A165601 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.

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, 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, 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

Offset

0,3

Links

A. Karttunen, Table of n, a(n) for n = 0..269535

Mathematica

Table[Sum[JacobiSymbol[i, 4n + 3], {i, 0, 2n + 1}], {n, 0, 100}] (* Indranil Ghosh, May 13 2017 *)

Prog

(Scheme)
(define (A165601 n) (let ((w (A004767 n))) (add (lambda (i) (jacobi-symbol i w)) 0 (/ (-1+ w) 2))))
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))
(define jacobi-symbol fix:jacobi-symbol)
(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)))))))))
(PARI) a(n) = sum(i=0, 2*n + 1, kronecker(i, 4*n + 3)); \\ Indranil Ghosh, May 13 2017
(Python)
from sympy import jacobi_symbol as J
def a(n): return sum([J(i, 4*n + 3) for i in range(2*n + 2)]) # Indranil Ghosh, May 13 2017

Crossrefs

Trisections: A165604, A165605, A165606.

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

Sequence in context: A370903 A392676 A153092 * A324030 A275821 A291674

Adjacent sequences: A165598 A165599 A165600 * A165602 A165603 A165604

Keyword

nonn

Author

Antti Karttunen, Oct 06 2009

Status

approved