A283939 - OEIS

%I #8 Dec 07 2019 12:18:28

%S 1,3,2,6,5,4,11,9,8,7,17,15,13,12,10,25,22,20,18,16,14,34,31,28,26,23,

%T 21,19,44,41,38,35,32,29,27,24,56,52,49,46,42,39,36,33,30,69,65,61,58,

%U 54,50,47,43,40,37,84,79,75,71,67,63,59,55,51,48,45,100

%N Interspersion of the signature sequence of sqrt(2).

%C Row n is the ordered sequence of numbers k such that A007336(k)=n.  As a sequence, A283939 is a permutation of the positive integers. As an array, A283939 is the joint-rank array (defined at A182801) of the numbers {i+j*r}, for i>=1, j>=1, where r = sqrt(2).  This is a transposable interspersion; i.e., every row intersperses all other rows, and every column intersperses all other columns.

%H Clark Kimberling, <a href="/A283939/b283939.txt">Antidiagonals n = 1..60, flattened</a>

%H Clark Kimberling and John E. Brown, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Kimberling/kimber67.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.

%e Northwest corner:

%e 1   3   6   11   17   25   34   44   56

%e 2   5   9   15   22   31   41   52   65

%e 4   8   13  20   28   38   49   61   75

%e 7   12  18  26   35   46   58   71   86

%e 10  16  23  32   42   54   67   81   97

%e 14  21  29  39   50   63   77   91   109

%t r = Sqrt[2]; z = 100;

%t s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*r];

%t u = Table[n + 1 + Sum[Floor[(n - k)/r], {k, 0, n}], {n, 0, z}] (* A022776, col 1 of A283939 *)

%t v = Table[s[n], {n, 0, z}] (* A022775, row 1 of A283939*)

%t w[i_, j_] := u[[i]] + v[[j]] + (i - 1)*(j - 1) - 1;

%t Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283939, array *)

%t p = Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283939, sequence *)

%o (PARI)

%o r = sqrt(2);

%o z = 100;

%o s(n) = if(n<1, 1, s(n - 1) + 1 + floor(n*r));

%o p(n) = n + 1 + sum(k=0, n, floor((n - k)/r));

%o u = v = vector(z + 1);

%o for(n=1, 101, (v[n] = s(n - 1)));

%o for(n=1, 101, (u[n] = p(n - 1)));

%o w(i, j) = u[i] + v[j] + (i - 1) * (j - 1) - 1;

%o tabl(nn) = {for(n=1, nn, for(k=1, n, print1(w(k, n - k + 1), ", "); );print(); ); };

%o tabl(10) \\ _Indranil Ghosh_, Mar 21 2017

%o (Python)

%o from sympy import sqrt

%o import math

%o def s(n): return 1 if n<1 else s(n - 1) + 1 +

%o int(math.floor(n*sqrt(2)))

%o def p(n): return n + 1 + sum([int(math.floor((n - k)/sqrt(2))) for k in range(0, n+1)])

%o v=[s(n) for n in range(0, 101)]

%o u=[p(n) for n in range(0, 101)]

%o def w(i,j): return u[i - 1] + v[j - 1] + (i - 1) * (j - 1) - 1

%o for n in range(1, 11):

%o ....print [w(k, n - k + 1) for k in range(1, n + 1)] # _Indranil Ghosh_, Mar 21 2017

%Y Cf. A007336, A002193, A022776, A283962.

%K nonn,tabl,easy

%O 1,2

%A _Clark Kimberling_, Mar 19 2017