A065936 a(n) is the integer (reduced squarefree) under the square root obtained when the inverse of a variant of Minkowski's question mark function is applied to the n-th ratio A007305(n+1)/A007306(n+1) in the left-hand subtree of Stern-Brocot tree and zero when it results a rational value.

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, 3, 17, 3, 37, 21, 13, 10, 37, 3, 401, 6, 13, 10, 401, 0, 17, 17, 0, 401, 10, 13, 6, 401, 3, 37, 10, 13, 21, 37, 3, 17, 3, 0, 37, 10, 0, 401, 506, 17, 5, 401, 37, 21610, 730, 5, 1373

Offset: 1

Author

Antti Karttunen, Dec 07 2001

Keywords

• nonn

Comments

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 Minkowski'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.

Examples

			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
		

Links

• Eric Weisstein's World of Mathematics, Quadratic Irrational Number.
• Index entries for sequences related to Minkowski's question mark function
• Index entries for sequences related to Stern's sequences

Crossrefs

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

Programs

Maple

[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;

Extensions

Description clarified by Antti Karttunen, Aug 26 2006