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Interspersion of the signature sequence of Pi (rectangular array by antidiagonals).
2

%I #21 Jan 22 2025 14:17:42

%S 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 17,10,92,77,63,50,39,29,20,13,118,100,84,69,56,44,33,24,16,147,127,

%U 108,91,76,62,49,38,28,19,179,157,136,116,99,83,68,55,43,32

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

%C 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 Clark Kimberling, <a href="/A283944/b283944.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 5 12 22 35 51 70 92 118

%e 2 7 15 26 40 57 77 100 127

%e 3 9 18 30 45 63 84 108 136

%e 4 11 21 34 50 69 91 115 145

%e 6 14 25 39 56 76 99 125 155

%e 8 17 29 44 62 83 107 134 165

%t r = Pi; 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}] (* A022796, col 1 of A283944 *)

%t v = Table[s[n], {n, 0, z}] (* A022795, row 1 of A283944 *)

%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}]] (* A283944, array*)

%t Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283944, sequence *)

%o (PARI)

%o \\ Produces the triangle when the array is read by antidiagonals

%o r = Pi;

%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 26 2017

%o (Python)

%o # Produces the triangle when the array is read by antidiagonals

%o import math

%o from mpmath import *

%o mp.dps = 100

%o def s(n): return 1 if n<1 else s(n - 1) + 1 + int(math.floor(n*pi))

%o def p(n): return n + 1 + sum([int(math.floor((n - k)/pi)) 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 26 2017

%Y Cf. A000796, A023133, A022796, A022795.

%K nonn,tabl,easy,changed

%O 1,2

%A _Clark Kimberling_, Mar 26 2017