A060884 - OEIS

%I #45 Dec 23 2024 13:06:08

%S 1,1,11,61,205,521,1111,2101,3641,5905,9091,13421,19141,26521,35855,

%T 47461,61681,78881,99451,123805,152381,185641,224071,268181,318505,

%U 375601,440051,512461,593461,683705,783871,894661,1016801,1151041,1298155,1458941,1634221

%N a(n) = n^4 - n^3 + n^2 - n + 1.

%C a(n) = Phi_10(n), where Phi_k is the k-th cyclotomic polynomial.

%C 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

%C 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

%C 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

%H Ray Chandler, <a href="/A060884/b060884.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from Harry J. Smith)

%H <a href="/index/Cy#CyclotomicPolynomialsValuesAtX">Index to values of cyclotomic polynomials of integer argument</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F G.f.: (1-4*x+16*x^2+6*x^3+5*x^4)/(1-x)^5. - _Emeric Deutsch_, Apr 01 2004

%F E.g.f.: exp(x)*(1 + 5*x^2 + 5*x^3 + x^4). - _Stefano Spezia_, Apr 22 2023

%p A060884 := proc(n)

%p         numtheory[cyclotomic](10,n) ;

%p end proc:

%p seq(A060884(n),n=0..20) ; # _R. J. Mathar_, Feb 07 2014

%t Table[1 + Fold[(-1)^(#2)*n^(#2) + #1 &, Range[0, 4]], {n, 0, 33}] (* or *)

%t 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 *)

%t 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 *)

%o (PARI) a(n) = { n^4 - n^3 + n^2 - n + 1 } \\ _Harry J. Smith_, Jul 13 2009

%Y Cf. A084849, A246392, A259257.

%K nonn,easy,changed

%O 0,3

%A _N. J. A. Sloane_, May 05 2001