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

1,3

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).

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

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 *)

Prog

(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

Crossrefs

Cf. A000330, A007310.

Sequence in context: A251690 A187752 A295181 * A163537 A219946 A117449

Adjacent sequences: A215570 A215571 A215572 * A215574 A215575 A215576

Keyword

nonn,easy

Author

Zak Seidov, Aug 16 2012

Status

approved