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A165460 The height at the 1/3 point of Jacobi-bridge, 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) (jacobi-symbol 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 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, 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

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Last modified June 24 21:50 EDT 2021. Contains 345433 sequences. (Running on oeis4.)