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%I #34 May 07 2018 18:44:44
%S 0,11,82,393,1364,3775,8886,18557,35368,62739,105050,167761,257532,
%T 382343,551614,776325,1069136,1444507,1918818,2510489,3240100,4130511,
%U 5206982,6497293,8031864,9843875,11969386,14447457,17320268,20633239,24435150,28778261,33718432,39315243,45632114
%N a(n) = n^5 + 5*n^3 + 5*n.
%C Also numbers of the form (n-th metallic mean)^5 - 1/(n-th metallic mean)^5, see link to Wikipedia.
%H Colin Barker, <a href="/A261391/b261391.txt">Table of n, a(n) for n = 0..1000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Metallic_mean">Metallic mean</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) = ( (n+sqrt(n^2+4))/2 )^5 - 1/( (n+sqrt(n^2+4))/2 )^5.
%F a(n) = -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). - _Colin Barker_, Aug 18 2015
%F G.f.: x*(11*x^4+16*x^3+66*x^2+16*x+11) / (x-1)^6. - _Colin Barker_, Aug 18 2015
%F E.g.f.: (x^5 + 15*x^4 + 70*x^3 + 120*x^2 + 71*x + 11)*e^x. - _G. C. Greubel_, Aug 21 2015
%t Array[#^5 + 5 #^3 + 5 # &, 34] (* _Michael De Vlieger_, Aug 18 2015 *)
%t Table[n^5 + 5*n^3 + 5*n, {n,0, 50}] (* _G. C. Greubel_, Aug 21 2015 *)
%t LinearRecurrence[{6,-15,20,-15,6,-1},{0,11,82,393,1364,3775},40] (* _Harvey P. Dale_, May 07 2018 *)
%o (PARI) concat(0, Vec(x*(11*x^4+16*x^3+66*x^2+16*x+11)/(x-1)^6 + O(x^100))) \\ _Colin Barker_, Aug 18 2015
%Y Cf. A001622, A014176, A098316, A098317, A098318, A176398, A176439, A176458, A176522.
%K nonn,easy
%O 0,2
%A _Raphael Ranna_, Aug 17 2015
%E Offset changed from 1 to 0, initial 0 added and b-file adapted from _Bruno Berselli_, Aug 25 2015
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