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A178415 Array T(n,k) of odd Collatz preimages read by antidiagonals. 3
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

T. D. Noe, T(n,k) for n=1..50, by antidiagonals

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 n>=2, when 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)

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

Sequence in context: A106607 A305082 A007042 * A249424 A076274 A058989

Adjacent sequences:  A178412 A178413 A178414 * A178416 A178417 A178418

KEYWORD

nonn,tabl

AUTHOR

T. D. Noe, May 28 2010

STATUS

approved

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Last modified December 1 21:42 EST 2020. Contains 338858 sequences. (Running on oeis4.)