

A165460


The height at the 1/3 point of Jacobibridge, computed for 12n+7. a(n) = Sum_{i=0..(4n+2)} J(i,12n+7), where J(i,m) is the Jacobi symbol.


8



2, 2, 6, 2, 8, 2, 10, 4, 10, 4, 10, 6, 14, 2, 4, 4, 18, 6, 14, 4, 12, 8, 22, 6, 16, 6, 20, 6, 2, 8, 18, 6, 28, 4, 20, 4, 30, 12, 14, 0, 14, 6, 28, 10, 28, 6, 32, 10, 16, 8, 26, 10, 26, 6, 24, 8, 36, 10, 28, 8, 26, 10, 30, 8, 0, 10, 32, 14, 18, 12, 0, 14, 44, 6, 32, 6, 38, 0, 32, 8, 22
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,1


COMMENTS

Conjecture: a(2n) = 2*A165605(2n) and a(2n+1) = (2/3)*A165605(2n+1).  Antti Karttunen, Oct 05 2009. (If true, then implies also the truth of conjecture in A165462.)


LINKS

A. Karttunen, Table of n, a(n) for n = 0..21845


MATHEMATICA

Table[Sum[JacobiSymbol[i, 12n + 7], {i, 0, 4n + 2}], {n, 0, 100}] (* Indranil Ghosh, May 13 2017 *)


PROG

(MIT Scheme:)
(define (A165460 n) (let ((w (A017605 n))) (add (lambda (i) (jacobisymbol i w)) 0 (/ (1+ w) 3))))
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))
(define jacobisymbol fix:jacobisymbol)
(define (fix:jacobisymbol p q) (if (not (and (fix:fixnum? p) (fix:fixnum? q) (fix:= 1 (fix:and q 1)))) (error "fix:jacobisymbol: 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, 4*n + 2, kronecker(i, 12*n + 7)); \\ Indranil Ghosh, May 13 2017
(Python)
from sympy import jacobi_symbol as J
def a(n): return sum([J(i, 12*n + 7) for i in range(4*n + 3)]) # Indranil Ghosh, May 13 2017


CROSSREFS

Cf. A165461, A165462, A165463, A165605.
Sequence in context: A221438 A327991 A193322 * A242649 A294876 A293514
Adjacent sequences: A165457 A165458 A165459 * A165461 A165462 A165463


KEYWORD

nonn


AUTHOR

Antti Karttunen, Oct 06 2009


STATUS

approved



