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%I #6 Jan 21 2018 04:07:52
%S 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,
%T 168,519,546,17,16,21,87,177,543,1275,2733,26,22,24,96,180,547,1518,
%U 4677,4920,40,23,30,99,181,555,1599,4893,11481,24603
%N Rectangular array R by antidiagonals: row n shows the positive integers whose base-3 digits have down-variation n, for n>=0. See Comments.
%C 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.
%C Every positive integer occurs exactly once in the array, so that as a sequence this is a permutation of the positive integers.
%C 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).
%e Northwest corner:
%e 1 2 4 5 8 13 14 17
%e 3 7 9 10 11 12 16 22
%e 6 15 18 19 20 21 24 30
%e 33 57 61 69 87 96 99 100
%e 60 141 168 177 180 181 182 183
%e 303 519 543 547 555 627 789 870
%t g[n_, b_] := Differences[IntegerDigits[n, b]];
%t b = 3; z = 200000; u = Table[-Total[Select[g[n, b], # < 0 &]], {n, 1, z}] ;
%t p[n_] := Position[u, n]; TableForm[Table[Take[Flatten[p[n]], 15], {n, 0, 9}]]
%t v[n_, k_] := p[k - 1][[n]];
%t Table[v[k, n - k + 1], {n, 10}, {k, n, 1, -1}] // Flatten
%Y Cf. A297444 (conjectured 1st column), A297440, A297442.
%K nonn,tabl,easy
%O 1,2
%A _Clark Kimberling_, Jan 20 2018
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