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A357097 A multiplication table for the rows of the extended Wythoff array. See comments for definition.

%I A357097 #28 Jan 05 2025 19:51:42
%S A357097 0,1,1,2,15,2,3,8,8,3,4,12,4,12,4,5,44,18,18,44,5,6,19,24,27,24,19,6,
%T A357097 7,62,28,96,96,28,62,7,8,26,34,42,128,42,34,26,8,9,30,14,51,56,56,51,
%U A357097 14,30,9,10,91,44,57,180,65,180,57,44,91,10,11,37,50,66,76,79,79,76,66,50,37,11
%N A357097 A multiplication table for the rows of the extended Wythoff array. See comments for definition.
%C A357097 Square array A(x,y), x >= 0, y >= 0, defined as follows:
%C A357097 (1) Extend the Wythoff array infinitely to the left, maintaining the Fibonacci recurrence (see A287870 examples). We denote this extended array as eW(n,m), n >= 0, m any integer, indexed such that eW(n,0) = n. From each row n, form the set of pairs S_n = {(eW(n,m+1),eW(n,m)) : integer m)}.
%C A357097 (2) Define addition and multiplication of pairs by (j1,k1) + (j2,k2) = (j1+j2, k1+k2) and (j1,k1) o (j2,k2) = (j1*j2 + k1*k2, j1*k2 + k1*j2 - k1*k2). (This defines a commutative ring with identity (1,0).)
%C A357097 (3) For nonnegative integers x and y, there is an integer z such that for every pair (j_x,k_x) in S_x and every pair (j_y,k_y) in S_y, (j_x,k_x) o (j_y,k_y) is in S_z. Define A(x,y) = z.
%C A357097 As a binary operation, A(.,.) is analogous to multiplication of coefficients in scientific numeric notation. The column position, m, used to define a pair in (1) above does not affect the eventual outcome, A(x,y), in (3), as no special pairs are selected from the pairs in S_x or S_y. The column position is analogous to the exponent. Notice also A(1,1) = 15 is substantially larger than A(2,2) = 4. This can be seen as analogous to 0.3 * 0.4 = 0.12 requiring more digits than 0.5 * 0.8 = 0.4.
%H A357097 Peter G. Anderson, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/52-5/Anderson.pdf">More Properties of the Zeckendorf Array</a>, Fib. Quart. 52-5 (2014), 15-21.
%H A357097 P. Arnoux, <a href="http://dx.doi.org/10.1016/0893-9659(89)90078-5">Some remarks about Fibonacci multiplication</a>, Appl. Math. Lett. 2 (1989), 319-320.
%H A357097 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/33-1/kimberling.pdf">The Zeckendorf array equals the Wythoff array</a>, Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 3-8.
%F A357097 A(x,y) = g(j1*j2 + k1*k2, j1*k2 + k1*j2 - k1*k2), where j1 = A035336(x+1), j2 = A035336(y+1), k1 = A003622(x+1), k2 = A003622(y+1) and g(j,k) = (if j = A000201(k+1) then k otherwise g(k,j-k)).
%F A357097 A(x,y) = A(y,x).
%F A357097 A(x,0) = x.
%F A357097 A(x, A(y,z)) = A(A(x,y), z).
%F A357097 A022344(A(x,y)) = A022344(x) * A022344(y).
%F A357097 A(A019586(x), A019586(y)) = A019586(A101330(x,y)). (conjectured)
%e A357097 Calculation for A(1,2). Rows 1 and 2 of A287870 (indexed from 0) start 1, 3, ... and 2, 4, ... . So we may use the pairs (3,1) and (4,2). The defined multiplication gives (3*4 + 1*2, 3*2 + 4*1 - 1*2) = (14,8). 8, 14 , ... is in row 8 of A287870, so A(1,2) = 8.
%e A357097 For A(1,1), we start as above to get (3*3 + 1*1, 3*1 + 3*1 - 1*1) = (10,5). In the more general case, we form a sequence using the Fibonacci recurrence (as ..., 5, 10, ... may be in the extension leftwards of A287870). This starts 5, 10, 5+10=15, 10+15=25, 15+25=40, ... . We observe 15, 25, 40, ... is in row 15. So A(1,1) = 15.
%e A357097 The top left corner of the array:
%e A357097   0   1   2    3    4    5    6    7    8    9
%e A357097   1  15   8   12   44   19   62   26   30   91
%e A357097   2   8   4   18   24   28   34   14   44   50
%e A357097   3  12  18   27   96   42   51   57   66  198
%e A357097   4  44  24   96  128   56  180   76   88  264
%e A357097   5  19  28   42   56   65   79   33  102  116
%e A357097   6  62  34   51  180   79  253  107  124  371
%e A357097   7  26  14   57   76   33  107   45  138  157
%e A357097   8  30  44   66   88  102  124  138  160  182
%e A357097   9  91  50  198  264  116  371  157  182  544
%o A357097 (PARI) lowerw(n) = (n+sqrtint(5*n^2))\2 ; \\ A000201
%o A357097 upperw(n) = (sqrtint(n^2*5)+n*3)\2; \\ A001950
%o A357097 compoundw(n) = (sqrtint(n^2*5)+n*3)\2 - 1; \\ A003622
%o A357097 wpair(p) = {my(x=p[2], y = p[1], z); while(1, my(n=1, ok=1); while(ok, my(xx = lowerw(n), yy = upperw(n)); if ((x == xx) && (y == yy), return([xx, yy])); if (xx > x, ok = 0); n++;); z = y; y += x; x = z;);}
%o A357097 row(p) = {my(x=p[1], y=p[2], u); while (1, my(n=1, ok=1); while(ok, my(xx = lowerw(n), yy = compoundw(n)); if ((x == xx) && (y == yy), return(n)); if (xx > x, ok = 0); n++;); u = x; x = y - u; y = u;);} \\ similar to A120873
%o A357097 wrow(p) = row(wpair(p));
%o A357097 prodpair(v1, v2) = my(j1=v1[1], j2 = v2[1], k1 = v1[2], k2 = v2[2]); [j1*j2 + k1*k2, j1*k2 + k1*j2 - k1*k2];
%o A357097 pair(n) = [lowerw(n+1), n];
%o A357097 T(n, k) = my(pn = pair(n), pk = pair(k), px = prodpair(pn, pk)); wrow(px)-1; \\ _Michel Marcus_, Sep 18 2022
%Y A357097 Cf. A022344 (norm), A035513, A120873, A287870, A348853.
%Y A357097 See the formula section for the relationships with A000201, A003622, A019586, A035336, A101330.
%K A357097 nonn,tabl
%O A357097 0,4
%A A357097 _Peter Munn_, Sep 11 2022