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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified December 3 08:07 EST 2021. Contains 349445 sequences. (Running on oeis4.)