A071934 -id:A071934 - cp's OEIS frontend

A071932 a(n) = 4*Sum_{i=1..n} K(i,i+1) - n, where K(x,y) is the Kronecker symbol (x/y).

1, 4, 7, 2, 5, 8, 11, 6, 1, 4, 7, 2, 5, 8, 11, 6, 9, 12, 15, 10, 5, 8, 11, 6, 1, 4, 7, 2, 5, 8, 11, 6, 9, 12, 15, 10, 13, 16, 19, 14, 9, 12, 15, 10, 5, 8, 11, 6, 9, 12, 15, 10, 5, 8, 11, 6, 1, 4, 7, 2, 5, 8, 11, 6, 9, 12, 15, 10, 13, 16, 19, 14, 9, 12, 15, 10, 13, 16, 19, 14, 17, 20, 23

Offset: 3

Benoit Cloitre, Jun 14 2002

a(n) > 0 for n > 2 and it seems that a(n)/log(n) is bounded: a(n) < 4*log(n) for n sufficiently large. Does lim_{n->infinity} a(n)/log(n) exist?

G. C. Greubel, Table of n, a(n) for n = 3..5000

Cf. A071933, A071934.

Table[4*Sum[KroneckerSymbol[j, j+1], {j, n}] - n, {n, 3, 80}] (* G. C. Greubel, Mar 17 2019 *)

for(n=3,100,print1(4*sum(i=1,n,kronecker(i,i+1))-n,","))

[4*sum(kronecker_symbol(j+1,j) for j in (1..n))-n for n in (3..80)] # G. C. Greubel, Mar 17 2019

A071933 a(n) = Sum_{i=1..n} K(i,i+1), where K(x,y) is the Kronecker symbol (x/y).

Original entry on oeis.org

1, 0, 1, 2, 3, 2, 3, 4, 5, 4, 3, 4, 5, 4, 5, 6, 7, 6, 7, 8, 9, 8, 7, 8, 9, 8, 7, 8, 9, 8, 9, 10, 11, 10, 11, 12, 13, 12, 13, 14, 15, 14, 13, 14, 15, 14, 13, 14, 15, 14, 15, 16, 17, 16, 15, 16, 17, 16, 15, 16, 17, 16, 17, 18, 19, 18, 19, 20, 21, 20, 21, 22, 23, 22, 21, 22, 23, 22, 23

Offset: 1

Benoit Cloitre, Jun 14 2002

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A071932, A071934.

Table[Sum[KroneckerSymbol[j, j+1], {j, 1, n}], {n, 1, 80}] (* G. C. Greubel, Mar 17 2019 *)

for(n=1,100,print1(sum(i=1,n,kronecker(i,i+1)),","))

[sum(kronecker_symbol(j,j+1) for j in (1..n)) for n in (1..80)] # G. C. Greubel, Mar 17 2019

a(n) = n/4 + O(n), asymptotically. [Perhaps O(n) should be o(n)? - N. J. A. Sloane, Mar 26 2019]

In fact we have a(n) = n/4 + O(log(n)). More precisely let c(n)=A037800(n) then we get a(8n)=2n+2+2c(n), a(8n+1)=2n+3+2c(n), a(8n+2)=2n+2+2c(n), a(8n+3)=2n+2+2c(n)+(-1)^n, a(8n+4)=2n+3+2c(n)+(-1)^n, a(8n+5)=2n+4+2c(n)+(-1)^n, a(8n+6)=2n+3+2c(n)+(-1)^n, a(8n+7)=2n+3+2c(n+1). - Benoit Cloitre, Mar 30 2019

cp's OEIS Frontend

A071932 a(n) = 4*Sum_{i=1..n} K(i,i+1) - n, where K(x,y) is the Kronecker symbol (x/y).

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