cp's OEIS Frontend

A158187 a(n) = 10*n^2 + 1.

%I A158187 #37 Feb 05 2021 09:42:39
%S A158187 1,11,41,91,161,251,361,491,641,811,1001,1211,1441,1691,1961,2251,
%T A158187 2561,2891,3241,3611,4001,4411,4841,5291,5761,6251,6761,7291,7841,
%U A158187 8411,9001,9611,10241,10891,11561,12251,12961,13691,14441,15211,16001,16811,17641
%N A158187 a(n) = 10*n^2 + 1.
%C A158187 Sequence found by reading the segment (1, 11) together with the line from 11, in the direction 11, 41, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - _Omar E. Pol_, Sep 10 2011
%C A158187 The identity (10n^2 + 1)^2 - (25n^2 + 5)*(2n)^2 = 1 can be written as a(n)^2 - A158445(n)*A005843(n)^2 = 1. - _Vincenzo Librandi_, Jan 03 2012
%H A158187 Vincenzo Librandi, <a href="/A158187/b158187.txt">Table of n, a(n) for n = 0..10000</a>
%H A158187 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158187 a(n) = A033583(n) + 1.
%F A158187 For n > 0: a(n) = A010010(n)/2.
%F A158187 From _Vincenzo Librandi_, Jan 03 2012: (Start)
%F A158187 G.f: x*(11 + 8*x + x^2)/(1-x)^3.
%F A158187 a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
%F A158187 From _Amiram Eldar_, Jul 15 2020: (Start)
%F A158187 Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(10))*coth(Pi/sqrt(10)))/2.
%F A158187 Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(10))*csch(Pi/sqrt(10)))/2. (End)
%F A158187 From _Amiram Eldar_, Feb 05 2021: (Start)
%F A158187 Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(10))*sinh(Pi/sqrt(5)).
%F A158187 Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(10))*csch(Pi/sqrt(10)). (End)
%F A158187 E.g.f.: exp(x)*(1 + 10*x + 10*x^2). - _Stefano Spezia_, Feb 05 2021
%t A158187 Table[10*n^2+1,{n,0,50}] (* _Vincenzo Librandi_, Jan 03 2012 *)
%o A158187 (PARI) a(n)=10*n^2+1 \\ _Charles R Greathouse IV_, Oct 16 2015
%Y A158187 Cf. A158445, A005843. - _Vincenzo Librandi_, Mar 19 2009
%K A158187 nonn,easy
%O A158187 0,2
%A A158187 _Reinhard Zumkeller_, Mar 13 2009