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A283944 Interspersion of the signature sequence of Pi (rectangular array by antidiagonals).

%I A283944 #21 Jan 22 2025 14:17:42
%S A283944 1,5,2,12,7,3,22,15,9,4,35,26,18,11,6,51,40,30,21,14,8,70,57,45,34,25,
%T A283944 17,10,92,77,63,50,39,29,20,13,118,100,84,69,56,44,33,24,16,147,127,
%U A283944 108,91,76,62,49,38,28,19,179,157,136,116,99,83,68,55,43,32
%N A283944 Interspersion of the signature sequence of Pi (rectangular array by antidiagonals).
%C A283944 Row n is the ordered sequence of numbers k such that A023133(k) = n. As a sequence, it is a permutation of the positive integers. This is a transposable interspersion; i.e., every row intersperses all other rows, and every column intersperses all other columns.
%H A283944 Clark Kimberling, <a href="/A283944/b283944.txt">Antidiagonals n = 1..60, flattened</a>
%H A283944 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 A283944 Northwest corner:
%e A283944 1  5   12  22   35   51   70   92    118
%e A283944 2  7   15  26   40   57   77   100   127
%e A283944 3  9   18  30   45   63   84   108   136
%e A283944 4  11  21  34   50   69   91   115   145
%e A283944 6  14  25  39   56   76   99   125   155
%e A283944 8  17  29  44   62   83   107  134   165
%t A283944 r = Pi; z = 100;
%t A283944 s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*r];
%t A283944 u = Table[n + 1 + Sum[Floor[(n - k)/r], {k, 0, n}], {n, 0, z}] (* A022796, col 1 of A283944 *)
%t A283944 v = Table[s[n], {n, 0, z}] (* A022795, row 1 of A283944 *)
%t A283944 w[i_, j_] := u[[i]] + v[[j]] + (i - 1)*(j - 1) - 1;
%t A283944 Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283944, array*)
%t A283944 Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283944, sequence *)
%o A283944 (PARI)
%o A283944 \\ Produces the triangle when the array is read by antidiagonals
%o A283944 r = Pi;
%o A283944 z = 100;
%o A283944 s(n) = if(n<1, 1, s(n - 1) + 1 + floor(n*r));
%o A283944 p(n) = n + 1 + sum(k=0, n, floor((n - k)/r));
%o A283944 u = v = vector(z + 1);
%o A283944 for(n=1, 101, (v[n] = s(n - 1)));
%o A283944 for(n=1, 101, (u[n] = p(n - 1)));
%o A283944 w(i, j) = u[i] + v[j] + (i - 1) * (j - 1) - 1;
%o A283944 tabl(nn) = {for(n=1, nn, for(k=1, n, print1(w(k, n - k + 1), ", "); ); print(); ); };
%o A283944 tabl(10) \\ _Indranil Ghosh_, Mar 26 2017
%o A283944 (Python)
%o A283944 # Produces the triangle when the array is read by antidiagonals
%o A283944 import math
%o A283944 from mpmath import *
%o A283944 mp.dps = 100
%o A283944 def s(n): return 1 if n<1 else s(n - 1) + 1 + int(math.floor(n*pi))
%o A283944 def p(n): return n + 1 + sum([int(math.floor((n - k)/pi)) for k in range(0, n+1)])
%o A283944 v=[s(n) for n in range(0, 101)]
%o A283944 u=[p(n) for n in range(0, 101)]
%o A283944 def w(i, j): return u[i - 1] + v[j - 1] + (i - 1) * (j - 1) - 1
%o A283944 for n in range(1, 11):
%o A283944     print([w(k, n - k + 1) for k in range(1, n + 1)]) # _Indranil Ghosh_, Mar 26 2017
%Y A283944 Cf. A000796, A023133, A022796, A022795.
%K A283944 nonn,tabl,easy
%O A283944 1,2
%A A283944 _Clark Kimberling_, Mar 26 2017