OFFSET
0,3
COMMENTS
a(n) = Phi_10(n), where Phi_k is the k-th cyclotomic polynomial.
Number of walks of length 5 between any two distinct nodes of the complete graph K_{n+1} (n>=1). Example: a(1)=1 because in the complete graph AB we have only one walk of length 5 between A and B: ABABAB. - Emeric Deutsch, Apr 01 2004
t^4-t^3+t^2-t+1 is the Alexander polynomial (with negative powers cleared) of the cinquefoil knot (torus knot T(5,2)). The associated Seifert matrix S is [[ -1, -1, 0, -1], [ 0, -1, 0, 0], [ -1, -1, -1, -1], [ 0, -1, 0, -1]]. a(n) = det(transpose(S)-n*S). Cf. A084849. - Peter Bala, Mar 14 2012
For odd n, a(n) * (n+1) / 2 also represents the first integer in a sum of n^5 consecutive integers that equals n^10. - Patrick J. McNab, Dec 26 2016
LINKS
Ray Chandler, Table of n, a(n) for n = 0..10000 (first 1001 terms from Harry J. Smith)
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
G.f.: (1-4*x+16*x^2+6*x^3+5*x^4)/(1-x)^5. - Emeric Deutsch, Apr 01 2004
E.g.f.: exp(x)*(1 + 5*x^2 + 5*x^3 + x^4). - Stefano Spezia, Apr 22 2023
MAPLE
A060884 := proc(n)
numtheory[cyclotomic](10, n) ;
end proc:
seq(A060884(n), n=0..20) ; # R. J. Mathar, Feb 07 2014
MATHEMATICA
Table[1 + Fold[(-1)^(#2)*n^(#2) + #1 &, Range[0, 4]], {n, 0, 33}] (* or *)
CoefficientList[Series[(1 - 4 x + 16 x^2 + 6 x^3 + 5 x^4)/(1 - x)^5, {x, 0, 33}], x] (* Michael De Vlieger, Dec 26 2016 *)
Table[n^4-n^3+n^2-n+1, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {1, 1, 11, 61, 205}, 40] (* Harvey P. Dale, Sep 08 2018 *)
PROG
(PARI) a(n) = { n^4 - n^3 + n^2 - n + 1 } \\ Harry J. Smith, Jul 13 2009
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
N. J. A. Sloane, May 05 2001
STATUS
approved