A060886 a(n) = n^4 - n^2 + 1.

1, 1, 13, 73, 241, 601, 1261, 2353, 4033, 6481, 9901, 14521, 20593, 28393, 38221, 50401, 65281, 83233, 104653, 129961, 159601, 194041, 233773, 279313, 331201, 390001, 456301, 530713, 613873, 706441, 809101, 922561, 1047553, 1184833, 1335181, 1499401

Offset: 0

Author

N. J. A. Sloane, May 05 2001

Keywords

• nonn
• easy

Comments

All positive divisors of a(n) are congruent to 1, modulo 12. Proof: If p is an odd prime different from 3 then n^4 - n^2 + 1 = 0 (mod p) implies: (a) (2n^2 - 1)^2 = -3 (mod p), whence p = 1 (mod 6); and (b) (n^2 - 1)^2 = -n^2 (mod p), whence p = 1 (mod 4). - Nick Hobson, Nov 13 2006

Appears to be the number of distinct possible sums of a set of n distinct integers between 1 and n^3. Checked up to n = 4. - Dylan Hamilton, Sep 21 2010

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000
• John Elias, Illustration of initial terms: chain-linked squares
• Index to values of cyclotomic polynomials of integer argument
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Programs

Magma

[n^4 - n^2 + 1: n in [0..40]]; /* or */ I:=[1,1,13, 73,241]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, Dec 20 2015

Maple

A060886 := proc(n)
        numtheory[cyclotomic](12,n) ;
end proc:
seq(A060886(n),n=0..20) ; # R. J. Mathar, Feb 07 2014

Mathematica

(Range[0, 29]^2 - 1/2)^2 + 3/4 (* Alonso del Arte, Dec 20 2015 *)
Table[n^4 - n^2 + 1, {n, 0, 25}] (* Vincenzo Librandi, Dec 20 2015 *)

PARI

a(n) = n^4 - n^2 + 1; \\ Harry J. Smith, Jul 14 2009

Formula

a(n) = Phi_12(n), where Phi_k is the k-th cyclotomic polynomial.

G.f.: (1-4*x+18*x^2+8*x^3+x^4)/(1-x)^5. - Colin Barker, Apr 21 2012

a(n) = (n^2 - 1/2)^2 + 3/4. - Alonso del Arte, Dec 20 2015

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5), for n>4. - Vincenzo Librandi, Dec 20 2015