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A167267 Interspersion of the signature sequence of (1+sqrt(5))/2.

%I A167267 #31 May 19 2021 09:29:46
%S A167267 1,3,2,7,5,4,12,10,8,6,19,16,14,11,9,28,24,21,18,15,13,38,34,30,26,23,
%T A167267 20,17,50,45,41,36,32,29,25,22,63,58,53,48,43,39,35,31,27,78,72,67,61,
%U A167267 56,51,46,42,37,33
%N A167267 Interspersion of the signature sequence of (1+sqrt(5))/2.
%C A167267 Row n is the ordered sequence of numbers k such that A084531(k)=n.  Is the difference sequence of column 1 equal to A019446? Is the difference sequence of row 1 essentially equal to A026351?
%C A167267 As a sequence, A167267 is a permutation of the positive integers. As an array, A167267 is the joint-rank array (defined at A182801) of the numbers {i+j*r}, for i>=1, j>=1, where r = golden ratio = (1+sqrt(5))/2. - _Clark Kimberling_, Nov 10 2012
%C A167267 This is a transposable interspersion; i.e., its transpose, A283734, is also an interspersion. - _Clark Kimberling_, Mar 16 2017
%D A167267 Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.
%H A167267 Clark Kimberling, <a href="/A167267/b167267.txt">Antidiagonals n = 1..60, flattened</a>
%H A167267 N. Carey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Carey/carey6.html">Lambda Words: A Class of Rich Words Defined Over an Infinite Alphabet</a>, J. Int. Seq. 16 (2013) #13.3.4
%F A167267 R(m,n) = sum{[(m-i+n+r)/r], i=1,2,...z(m,n)}, where r = (1+sqrt(5))/2 and z(m,n) = m + [(n-1)*r].  - _Clark Kimberling_, Nov 10 2012
%e A167267 Northwest corner:
%e A167267 1....3....7....12...19...28...38
%e A167267 2....5....10...16...24...34...45
%e A167267 4....8....14...21...30...41...53
%e A167267 6....11...18...26...36...48...61
%e A167267 9....15...23...32...43...56...70
%e A167267 13...20...29...39...51...65...80
%t A167267 v = GoldenRatio;
%t A167267 x = Table[Sum[Ceiling[i*v], {i, q}], {q, 0, end = 35}];
%t A167267 y = Table[Sum[Ceiling[i*1/v], {i, q}], {q, 0, end}];
%t A167267 tot[p_, q_] := x[[p + 1]] + p q + 1 + y[[q + 1]]
%t A167267 row[r_] := Table[tot[n, r], {n, 0, (end - 1)/v}]
%t A167267 Grid[Table[row[n], {n, 0, (end - 1)}]]
%t A167267 (* _Norman Carey_, Jul 03 2012 *)
%o A167267 (PARI)
%o A167267 \\ Produces the triangle when the array is read by antidiagonals
%o A167267 r = (1+sqrt(5))/2;
%o A167267 z = 100;
%o A167267 s(n) = if(n<1, 1, s(n - 1) + 1 + floor(n*r));
%o A167267 p(n) = n + 1 + sum(k=0, n, floor((n - k)/r));
%o A167267 u = v = vector(z + 1);
%o A167267 for(n=1, 101, (v[n] = s(n - 1)));
%o A167267 for(n=1, 101, (u[n] = p(n - 1)));
%o A167267 w(i, j) = v[i] + u[j] + (i - 1) * (j - 1) - 1;
%o A167267 tabl(nn) = {for(n=1, nn, for(k=1, n, print1(w(n - k + 1, k), ", "); ); print(); ); };
%o A167267 tabl(10) \\ _Indranil Ghosh_, Mar 26 2017
%o A167267 (Python)
%o A167267 # Produces the triangle when the array is read by antidiagonals
%o A167267 import math
%o A167267 from sympy import sqrt
%o A167267 r=(1 + sqrt(5))/2
%o A167267 def s(n): return 1 if n<1 else s(n - 1) + 1 + int(math.floor(n*r))
%o A167267 def p(n): return n + 1 + sum(int(math.floor((n - k)/r)) for k in range(n+1))
%o A167267 v=[s(n) for n in range(101)]
%o A167267 u=[p(n) for n in range(101)]
%o A167267 def w(i, j): return u[i - 1] + v[j - 1] + (i - 1) * (j - 1) - 1
%o A167267 for n in range(1, 11):
%o A167267     print([w(k, n - k + 1) for k in range(1, n + 1)]) # _Indranil Ghosh_, Mar 26 2017
%Y A167267 Cf. A001622, A084531, A084531, A283734.
%K A167267 nonn,tabl
%O A167267 1,2
%A A167267 _Clark Kimberling_, Oct 31 2009