%I #12 Jul 03 2021 11:21:33
%S 1,2,3,4,5,6,8,7,10,11,13,9,16,15,20,26,12,18,19,34,33,40,14,21,29,46,
%T 47,60,80,17,23,30,56,61,100,101,121,22,24,32,57,87,142,141,182,242,
%U 25,28,35,59,92,168,181,304,303,364,27,31,38,62,96,173,263
%N Rectangular array R by antidiagonals: row n shows the positive integers whose base-3 digits have total 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 the linear recurrence relation c(n) = c(n-1) + 2*c(n-2) - 2*c(n-3) + 3*c(n-4) - 3*c(n-5).
%e Northwest corner:
%e 1 2 4 8 13 26 40 80
%e 3 5 7 9 12 14 17 22
%e 6 10 16 18 21 23 24 28
%e 11 15 19 29 30 32 35 38
%e 20 34 46 56 57 59 62 65
%e 33 47 61 87 92 96 99 102
%t a[n_, b_] := Differences[IntegerDigits[n, b]];
%t b = 3; z = 250000; t = Table[a[n, b], {n, 1, z}];
%t u = Map[Total, Map[Abs, t]]; p[n_] := Position[u, n];
%t 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, 12}, {k, n, 1, -1}] // Flatten
%Y Cf. A007089, A297330 (guide), A297443 (conjectured 1st column), A297441, A297442.
%K nonn,tabl,base,easy
%O 1,2
%A _Clark Kimberling_, Jan 20 2018