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A178415 Array T(n,k) of odd Collatz preimages read by antidiagonals. 16
1, 3, 5, 9, 13, 21, 7, 37, 53, 85, 17, 29, 149, 213, 341, 11, 69, 117, 597, 853, 1365, 25, 45, 277, 469, 2389, 3413, 5461, 15, 101, 181, 1109, 1877, 9557, 13653, 21845, 33, 61, 405, 725, 4437, 7509, 38229, 54613, 87381, 19, 133, 245, 1621, 2901, 17749, 30037 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Every odd number occurs uniquely in this array. See A178414.
LINKS
Eric Weisstein's World of Mathematics, Collatz Problem
FORMULA
From Bob Selcoe, Apr 09 2015 (Start):
T(n,k) = 4*T(n,k-1) + 1.
T(n,k) = T(1,k) + 2^(2k+1)*(n-1)/2 when n is odd;
T(n,k) = T(2,k) + 4^k*(n-2)/2 when n >= 2 and n is even. So equivalently:
T(n,k) = T(n-2,k) + 2^(2k+1) when n is odd; and
T(n,k) = T(n-2,k) + 4^k when n is even.
Let j be the n-th positive odd number coprime with 3. Then:
T(n,k) = (j*4^k - 1)/3 when j == 1 (mod 3); and
T(n,k) = (j*2^(2k-1) - 1)/3 when j == 2 (mod 3).
(End)
From Wolfdieter Lang, Sep 18 2021: (Start)
T(n, k) = ((3*n - 1)*4^k - 2)/6 if n is even, and ((3*n - 2)*4^k - 1)/3 if n is odd, for n >= 1 and k >= 1. Also for n = 0: -A007583(k-1), with A007583(-1) = 1/2, and for k = 0: A022998(n-1)/2, with A022998(-1) = -1.
O.g.f. for array T (with row n = 0 and column k = 0; z for rows and x for columns): G(z, x) = (1/(2*(1-x)*(1-4*x)*(1-z^2)^2)) * ((2*x-4)*z^3 + (3-5*x)*z^2 + 2*x*z + 3*x - 1). (End)
EXAMPLE
Array T begins:
. 1 5 21 85 341 1365 5461 21845 87381 349525
. 3 13 53 213 853 3413 13653 54613 218453 873813
. 9 37 149 597 2389 9557 38229 152917 611669 2446677
. 7 29 117 469 1877 7509 30037 120149 480597 1922389
. 17 69 277 1109 4437 17749 70997 283989 1135957 4543829
. 11 45 181 725 2901 11605 46421 185685 742741 2970965
. 25 101 405 1621 6485 25941 103765 415061 1660245 6640981
. 15 61 245 981 3925 15701 62805 251221 1004885 4019541
. 33 133 533 2133 8533 34133 136533 546133 2184533 8738133
. 19 77 309 1237 4949 19797 79189 316757 1267029 5068117
- L. Edson Jeffery, Mar 11 2015
From Bob Selcoe, Apr 09 2015 (Start):
n=5, j=13: T(5,3) = 277 = (13*4^3 - 1)/3;
n=6, j=17: T(6,4) = 725 = (17*2^7 - 1)/3.
(End)
MATHEMATICA
t[n_, 1] := t[n, 1] = If[OddQ[n], 4n-3, 2n-1]; t[n_, k_] := t[n, k] = 4*t[n, k-1]+1; Flatten[Table[t[n-i+1, i], {n, 20}, {i, n}]]
CROSSREFS
Rows of array: -A007583(k-1) (n=0), A002450 (n=1), A072197(k-1) (n=2), A206374(n=3), A072261 (n=4), A323824 (n=5), A072262 (n=6), A330246 (n=7), A072201 (n=8), ...
Columns of array: A022998(n-1)/2 (k=0), A178414 (k=1), ...
Cf. A347834 (permuted rows of the array).
Sequence in context: A106607 A305082 A007042 * A249424 A076274 A058989
KEYWORD
nonn,tabl,easy
AUTHOR
T. D. Noe, May 28 2010
STATUS
approved

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Last modified June 17 15:39 EDT 2024. Contains 373458 sequences. (Running on oeis4.)