A061167 - OEIS

%I #42 Sep 08 2022 08:45:03

%S 0,0,30,240,1020,3120,7770,16800,32760,59040,99990,161040,248820,

%T 371280,537810,759360,1048560,1419840,1889550,2476080,3199980,4084080,

%U 5153610,6436320,7962600,9765600,11881350,14348880,17210340,20511120,24299970,28629120

%N a(n) = n^5 - n.

%C (b^2+c^2)/(bc+1) is an integer if {b,c} are of the form {0,n}, {n,n^3}, {n^3,n^5-n}, {n^5-n,n^7-2n^3}, {n^7-2n^3,n^9-3n^5+n}, etc. for some n, in which case the division results in n^2. Cf. A052530.

%C Convolution of A033429 by A033581. - _R. J. Mathar_, Aug 19 2008

%H Vincenzo Librandi, <a href="/A061167/b061167.txt">Table of n, a(n) for n = 0..1000</a>

%H D. Zagier, <a href="http://www-groups.mcs.st-andrews.ac.uk/~john/Zagier/Problems.html">Problems posed at the St Andrews Colloquium, 1996</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) = 30*A033455(n-1). [Corrected by _Bernard Schott_, Mar 16 2021]

%F a(n) = -n*A024002(n).

%F a(n) = A000584(n) - n.

%F O.g.f.: 30x^2(1+x)^2/(1-x)^6. - _R. J. Mathar_, Aug 19 2008

%F a(n) = n * (n-1) * (n+1) * (n^2+1). - _Bernard Schott_, Mar 16 2021

%F E.g.f.: exp(x)*x^2*(15 + 25*x + 10*x^2 + x^3). - _Stefano Spezia_, Dec 27 2021

%e a(2) = 32 - 2 = 30.

%t Table[n^5 - n, {n, 0, 40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 20 2012 *)

%o (Magma) [n^5-n: n in [0..40]]; // _Vincenzo Librandi_, May 02 2011

%Y Subsequence of A249674.

%Y Cf. A000584, A024002, A033429, A033455, A033581, A052530.

%K easy,nonn

%O 0,3

%A _Henry Bottomley_, Apr 18 2001