%I #27 Apr 23 2023 07:25:52
%S -1,0,21,182,819,2604,6665,14706,29127,53144,90909,147630,229691,
%T 344772,501969,711914,986895,1340976,1790117,2352294,3047619,3898460,
%U 4929561,6168162,7644119,9390024,11441325,13836446,16616907,19827444,23516129,27734490,32537631,37984352,44137269
%N a(n) = n^5 - n^4 + n^3 - n^2 + n - 1.
%C Number of walks of length 6 between any two distinct nodes of the complete graph K_{n+1} (n>=1). - _Emeric Deutsch_, Apr 01 2004
%C For odd n, a(n) * (n+1) / 2 + 1 also represents the first integer in a sum of n^6 consecutive integers that equals n^12. - _Patrick J. McNab_, Dec 26 2016
%H Harry J. Smith, <a href="/A062159/b062159.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F a(n) = round(n^6/(n+1)) for n>2 = A062160(n,6).
%F G.f.: (76x^3 + 6x^2 + 27x^4 + 6x^5 + 6x - 1)/(1-x)^6 (for the signed sequence). - _Emeric Deutsch_, Apr 01 2004
%F a(n) = (n^6 - 1)/(n+1). a(n) = (n-1)(n^2 - n + 1)(n^2 + n + 1) = (n-1)*A002061(n)*A002061(n+1). - _Alexander Adamchuk_, Apr 12 2006
%F a(0)=-1, a(1)=0, a(2)=21, a(3)=182, a(4)=819, a(5)=2604, a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - _Harvey P. Dale_, Dec 20 2015
%F E.g.f.: exp(x)*(x^5 + 9*x^4 + 20*x^3 + 10*x^2 + x - 1). - _Stefano Spezia_, Apr 22 2023
%e a(4) = 4^5 - 4^4 + 4^3 - 4^2 + 4 - 1 = 1024 - 256 + 64 - 16 + 4 - 1 = 819.
%p A062159:=n->n^5-n^4+n^3-n^2+n-1; seq(A062159(k), k=0..100); # _Wesley Ivan Hurt_, Nov 06 2013
%t Table[n^5-n^4+n^3-n^2+n-1, {n,0,100}] (* _Wesley Ivan Hurt_, Nov 06 2013 *)
%t LinearRecurrence[{6,-15,20,-15,6,-1},{-1,0,21,182,819,2604},40] (* _Harvey P. Dale_, Dec 20 2015 *)
%o (PARI) { for (n=0, 1000, write("b062159.txt", n, " ", n*(n*(n*(n*(n - 1) + 1) - 1) + 1) - 1) ) } \\ _Harry J. Smith_, Aug 02 2009
%Y Cf. A023443, A002061, A062158, A060884, A060888.
%Y Cf. A002061.
%K easy,sign
%O 0,3
%A _Henry Bottomley_, Jun 08 2001
%E More terms from _Emeric Deutsch_, Apr 01 2004