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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. 13
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 (list; graph; refs; listen; history; text; internal format)
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

(MIT 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 xrange(2*n + 2)]) # Indranil Ghosh, May 13 2017

CROSSREFS

Trisections: A165604, A165605, A165606.

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

Sequence in context: A230406 A214254 A153092 * A275821 A291674 A265157

Adjacent sequences:  A165598 A165599 A165600 * A165602 A165603 A165604

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 06 2009

STATUS

approved

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Last modified December 14 00:21 EST 2017. Contains 295976 sequences.