1,2

Note: the underlying function N2Qv (see the Maple code) maps natural numbers 1, 2, 3, 4, 5, ..., through all the positive rationals in the open range (0,1): 1/2, 1/3, 2/3, 1/4, 2/5, 3/5, ... bijectively to the union of positive rationals and quadratic surds. A065937 gives similar mapping involving the inverse of the standard Minkowki's question mark function.

Note the symmetry of rows 0; 5,5; 0,2,2,0; 2,3,0,3,3,0,3,2; 5,13,17,2,17,37,5,13,13,5,37,17,2,17,13,5; ... emanating from the symmetry present in A007306.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for sequences related to Minkowski's question mark function

Index entries for sequences related to Stern's sequences

The first few values for this mapping are N2Qv(1) = 1, N2Qv(2) = (sqrt(5)-1)/2, N2Qv(3) = (sqrt(5)+1)/2, N2Qv(4) = 1/2, N2Qv(5) = sqrt(2)/2, N2Qv(6) = sqrt(2), N2Qv(7) = 2, N2Qv(8) = sqrt(2)-1

[seq(find_sqrt(N2Qv(j)), j=1..512)];

N2Qv := proc(n) local m; m := n + 2^floor_log_2(n); Inverse_of_Variant_of_MinkowskisQMark(A007305(m+1)/A047679(m-1)); end;

Inverse_of_Variant_of_MinkowskisQMark := proc(r) local x, y, b, d, k, s, i, q; x := numer(r); y := denom(r); if(y = 2*x) then RETURN(1); fi; b := []; d := []; k := 0; s := 0; i := 0; while(x <> 0) do q := floor(x/y); if(i > 0) then b := [op(b), q]; d := [op(d), x]; fi; x := 2*(x-(q*y)); if(member(x, d, 'k') and (k > 1) and (b[k] <> b[k-1]) and (q <> floor(x/y))) then s := eval_periodic_confrac_tail(list2runcounts(b[k..nops(b)])); b := b[1..(k-1)]; break; fi; i := i+1; od; if(0 = k) then b := b[1..(nops(b)-1)]; b := [op(b), b[nops(b)]]; fi; if(r < (1/2)) then RETURN(factor(eval_confrac([0, op(list2runcounts(b))], s))); else RETURN(factor(eval_confrac(list2runcounts(b), s))); fi; end;

eval_confrac := proc(c, z) local x, i; x := z; for i in reverse(c) do x := (`if`((0=x), x, (1/x)))+i; od; RETURN(x); end;

eval_periodic_confrac_tail := proc(c) local x, i, u, r; x := (eval_confrac(c, u) - u) = 0; r := [solve(x, u)]; RETURN(max(r[1], r[2])); end;

list2runcounts := proc(b) local a, p, y, c; if(0 = nops(b)) then RETURN([]); fi; a := []; c := 0; p := b[1]; for y in b do if(y <> p) then a := [op(a), c]; c := 0; p := y; fi; c := c+1; od; RETURN([op(a), c]); end;

find_sqrt := proc(x) local n, i, y; n := nops(x); if(n < 2) then RETURN(0); fi; if((2 = n) and (`^` = op(0, x)) and (1/2 = op(2, x))) then RETURN(op(1, x)); else for i from 0 to n do y := find_sqrt(op(i, x)); if(y <> 0) then RETURN(y); fi; od; RETURN(0); fi; end;

a(n) = A065937(A065934(n)). Positions of the zeros are given by A065810. Positions of sqrt(n) in this mapping: A065938.

Sequence in context: A144776 A065937 A115144 this_sequence A021649 A144184 A119380

Adjacent sequences: A065933 A065934 A065935 this_sequence A065937 A065938 A065939

nonn

Antti Karttunen Dec 07 2001. Description clarified by Antti Karttunen (his-firstname.his-surname(AT)gmail.com), Aug 26 2006