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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. 3
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 (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..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.  A297555 (conjectured 1st column), A297551, A297553.

Sequence in context: A284572 A157182 A292145 * A286555 A243353 A182194

Adjacent sequences:  A297549 A297550 A297551 * A297553 A297554 A297555

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Jan 21 2018

STATUS

approved

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Last modified September 19 12:17 EDT 2020. Contains 337178 sequences. (Running on oeis4.)