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A083044 Square table read by antidiagonals forms a permutation of the natural numbers: T(n,0) = floor(n*x/(x-1))+1, T(n,k+1) = ceiling(x*T(n,k)), where x=3/2, n >= 0, k >= 0. 19
1, 2, 4, 3, 6, 7, 5, 9, 11, 10, 8, 14, 17, 15, 13, 12, 21, 26, 23, 20, 16, 18, 32, 39, 35, 30, 24, 19, 27, 48, 59, 53, 45, 36, 29, 22, 41, 72, 89, 80, 68, 54, 44, 33, 25, 62, 108, 134, 120, 102, 81, 66, 50, 38, 28, 93, 162, 201, 180, 153, 122, 99, 75, 57, 42, 31, 140, 243 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
First row is A061419, first column is T(n,0) = A016777(n) = 3n+1 (namely the numbers not of the form ceiling(3*k/2) for any natural number k, in increasing order), main diagonal is A083045, antidiagonal sums give A083046. [further detail on first column added by Glen Whitney, Aug 03 2018]
A083044 is the dispersion of the sequence A007494 of positive integers congruent to (0 or 2) mod 3; see A191655. - Clark Kimberling, Jun 10 2011
If T(n+1,k) - T(n,k) = 2m, then T(n+1,k+1) - T(n,k+1) = ceiling(3T(n+1,k)/2) - ceiling(3T(n,k)/2) = ceiling(3T(n,k)/2 + 3m) - ceiling(3T(n,k)/2) = 3m. Similarly, if T(n+1,k) - T(n,k) = 2m+1, then T(n+1,k+1) - T(n,k+1) = ceiling(3T(n,k)/2 + 3m + 3/2) - ceiling(3T(n,k)/2) = {3m+1 or 3m+2, according to whether T(n,k) is even or odd}. The first differences of the first column T(n,0) are periodic: (3)*. The parities of the first column T(n,0) are periodic: (odd,even)*. Hence by induction using the prior two observations, the first differences and parities of every column will be periodic; e.g., for the second column T(n,2): the first differences are (4,5)* and the parities are (even,even,odd,odd)*; for the third column T(n,3): (6,8,6,7)* and (odd,odd,odd,odd,even,even,even,even)*; for the fourth column T(n,4): (9,12,9,10,9,12,9,11)* and (o,e,e,o,o,e,e,o,e,o,o,e,e,o,o,e)*. Is the period length of the first differences of column k always 2^{k-1}? And is the period length of parities always 2^k? Does every integer > 2 occur as T(n+1,k) - T(n,k) for some n and k? Is the smallest first difference in column k always A061418(k+1)? And is the largest first difference in column k always A061419(k+2)? - Glen Whitney, Aug 03 2018
Consider the following two-player game: Start with two nonempty piles of counters. Players alternate taking turns consisting of first discarding one of the piles and then dividing the remaining pile into two nonempty piles. The smaller pile may always be discarded; the larger pile may only be discarded if the smaller pile is at least half as large. The player who cannot move (because the configuration has reached two piles of one counter each) loses. Then the numbers c for which two piles of size c is a losing configuration (for the player whose turn it is) are exactly T(4,k) for k > 1, together with 1,3,5, and 9. - Glen Whitney, Aug 03 2018
LINKS
FORMULA
T(A163491(n)-1, A087088(n)-1) = n. - Peter Munn, Jul 16 2020 [corrected Peter Munn, Aug 02 2020]
EXAMPLE
Table begins:
1 2 3 5 8 12 18 27 41 62 93 140 ...
4 6 9 14 21 32 48 72 108 162 243 365 ...
7 11 17 26 39 59 89 134 201 302 453 680 ...
10 15 23 35 53 80 120 180 270 405 608 912 ...
13 20 30 45 68 102 153 230 345 518 777 1166 ...
16 24 36 54 81 122 183 275 413 620 930 1395 ...
19 29 44 66 99 149 224 336 504 756 1134 1701 ...
22 33 50 75 113 170 255 383 575 863 1295 1943 ...
25 38 57 86 129 194 291 437 656 984 1476 2214 ...
28 42 63 95 143 215 323 485 728 1092 1638 2457 ...
31 47 71 107 161 242 363 545 818 1227 1841 2762 ...
CROSSREFS
Row in which a number occurs: A163491.
Column in which a number occurs: A087088.
Sequence in context: A083050 A194030 A353658 * A361995 A126714 A035506
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Apr 18 2003
STATUS
approved

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Last modified June 17 20:03 EDT 2024. Contains 373464 sequences. (Running on oeis4.)