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%I #10 Jul 03 2021 10:59:18
%S 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,
%T 35,51,42,20,25,33,44,50,77,76,63,22,27,39,49,72,99,114,115,85,26,31,
%U 45,55,78,113,141,140,204,170,37,32,48,67,92,119,179,205
%N Rectangular array R by antidiagonals: row n shows the positive integers whose base-4 digits have total 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 a homogeneous linear recurrence relation.
%e Northwest corner:
%e 1 2 3 5 10 15 21 42
%e 4 6 9 11 14 16 20 22
%e 7 8 13 17 23 25 27 31
%e 12 18 24 30 33 39 45 48
%e 19 29 34 44 49 55 67 71
%e 28 35 50 72 78 92 98 108
%t a[n_, b_] := Differences[IntegerDigits[n, b]];
%t b = 4; z = 250000; t = Table[a[n, b], {n, 1, z}];
%t u = Map[Total, Map[Abs, t]]; p[n_] := Position[u, n];
%t 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, 12}, {k, n, 1, -1}] // Flatten
%Y Cf. A007090, A297554 (conjectured 1st column), A297552, A297553.
%K nonn,tabl,base,easy
%O 1,2
%A _Clark Kimberling_, Jan 21 2018
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