A297441 Rectangular array R by antidiagonals: row n shows the positive integers whose base-3 digits have down-variation n, for n>=0. See Comments.

1, 2, 3, 4, 7, 6, 5, 9, 15, 33, 8, 10, 18, 57, 60, 13, 11, 19, 61, 141, 303, 14, 12, 20, 69, 168, 519, 546, 17, 16, 21, 87, 177, 543, 1275, 2733, 26, 22, 24, 96, 180, 547, 1518, 4677, 4920, 40, 23, 30, 99, 181, 555, 1599, 4893, 11481, 24603

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) + 9*c(n-2) - 9*c(n-3).

Links

Table of n, a(n) for n=1..55.

Example

Northwest corner:
    1      2      4      5      8     13     14     17
    3      7      9     10     11     12     16     22
    6     15     18     19     20     21     24     30
   33     57     61     69     87     96     99    100
   60    141    168    177    180    181    182    183
  303    519    543    547    555    627    789    870

Mathematica

g[n_, b_] := Differences[IntegerDigits[n, b]];
b = 3; 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. A297444 (conjectured 1st column), A297440, A297442.

Sequence in context: A122155 A106454 A270195 * A292959 A292957 A338644

Adjacent sequences: A297438 A297439 A297440 * A297442 A297443 A297444

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Jan 20 2018

Status

approved