A297551 Rectangular array R by antidiagonals: row n shows the positive integers whose base-4 digits have total variation n, for n>=0. See Comments.

1, 2, 4, 3, 6, 7, 5, 9, 8, 12, 10, 11, 13, 18, 19, 15, 14, 17, 24, 29, 28, 21, 16, 23, 30, 34, 35, 51, 42, 20, 25, 33, 44, 50, 77, 76, 63, 22, 27, 39, 49, 72, 99, 114, 115, 85, 26, 31, 45, 55, 78, 113, 141, 140, 204, 170, 37, 32, 48, 67, 92, 119, 179, 205

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 a homogeneous linear recurrence relation.

Links

Table of n, a(n) for n=1..64.

Example

Northwest corner:
   1      2      3      5     10     15     21     42
   4      6      9     11     14     16     20     22
   7      8     13     17     23     25     27     31
  12     18     24     30     33     39     45     48
  19     29     34     44     49     55     67     71
  28     35     50     72     78     92     98    108

Mathematica

a[n_, b_] := Differences[IntegerDigits[n, b]];
b = 4; 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

Crossrefs

Cf. A007090, A297554 (conjectured 1st column), A297552, A297553.

Sequence in context: A194277 A226246 A216623 * A297673 A083050 A194030

Adjacent sequences: A297548 A297549 A297550 * A297552 A297553 A297554

Keyword

nonn,tabl,base,easy

Author

Clark Kimberling, Jan 21 2018

Status

approved