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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. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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 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.

LINKS

Table of n, a(n) for n=1..77.

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

EXAMPLE

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

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;

CROSSREFS

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

Sequence in context: A286979 A176184 A176144 * A021649 A200257 A236023

Adjacent sequences:  A065933 A065934 A065935 * A065937 A065938 A065939

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 07 2001

EXTENSIONS

Description clarified by Antti Karttunen, Aug 26 2006

STATUS

approved

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Last modified December 13 16:57 EST 2017. Contains 295959 sequences.