A297553 Rectangular array R by antidiagonals: row n shows the positive integers whose base-4 digits have up-variation n, for n>=0. See Comments.

1, 2, 6, 3, 11, 7, 4, 17, 18, 19, 5, 22, 23, 35, 99, 8, 24, 27, 51, 114, 115, 9, 25, 28, 67, 119, 291, 307, 10, 26, 29, 71, 179, 306, 563, 1587, 12, 33, 30, 75, 275, 311, 819, 1827, 1843, 13, 38, 31, 76, 290, 371, 1075, 1842, 4659, 4915

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      4      5      8      9     10
    6     11     17     22     24     25     26     33
    7     18     23     27     28     29     30     31
   19     35     51     67     71     75     76     77
   99    114    119    179    275    290    295    305
  115    291    306    311    371    435    451    455
  307    563    819   1075   1139   1203   1219   1223

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, A297556 (conjectured 1st column), A297551, A297552.

Sequence in context: A295644 A079178 A322365 * A347558 A317490 A256466

Adjacent sequences: A297550 A297551 A297552 * A297554 A297555 A297556

Keyword

nonn,tabl,base,easy

Author

Clark Kimberling, Jan 21 2018

Status

approved