# Tables of Permutation of Natural Numbers

Paul D Hanna pauldhanna at juno.com
Fri Apr 18 08:52:20 CEST 2003

```Is the following conjecture a known theorem?

Given a real x>1, define a square table T(n,k), with n>=0 and k>=0, by
T(n,0) = floor( n*x/(x-1) ) + 1,
T(n,k+1) = ceil( x*T(n,k) );
then the table, when read by anti-diagonals,
forms a permutation of the natural numbers for all x > 1.

It seems surprising that all of the natural numbers would
occur once and only once in the table for all x>1.

The main diagonal and the sum of anti-diagonals also form
interesting sequences.  Wonder what their asymptotics are.

Three examples are given below.

Thanks,
Paul D Hanna
-----------------------------------------------------------------

EXAMPLE 1: x=3/2.

New EIS entry A083044:
Square table read by anti-diagonals forms a permutation of the natural
numbers:
T(n,0)=floor(n*x/(x-1))+1, T(n,k+1)=ceil(x*T(n,k)),
where x=3/2, n>=0, k>=0.

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,
122,99,75,57,42,31,140,243,302,270,230,183,149,113,86,63,47,
34,210,365,453,405,345,275,224,170,129,95,71,51,37,315,548,
680,608,518,413,336,255,194,143,...

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 ...
...

First row is A061419, first column is T(n,0)=3n+1.

Main diagonal (A083045):

1,6,17,35,68,122,224,383,656,1092,1841,2978,4859,7835,12776,
20291,32664,51422,82485,129720,204821,319482,506060,789872,
1237733,1927494,3024318,4687259,7274921,11271293,17578760,
27133793,42125475,64909160,100763060,155235216,239286168,
367459469,568391688,873351174,1345020897,2062173054,
3183907319,4866700238,7470646523,11439387716,17605111616,
26873568347,41292926031,63000698543,96732311610,...

Sum of anti-diagonals (A083046):

1,6,16,35,67,118,197,319,506,789,1215,1860,2830,4290,6481,
9771,14708,22120,33239,49921,74946,112490,168807,253286,
380008,570095,855228,1282931,1924490,2886828,4330341,
6495613,9743524,14615392,21923196,32884907,49327480,
73991342,110987136,166480833,249721379,374582202,561873440,
842810301,1264215596,1896323542,2844485462,4266728350,
6400092681,9600139182,14400208933,...
-----------------------------------------------------------------

EXAMPLE 2: x=(sqrt(5)+1)/2.

New EIS entry A083047:
Square table read by anti-diagonals forms a permutation of the natural
numbers:
T(n,0)=floor(n*x/(x-1))+1, T(n,k+1)=ceil(x*T(n,k)),
where x=(sqrt(5)+1)/2, n>=0, k>=0.

1,2,3,4,5,6,7,9,10,8,12,15,17,13,11,20,25,28,22,18,14,
33,41,46,36,30,23,16,54,67,75,59,49,38,26,19,88,109,122,
96,80,62,43,31,21,143,177,198,156,130,101,70,51,34,24,
232,287,321,253,211,164,114,83,56,39,27,376,465,520,410,
342,266,185,135,91,64,44,...

Table begins:
1  2  4  7   12  20  33  54  88   143  232  376  ...
3  5  9  15  25  41  67  109 177  287  465  753  ...
6  10 17 28  46  75  122 198 321  520  842  1363 ...
8  13 22 36  59  96  156 253 410  664  1075 1740 ...
11 18 30 49  80  130 211 342 554  897  1452 2350 ...
14 23 38 62  101 164 266 431 698  1130 1829 2960 ...
16 26 43 70  114 185 300 486 787  1274 2062 3337 ...
19 31 51 83  135 219 355 575 931  1507 2439 3947 ...
21 34 56 91  148 240 389 630 1020 1651 2672 4324 ...
24 39 64 104 169 274 444 719 1164 1884 3049 4934 ...
27 44 72 117 190 308 499 808 1308 2117 3426 5544 ...
...

First column is A026352, a(n)=floor(n*x/(x-1))+1, x=(sqrt(5)+1)/2:

1,3,6,8,11,14,16,19,21,24,27,29,32,35,37,40,42,45,48,50,53,
55,58,61,63,66,69,71,74,76,79,82,84,87,90,92,95,97,100,103,
105,108,110,113,116,118,121,124,126,129,131,...

Main diagonal (A083048):

1,5,17,36,80,164,300,575,1020,1884,3426,5921,10568,18697,
31850,55716,94332,163579,282388,474625,814328,1363979,
2328358,3963781,6609951,11209355,18969158,31524783,53186480,
88235842,148471479,249459365,412861197,692181267,1159060986,
1914488240,3200041199,5280109581,8811311063,14690495223,
24203215020,40296527710,66336054634,110305206151,183285099665,
301369047866,500211631630,829724432620,1362887344350,
2258514337169,3707669252792,...

Sum of anti-diagonals (A083049):

1,5,15,34,68,127,225,387,652,1084,1787,2927,4775,7769,12616,
20462,33160,53709,86962,140769,227834,368711,596658,965488,
1562270,2527887,4090292,6618319,10708756,17327225,28036136,
45363522,73399824,118763517,192163518,310927217,503090922,
814018331,1317109450,2131127984,3448237642,5579365839,
9027603699,14606969761,23634573689,38241543684,61876117612,
100117661541,161993779403,262111441199,424105220862,...
-----------------------------------------------------------------

EXAMPLE 3: x=sqrt(2).

New EIS entry A083050:
Square table read by anti-diagonals forms a permutation of the natural
numbers:
T(n,0)=floor(n*x/(x-1))+1, T(n,k+1)=ceil(x*T(n,k)),
where x=sqrt(2), n>=0, k>=0.

1,2,4,3,6,7,5,9,10,11,8,13,15,16,14,12,19,22,23,20,18,17,27,
32,33,29,26,21,25,39,46,47,42,37,30,24,36,56,66,67,60,53,43,
34,28,51,80,94,95,85,75,61,49,40,31,73,114,133,135,121,107,
87,70,57,44,35,104,162,189,191,172,152,124,99,81,63,50,38,
148,230,268,271,244,215,176,141,115,90,71,54,41,210,326,380,
384,346,305,249,200,163,128,...

Table begins:
1  2  3  5   8   12  17  25  36  51  73   104 ...
4  6  9  13  19  27  39  56  80  114 162  230 ...
7  10 15 22  32  46  66  94  133 189 268  380 ...
11 16 23 33  47  67  95  135 191 271 384  544 ...
14 20 29 42  60  85  121 172 244 346 490  693 ...
18 26 37 53  75  107 152 215 305 432 611  865 ...
21 30 43 61  87  124 176 249 353 500 708  1002 ...
24 34 49 70  99  141 200 283 401 568 804  1138 ...
28 40 57 81  115 163 231 327 463 655 927  1311 ...
31 44 63 90  128 182 258 365 517 732 1036 1466 ...
35 50 71 101 143 203 288 408 577 817 1156 1635 ...
...

First column (A083051) is a(n)=floor(n*x/(x-1))+1, x=sqrt(2):

1,4,7,11,14,18,21,24,28,31,35,38,41,45,48,52,55,59,62,65,69,
72,76,79,82,86,89,93,96,100,103,106,110,113,117,120,123,127,
130,134,137,140,144,147,151,154,158,161,164,168,171,175,178,
181,185,188,192,195,199,202,205,209,212,216,219,222,226,229,
233,236,239,243,246,...

Main diagonal (A083052):

1,6,15,33,60,107,176,283,463,732,1156,1773,2702,4170,6333,
9611,14365,21842,32281,48041,71690,106136,158196,232380,
340846,504318,740949,1089909,1590748,2348540,3411262,4978297,
7278002,10600284,15494846,22455796,32552694,47462891,68901339,
100226011,144922573,209398564,304247370,440319636,637914484,
920016532,1335356274,1922485613,2774488698,4009863909,5781736941,...

Sum of anti-diagonals (A083053):

1,6,16,35,66,114,185,290,443,661,976,1425,2064,2972,4259,
6083,8667,12327,17506,24834,35203,49869,70615,99959,141462,
200159,283173,400577,566616,801435,1133522,1603168,2267350,
3206653,4535033,6413648,9070416,12827656,18141199,25655688,
36282784,51311773,72565977,102623965,145132384,205248368,
290265215,410497200,580530904,820994882,1161062298,...
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```

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