A099730 Array read by antidiagonals. Rows contain odd numbers reaching same odd successor in Collatz function iteration.

1, 5, 3, 21, 13, 7, 85, 53, 29, 9, 341, 213, 117, 37, 11, 1365, 853, 469, 149, 45, 15, 5461, 3413, 1877, 597, 181, 61, 17, 21845, 13653, 7509, 2389, 725, 245, 69, 19, 87381, 54613, 30037, 9557, 2901, 981, 277, 77, 23

Offset: 1

Author

Lambert Klasen (lambert.klasen(AT)gmx.de), Nov 09 2004

Keywords

• nonn
• tabl

Comments

All numbers that end in 3 will begin with numbers from previous row (for example, 3413 is 341&3). - Jean-Bernard François, Sep 09 2013

The sequence is a permutation of the odd positive integers. - Bob Selcoe, Jul 26 2015

Examples

			t(1, 2) = 53 = 4*13+1, t(2, 5) = 7509 = 4*1877+1.
Array begins:
1 5 21 85 341 1365 5461 21845 87381 ...
3 13 53 213 853 3413 13653 54613 218453 ...
7 29 117 469 1877 7509 30037 120149 480597 ...
9 37 149 597 2389 9557 38229 152917 611669 ...
11 45 181 725 2901 11605 46421 185685 742741 ...
15 61 245 981 3925 15701 62805 251221 1004885 ...
17 69 277 1109 4437 17749 70997 283989 1135957 ...
19 77 309 1237 4949 19797 79189 316757 1267029 ...
...
Construct array by writing odd numbers in columns, taking first overflow after two steps and then an overflow each fourth step (for each column).
		

Crossrefs

First row = A002450 (except leading zero), second row = A072197, third row = A072261.

Programs

Mathematica

t[n_, k_] := 2^(2*(k + 1) - 1)*(n + Quotient[n + 1, 3]) + (4^(k + 1) - 1)/3; Table[t[n - k, k], {n, 0, 8}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Sep 09 2013, after Maon Wenders *)

PARI

g(n)=(n+1)\3
T(n,k)=2^(2*(k+1)-1)*(n+g(n))+(4^(k+1)-1)/3
for(i=0,20,for(j=0,10,print1(T(i,j), ", "));print())\\ Maon Wenders, Jul 15 2012

Formula

Let g(n)= floor((n+1)/3), then T(n,k) = 2^(2*(k+1)-1) *(n+g(n)) + (4^(k+1)-1)/3. - Maon Wenders, Jul 15 2012

t(n, k) = 4*t(n, k-1) + 1. - Jean-Bernard François, Sep 09 2013