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%I #58 Sep 08 2022 08:45:55
%S 0,5,17,36,62,95,135,182,236,297,365,440,522,611,707,810,920,1037,
%T 1161,1292,1430,1575,1727,1886,2052,2225,2405,2592,2786,2987,3195,
%U 3410,3632,3861,4097,4340,4590,4847,5111,5382,5660,5945,6237,6536,6842,7155,7475
%N a(n) = n*(7*n+3)/2.
%C This sequence is related to A050409 by A050409(n) = n*a(n) - Sum_{i=0..n-1} a(i).
%H Vincenzo Librandi, <a href="/A186029/b186029.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: x*(5+2*x)/(1-x)^3.
%F a(n) - a(-n) = A008585(n).
%F a(n) + a(-n) = A033582(n).
%F n*a(n+1) - (n+1)*a(n) = A024966(n). - _Bruno Berselli_, May 30 2012
%F n*a(n+2) - (n+2)*a(n) = A067727(n) for n>0. - _Bruno Berselli_, May 30 2012
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=5, a(2)=17. - _Philippe Deléham_, Mar 26 2013
%F a(n) = A174738(7*n+4). - _Philippe Deléham_, Mar 26 2013
%F E.g.f.: (1/2)*(7*x^2 + 10*x)*exp(x). - _G. C. Greubel_, Jul 17 2017
%e From _Ilya Gutkovskiy_, Mar 31 2016: (Start)
%e . o o o o o o o o o o o o
%e . o o
%e . o o o o o o o o o o o o o o o o o o o o o o o o
%e . o o o o o o o o o o o o
%e . o o o o o o o o o o o o o o o o o o o o
%e . o o o o o o o o o o o o o o o o o o o o o o o o
%e . o o o o o o
%e . o o o o o o o o o o o o o o o o o o o o
%e .
%e . n=1 n=2 n=3 n=4
%e (End)
%t Table[(n - 1) (7 n - 4)/2, {n, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 06 2011 *)
%t LinearRecurrence[{3,-3,1},{0,5,17},50] (* _Harvey P. Dale_, Sep 07 2022 *)
%o (Magma) [n*(7*n+3)/2: n in [0..44]];
%o (PARI) a(n)=n*(7*n+3)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. numbers of the form n*(d*n+10-d)/2 indexed in A140090.
%Y Cf. A017041 (first differences).
%Y Cf. A001106, A022264, A022265, A024966, A050409, A067727, A174738, A179986, A218471.
%K nonn,easy
%O 0,2
%A _Bruno Berselli_, Feb 11 2011
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