A297552 - OEIS

%I #9 Jul 03 2021 10:59:11

%S 1,2,4,3,9,8,5,14,13,12,6,16,24,28,76,7,17,29,44,136,140,10,18,32,48,

%T 141,200,204,11,19,33,49,156,205,460,1228,15,20,34,50,196,220,716,

%U 2188,2252,21,25,35,51,201,396,780,2248,3212,3276

%N Rectangular array R by antidiagonals:  row n shows the positive integers whose base-4 digits have down-variation n, for n>=0.  See Comments.

%C Suppose that a number n has base-b digits b(m), b(m-1), ..., b(0).  The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1).  The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). See A297330 for a guide to related sequences and partitions of the natural numbers.

%C Every positive integer occurs exactly once in the array, so that as a sequence this is a permutation of the positive integers.

%C Conjecture:  each column, after some number of initial terms, satisfies a homogeneous linear recurrence relation.

%e Northwest corner:

%e     1      2      3      5      6      7     10     15

%e     4      9     14     16     17     18     19     20

%e     8     13     24     29     32     33     34     35

%e    12     28     44     48     49     50     51     52

%e    76    136    141    156    196    201    206    216

%e   140    200    205    220    396    456    461    476

%e   204    460    716    780    796    812    816    817

%t g[n_, b_] := Differences[IntegerDigits[n, b]];

%t b = 4; z = 200000; u = Table[-Total[Select[g[n, b], # < 0 &]], {n, 1, z}] ;

%t p[n_] := Position[u, n]; TableForm[Table[Take[Flatten[p[n]], 15], {n, 0, 9}]]

%t v[n_, k_] := p[k - 1][[n]];

%t Table[v[k, n - k + 1], {n, 10}, {k, n, 1, -1}] // Flatten

%Y Cf. A007090, A297555 (conjectured 1st column), A297551, A297553.

%K nonn,tabl,base,easy

%O 1,2

%A _Clark Kimberling_, Jan 21 2018