OFFSET
1,2
COMMENTS
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.
Every positive integer occurs exactly once in the array, so that as a sequence this is a permutation of the positive integers.
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).
EXAMPLE
Northwest corner:
1 2 4 8 13 26 40 80
3 5 7 9 12 14 17 22
6 10 16 18 21 23 24 28
11 15 19 29 30 32 35 38
20 34 46 56 57 59 62 65
33 47 61 87 92 96 99 102
MATHEMATICA
a[n_, b_] := Differences[IntegerDigits[n, b]];
b = 3; z = 250000; t = Table[a[n, b], {n, 1, z}];
u = Map[Total, Map[Abs, t]]; p[n_] := Position[u, n];
TableForm[Table[Take[Flatten[p[n]], 15], {n, 0, 9}]]
v[n_, k_] := p[k - 1][[n]]
Table[v[k, n - k + 1], {n, 12}, {k, n, 1, -1}] // Flatten
KEYWORD
AUTHOR
Clark Kimberling, Jan 20 2018
STATUS
approved