A215573 a(n) = n*(n+1)*(2n+1)/6 modulo n.

0, 1, 2, 2, 0, 1, 0, 4, 6, 5, 0, 2, 0, 7, 10, 8, 0, 3, 0, 10, 14, 11, 0, 4, 0, 13, 18, 14, 0, 5, 0, 16, 22, 17, 0, 6, 0, 19, 26, 20, 0, 7, 0, 22, 30, 23, 0, 8, 0, 25, 34, 26, 0, 9, 0, 28, 38, 29, 0, 10, 0, 31, 42, 32, 0, 11, 0, 34, 46, 35, 0, 12, 0, 37, 50, 38

Offset: 1

Author

Zak Seidov, Aug 16 2012

Keywords

• nonn
• easy

Comments

a(n) = 0 for n = 6k +- 1, that is, A007310 (numbers congruent to 1 or 5 mod 6).

Graph consists of 4 linear patterns.

Links

• Zak Seidov, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

Crossrefs

Cf. A000330, A007310, A048153.

Programs

Maple

seq(modp(n*(n+1)*(2*n+1)/6,n),n=1..100); # Muniru A Asiru, Feb 07 2019

Mathematica

Table[Mod[(n(n+1)(2n+1))/6,n],{n,80}] (* or *) LinearRecurrence[{0,0,0,0,0,2,0,0,0,0,0,-1},{0,1,2,2,0,1,0,4,6,5,0,2},80] (* Harvey P. Dale, Aug 25 2023 *)

PARI

a(n)=n*(n+1)*(2*n+1)/6 % n; \\ Michel Marcus, Oct 19 2013

PARI

concat(0, Vec(x^2*(1 + 2*x + 2*x^2 + x^4 + 2*x^6 + 2*x^7 + x^8) / ((1 - x)^2*(1 + x)^2*(1 - x + x^2)^2*(1 + x + x^2)^2) + O(x^80))) \\ Colin Barker, Feb 07 2019

Python

def A215573(n): return n*(n-1)*((n<<1)-1)//6%n # Chai Wah Wu, Jun 03 2024

Formula

a(n) = A000330(n) mod n.

From Colin Barker, Feb 07 2019: (Start)

G.f.: x^2*(1 + 2*x + 2*x^2 + x^4 + 2*x^6 + 2*x^7 + x^8) / ((1 - x)^2*(1 + x)^2*(1 - x + x^2)^2*(1 + x + x^2)^2).

a(n) = 2*a(n-6) - a(n-12) for n>12. (End)

a(6*n) = n, a(6*n+1) = 0, a(6*n+2) = 3*n+1, a(6*n+3) = 4*n+2, a(6*n+4) = 3*n+2, a(6*n+5) = 0. - Philippe Deléham, Mar 05 2023

a(n) = A048153(n) mod n. - Alois P. Heinz, Jun 03 2024

a(n) = A000330(n-1) mod n. - Chai Wah Wu, Jun 03 2024

Sum_{k=1..n} a(k) ~ (11/72) * n^2. - Amiram Eldar, Apr 05 2025