A165460 - OEIS

%I #13 Dec 07 2019 12:18:26

%S 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,

%T 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,

%U 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

%N 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.

%C 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.)

%H A. Karttunen, <a href="/A165460/b165460.txt">Table of n, a(n) for n = 0..21845</a>

%t Table[Sum[JacobiSymbol[i, 12n + 7], {i, 0, 4n + 2}], {n, 0, 100}] (* _Indranil Ghosh_, May 13 2017 *)

%o (MIT Scheme:)

%o (define (A165460 n) (let ((w (A017605 n))) (add (lambda (i) (jacobi-symbol i w)) 0 (/ (-1+ w) 3))))

%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, 4*n + 2, kronecker(i, 12*n + 7)); \\ _Indranil Ghosh_, May 13 2017

%o (Python)

%o from sympy import jacobi_symbol as J

%o def a(n): return sum([J(i, 12*n + 7) for i in range(4*n + 3)]) # _Indranil Ghosh_, May 13 2017

%Y Cf. A165461, A165462, A165463, A165605.

%K nonn

%O 0,1

%A _Antti Karttunen_, Oct 06 2009