A165601 - OEIS

%I #11 Dec 07 2019 12:18:26

%S 1,1,3,2,3,3,1,3,6,4,3,5,6,4,9,2,3,7,2,5,9,6,6,8,0,5,9,8,6,10,6,5,15,

%T 2,9,10,0,7,12,10,3,11,6,2,15,8,6,13,12,9,12,0,9,14,12,7,15,12,6,15,1,

%U 6,21,12,12,13,6,11,0,6,9,14,12,8,24,10,9,19,0,10,12,12,9,18,18,1,15

%N Midpoint height of Jacobi-bridge, computed for 4n+3. a(n) = Sum_{i=0..(2n+1)} J(i,4n+3), where J(i,m) is the Jacobi symbol.

%H A. Karttunen, <a href="/A165601/b165601.txt">Table of n, a(n) for n = 0..269535</a>

%t Table[Sum[JacobiSymbol[i, 4n + 3], {i, 0, 2n + 1}], {n, 0, 100}] (* _Indranil Ghosh_, May 13 2017 *)

%o (MIT Scheme:)

%o (define (A165601 n) (let ((w (A004767 n))) (add (lambda (i) (jacobi-symbol i w)) 0 (/ (-1+ w) 2))))

%o (define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

%o (define jacobi-symbol fix:jacobi-symbol)

%o (define (fix:jacobi-symbol p q) (if (not (and (fix:fixnum? p) (fix:fixnum? q) (fix:= 1 (fix:and q 1)))) (error "fix:jacobi-symbol: args must be fixnums, and 2. arg should be odd: " p q) (let loop ((p p) (q q) (s 0)) (cond ((fix:zero? p) 0) ((fix:= 1 p) (fix:- 1 (fix:and s 2))) ((fix:= 1 (fix:and p 1)) (loop (fix:remainder q p) p (fix:xor s (fix:and p q)))) (else (loop (fix:lsh p -1) q (fix:xor s (fix:xor q (fix:lsh q -1)))))))))

%o (PARI) a(n) = sum(i=0, 2*n + 1, kronecker(i, 4*n + 3)); \\ _Indranil Ghosh_, May 13 2017

%o (Python)

%o from sympy import jacobi_symbol as J

%o def a(n): return sum([J(i, 4*n + 3) for i in range(2*n + 2)]) # _Indranil Ghosh_, May 13 2017

%Y Trisections: A165604, A165605, A165606.

%Y Cf. A165602, A165603, A165460, A166045, A166046, A166047.

%K nonn

%O 0,3

%A _Antti Karttunen_, Oct 06 2009