login
a(n) = 9*n - a(n-1), with n>1, a(1)=1.
8

%I #41 Aug 04 2024 20:19:15

%S 1,17,10,26,19,35,28,44,37,53,46,62,55,71,64,80,73,89,82,98,91,107,

%T 100,116,109,125,118,134,127,143,136,152,145,161,154,170,163,179,172,

%U 188,181,197,190,206,199,215,208,224,217,233,226,242,235,251,244,260,253

%N a(n) = 9*n - a(n-1), with n>1, a(1)=1.

%H Vincenzo Librandi, <a href="/A166524/b166524.txt">Table of n, a(n) for n = 1..1000</a>

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

%F G.f.: x*(1+16*x-8*x^2) / ( (1+x)*(1-x)^2 ). - _R. J. Mathar_, Mar 08 2011

%F From _G. C. Greubel_, May 16 2016: (Start)

%F E.g.f.: (1/4)*(23*exp(-x) + 9*(1 + 2*x)*exp(x) - 32).

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

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 1/8 + cot(Pi/9)*Pi/9. - _Amiram Eldar_, Feb 24 2023

%F a(n) = (1/4)*(18*n + 9 + 23*(-1)^n). - _G. C. Greubel_, Aug 04 2024

%t CoefficientList[Series[(1 +16 x -8 x^2)/((1+x) (1-x)^2), {x,0,80}], x] (* _Vincenzo Librandi_, Sep 13 2013 *)

%t LinearRecurrence[{1,1,-1},{1,17,10},60] (* _Harvey P. Dale_, Dec 24 2014 *)

%o (Magma) [n eq 1 select 1 else 9*n-Self(n-1): n in [1..80]]; // _Vincenzo Librandi_, Sep 13 2013

%o (SageMath)

%o def A166524(n): return (9*n - 7 + 23*((n+1)%2))//2

%o [A166524(n) for n in range(1, 101)] # _G. C. Greubel_, Aug 04 2024

%Y Cf. A166519, A166520, A166521, A166522, A166523, A166525, A166526.

%K nonn,easy

%O 1,2

%A _Vincenzo Librandi_, Oct 16 2009