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a(n) = (9*n+1)*(9*n+8).
1

%I #43 Oct 18 2024 19:39:38

%S 8,170,494,980,1628,2438,3410,4544,5840,7298,8918,10700,12644,14750,

%T 17018,19448,22040,24794,27710,30788,34028,37430,40994,44720,48608,

%U 52658,56870,61244,65780,70478,75338,80360,85544,90890,96398,102068,107900,113894,120050

%N a(n) = (9*n+1)*(9*n+8).

%H T. D. Noe, <a href="/A001534/b001534.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 a(n) = 162*n + a(n-1) with a(0)=8. - _Vincenzo Librandi_, Nov 12 2010

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0)=8, a(1)=170, a(2)=494. - _Harvey P. Dale_, Aug 20 2011

%F G.f.: -((2*(x*(4*x+73)+4))/(x-1)^3). - _Harvey P. Dale_, Aug 20 2011

%F Sum_{n>=0} 1/a(n) = (Psi(8/9)-Psi(1/9))/63 = 0.13700722.. - _R. J. Mathar_, May 30 2022

%F Sum_{n>=0} 1/a(n) = cot(Pi/9)*Pi/63. - _Amiram Eldar_, Sep 10 2022

%F From _Amiram Eldar_, Feb 19 2023: (Start)

%F a(n) = A017173(n)*A017257(n).

%F Product_{n>=0} (1 - 1/a(n)) = cosec(Pi/9)*cos(sqrt(53)*Pi/18).

%F Product_{n>=0} (1 + 1/a(n)) = cosec(Pi/9)*cos(sqrt(5)*Pi/6). (End)

%F E.g.f.: exp(x)*(8 + 81*x*(2 + x)). - _Elmo R. Oliveira_, Oct 18 2024

%t f[n_]:=Module[{n9=9n},(n9+1)(n9+8)];Array[f,40,0] (* or *) LinearRecurrence[ {3,-3,1},{8,170,494},50] (* _Harvey P. Dale_, Aug 20 2011 *)

%o (PARI) a(n)=(9*n+1)*(9*n+8) \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A017173, A017257.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_