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a(n) = (2*n + 1)*(5*n + 6).
4

%I #23 Oct 27 2024 18:58:54

%S 6,33,80,147,234,341,468,615,782,969,1176,1403,1650,1917,2204,2511,

%T 2838,3185,3552,3939,4346,4773,5220,5687,6174,6681,7208,7755,8322,

%U 8909,9516,10143,10790,11457,12144,12851,13578,14325,15092,15879,16686,17513

%N a(n) = (2*n + 1)*(5*n + 6).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = A153126(2*n+1) = A000566(2*(n+1)) - 1.

%F a(n) = a(n-1) + 20*n + 7 (with a(0)=6). - _Vincenzo Librandi_, Dec 27 2010

%F G.f.: (-6-15*x+x^2)/(-1+x)^3 - _Harvey P. Dale_, Jun 07 2021

%F Sum_{n>=0} 1/a(n) = 5/7 - sqrt(1+2/sqrt(5))*Pi/14 - sqrt(5)*log(phi)/14 - 5*log(5)/28 + 2*log(2)/7, where phi is the golden ratio (A001622). - _Amiram Eldar_, Aug 23 2022

%F From _Elmo R. Oliveira_, Oct 27 2024: (Start)

%F E.g.f.: exp(x)*(6 + 27*x + 10*x^2).

%F a(n) = A005408(n)*A016861(n+1).

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

%t Table[(2n+1)(5n+6),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{6,33,80},50] (* _Harvey P. Dale_, Jun 07 2021 *)

%o (PARI) a(n)=(2*n+1)*(5*n+6) \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A000566, A001622, A005408, A016861, A019952, A033571, A153126.

%K nonn,easy,changed

%O 0,1

%A _Reinhard Zumkeller_, Dec 20 2008