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A283941 Interspersion of the signature sequence of sqrt(5).

%I A283941 #34 Jun 13 2025 18:17:09
%S A283941 1,4,2,9,6,3,16,12,8,5,25,20,15,11,7,37,30,24,19,14,10,51,43,35,29,23,
%T A283941 18,13,67,58,49,41,34,28,22,17,85,75,65,56,47,40,33,27,21,106,94,83,
%U A283941 73,63,54,46,39,32,26,129,116,103,92,81,71,61,53,45,38,31
%N A283941 Interspersion of the signature sequence of sqrt(5).
%C A283941 Row n is the ordered sequence of numbers k such that A023117(k)=n.  As a sequence, A283941 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 A283941 Clark Kimberling, <a href="/A283941/b283941.txt">Antidiagonals n = 1..60, flattened</a>
%H A283941 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 A283941 Northwest corner:
%e A283941   1   4   9   16  25  37  51  67
%e A283941   2   6   12  20  30  43  58  76
%e A283941   3   8   15  24  35  49  65  83
%e A283941   5   11  19  29  41  56  73  92
%e A283941   7   14  23  34  47  63  81  101
%e A283941   10  18  28  40  54  71  90  111
%t A283941 r = Sqrt[5]; z = 100;
%t A283941 s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*r];
%t A283941 u = Table[n + 1 + Sum[Floor[(n - k)/r], {k, 0, n}], {n, 0, z}] (* A022780 , col 1 of A283941 *)
%t A283941 v = Table[s[n], {n, 0, z}] (* A022779, row 1 of A283941*)
%t A283941 w[i_, j_] := u[[i]] + v[[j]] + (i - 1)*(j - 1) - 1;
%t A283941 Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283941, array *)
%t A283941 Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283941, sequence *)
%o A283941 (PARI)
%o A283941 r = sqrt(5);
%o A283941 z = 100;
%o A283941 s(n) = if(n<1, 1, s(n - 1) + 1 + floor(n*r));
%o A283941 p(n) = n + 1 + sum(k=0, n, floor((n - k)/r));
%o A283941 u = v = vector(z + 1);
%o A283941 for(n=1, 101, (v[n] = s(n - 1)));
%o A283941 for(n=1, 101, (u[n] = p(n - 1)));
%o A283941 w(i, j) = u[i] + v[j] + (i - 1) * (j - 1) - 1;
%o A283941 tabl(nn) = {for(n=1, nn, for(k=1, n, print1(w(k, n - k + 1), ", "); );
%o A283941 print(); ); };
%o A283941 tabl(20) \\ _Indranil Ghosh_, Mar 21 2017
%o A283941 (Python)
%o A283941 from sympy import sqrt
%o A283941 import math
%o A283941 def s(n): return 1 if n<1 else s(n - 1) + 1 + int(math.floor(n*sqrt(5)))
%o A283941 def p(n): return n + 1 + sum([int(math.floor((n - k)/sqrt(5))) for k in range(0, n+1)])
%o A283941 v=[s(n) for n in range(0, 101)]
%o A283941 u=[p(n) for n in range(0, 101)]
%o A283941 def w(i,j): return u[i - 1] + v[j - 1] + (i - 1) * (j - 1) - 1
%o A283941 for n in range(1, 11):
%o A283941     print([w(k, n - k + 1) for k in range(1, n + 1)]) # _Indranil Ghosh_, Mar 21 2017
%Y A283941 Cf. A002163, A023117, A022779, A022780.
%K A283941 nonn,tabl,easy
%O A283941 1,2
%A A283941 _Clark Kimberling_, Mar 19 2017
%E A283941 Edited by _Clark Kimberling_, Feb 27 2018