

A060886


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


6



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
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OFFSET

0,3


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
a(n) = Phi_12(n), where Phi_k is the kth cyclotomic polynomial.


LINKS

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index to values of cyclotomic polynomials of integer argument
Index entries for linear recurrences with constant coefficients, signature (5,10,10,5,1).


FORMULA

G.f.: (14*x+18*x^2+8*x^3+x^4)/(1x)^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(n1)10*a(n2)+10*a(n3)5*a(n4)+a(n5), for n>4.  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 *)


PROG

(PARI) { for (n=0, 1000, write("b060886.txt", n, " ", n^4  n^2 + 1); ) } \\ Harry J. Smith, Jul 14 2009
(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(n1)10*Self(n2)+10*Self(n3)5*Self(n4)+Self(n5): n in [1..40]]; // Vincenzo Librandi, Dec 20 2015


CROSSREFS

Sequence in context: A084218 A175361 A125258 * A081586 A143008 A107963
Adjacent sequences: A060883 A060884 A060885 * A060887 A060888 A060889


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane, May 05 2001


STATUS

approved



