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A262324 Conway's triangle of "happy factorizations" (flattened).

%I A262324 #21 Sep 20 2015 04:09:39
%S A262324 0,0,1,1,1,2,1,3,2,2,1,5,2,3,7,1,2,4,3,3,1,10,1,11,3,4,1,13,7,2,3,5,4,
%T A262324 4,1,17,2,9,1,19,4,5,3,7,2,11,23,1,4,6,5,5,1,26,1,27,7,4,1,29,5,6,31,
%U A262324 1,16,2,11,3,17,2,5,7,6,6,1,37,2,19,3,13,2,20,1,41,6,7,1,43,11,4,5,9,23,2,47,1,6,8,7,7
%N A262324 Conway's triangle of "happy factorizations" (flattened).
%C A262324 Conway's triangle is listed by increasing couple products, with duplicate squares removed.
%H A262324 J. H. Conway, <a href="http://www.cs.uwaterloo.ca/journals/JIS/happy.html">On Happy Factorizations</a>, J. Integer Sequences, Vol. 1, 1998, #1.
%e A262324 Triangle begins:
%e A262324 {0,0},
%e A262324 {1,1},
%e A262324 {1,2},   {1,3},  {2,2},
%e A262324 {1,5},   {2,3},  {7,1},  {2,4}, {3,3},
%e A262324 {1,10}, {1,11},  {3,4}, {1,13}, {7,2},  {3,5},  {4,4},
%e A262324 {1,17},  {2,9}, {1,19},  {4,5}, {3,7}, {2,11}, {23,1}, {4,6}, {5,5},
%e A262324 ...
%e A262324 The original triangle (adapted and truncated):
%e A262324                            ...
%e A262324                       5^2  ...
%e A262324                  4^2  1*26 ...
%e A262324             3^2  1*17 1*27 ...
%e A262324         2^2 1*10 2*9  7*4  ...
%e A262324     1^2 1*5 1*11 1*19 1*29 ...
%e A262324 0^2 1*2 2*3 3*4  4*5  5*6  ...
%e A262324 1^2 1*3 7*1 1*13 3*7  31*1 ...
%e A262324     2^2 2*4 7*2  2*11 16*2 ...
%e A262324         3^2 3*5  23*1 11*3 ...
%e A262324             4^2  4*6  17*2 ...
%e A262324                  5^2  5*7  ...
%e A262324                       6^2  ...
%e A262324                            ...
%t A262324 f[0] = {0, 0}; f[32] = {16, 2}(* to speed up *); f[n_] := Do[c = n/b; If[b == c, Return[{b, b}]]; r1 = Reduce[r >= 0 && s >= 0 && c > 1 && b*r^2 + 1 == c*s^2, {r, s}, Integers]; If[r1 =!= False, Return[{b, c}]]; r2 = Reduce[r >= 0 && s >= 0 && r == 2x + 1 && s == 2y + 1 && b*r^2 + 2 == c *s^2, {r, s, x, y}, Integers]; If[r2 =!= False, Return[{b, c}]], {b, Divisors[n]}]; Table[Print["f(", n, ") = ", fn = f[n]]; fn, {n, 0, 49}] // Flatten
%Y A262324 Cf. A007966, A007969, A007970, A191854, A191855, A191856.
%K A262324 tabf,nonn
%O A262324 0,6
%A A262324 _Jean-François Alcover_, Sep 18 2015