A158536 - OEIS

%I #31 Jan 15 2025 21:19:17

%S 11,132,495,1100,1947,3036,4367,5940,7755,9812,12111,14652,17435,

%T 20460,23727,27236,30987,34980,39215,43692,48411,53372,58575,64020,

%U 69707,75636,81807,88220,94875,101772,108911,116292,123915,131780,139887,148236,156827,165660

%N a(n) = 121*n^2 + 11.

%C The identity (22*n^2 + 1)^2 - (121*n^2 + 11)*(2*n)^2 = 1 can be written as A158537(n)^2 -a(n)*A005843(n)^2 = 1.

%H Vincenzo Librandi, <a href="/A158536/b158536.txt">Table of n, a(n) for n = 0..10000</a>

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

%F From  _R. J. Mathar_, Oct 16 2009: (Start)

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

%F G.f.: 11*(1+9*x+12*x^2)/(1-x)^3. (End)

%F From _Amiram Eldar_, Mar 06 2023: (Start)

%F Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(11))*Pi/sqrt(11) + 1)/22.

%F Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(11))*Pi/sqrt(11) + 1)/22. (End)

%F E.g.f.: 11*exp(x)*(1 + 11*x + 11*x^2). - _Elmo R. Oliveira_, Jan 15 2025

%t 121Range[0,40]^2+11 (* _Harvey P. Dale_, Mar 04 2011 *)

%t LinearRecurrence[{3, -3, 1}, {11, 132, 495}, 50] (* _Vincenzo Librandi_, Feb 12 2012 *)

%o (Magma) I:=[11, 132, 495]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Feb 12 2012

%o (PARI) for(n=1, 40, print1(121*n^2+11", ")); \\ _Vincenzo Librandi_, Feb 12 2012

%Y Cf. A005843, A158537.

%K nonn,less,easy

%O 0,1

%A _Vincenzo Librandi_, Mar 21 2009

%E a(0) added by _R. J. Mathar_, Oct 16 2009