A297552 Rectangular array R by antidiagonals: row n shows the positive integers whose base-4 digits have down-variation n, for n>=0. See Comments.

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, 141, 200, 204, 11, 19, 33, 49, 156, 205, 460, 1228, 15, 20, 34, 50, 196, 220, 716, 2188, 2252, 21, 25, 35, 51, 201, 396, 780, 2248, 3212, 3276

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..55.

Example

Northwest corner:
    1      2      3      5      6      7     10     15
    4      9     14     16     17     18     19     20
    8     13     24     29     32     33     34     35
   12     28     44     48     49     50     51     52
   76    136    141    156    196    201    206    216
  140    200    205    220    396    456    461    476
  204    460    716    780    796    812    816    817

Mathematica

g[n_, b_] := Differences[IntegerDigits[n, b]];
b = 4; z = 200000; u = Table[-Total[Select[g[n, b], # < 0 &]], {n, 1, z}] ;
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, 10}, {k, n, 1, -1}] // Flatten

Crossrefs

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

Sequence in context: A157182 A343232 A292145 * A286555 A243353 A182194

Adjacent sequences: A297549 A297550 A297551 * A297553 A297554 A297555

Keyword

nonn,tabl,base,easy

Author

Clark Kimberling, Jan 21 2018

Status

approved