (A254051bi (A002260 n) (A004736 n)))
%o (define (A254051bi row col) (/ (+ 3 (* (A000244 row) (- (* 2 (A032766 col)) 1))) 6))
%o (define (A254051 n) (A191450biv2 (A004736 n) (A002260 n))) ;; As transpose of A191450
%o (define (A254051bi row col) (/ (+ 1 (A003961 (* (A000079 (- row 1)) (+ -1 (* 2 (A249745 col)))))) 2))
%o (define (A254051bi row col) (/ (+ 1 (A003961 (* (A000079 (- row 1)) (A254050 col)))) 2))
%o (define (A254051bi row col) (/ (+ 1 (* (A000244 (- row 1)) (A007310 col))) 2))
%Y Inverse: A254052.
%Y Transpose: A191450.
%Y Row 1: A032766.
%Y Cf. A007051, A057198, A199109, A199113 (columns 1-4).
%Y Cf. A254046 (row index of n in this array, see also A253786), A253887 (column index).
%Y Array A135765(n,k) = 2*A(n,k) - 1.
%Y Other related arrays: A254055, A254101, A254102.
%Y Cf. also A000079, A000244, A003961, A007310, A032766, A002260, A004736, A038722, A165355, A254050.
%Y Related permutations: A048673, A254053, A183209, A249745, A254103, A254104.
%K nonn,tabl
%O 1,2
%A _L. Edson Jeffery_ & _Antti Karttunen_, Jan 24 2015
|
