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A007533 a(n) = (5*n + 1)^2 + 4*n + 1.

%I A007533 M2162 #49 Oct 31 2024 15:06:43
%S A007533 2,41,130,269,458,697,986,1325,1714,2153,2642,3181,3770,4409,5098,
%T A007533 5837,6626,7465,8354,9293,10282,11321,12410,13549,14738,15977,17266,
%U A007533 18605,19994,21433,22922,24461,26050,27689,29378,31117,32906,34745,36634,38573,40562,42601
%N A007533 a(n) = (5*n + 1)^2 + 4*n + 1.
%C A007533 Also, numbers of the form (3*k + 1)^2 + (4*k + 1)^2. - _Bruno Berselli_, Dec 11 2011
%C A007533 The continued fraction expansion of sqrt(a(n)) is [5n+1; {2, 2, 10n+2}]. For n=0, this collapses to [1; {2}]. - _Magus K. Chu_, Aug 27 2022
%D A007533 W. Sierpiński, Elementary Theory of Numbers. Państ. Wydaw. Nauk., Warsaw, 1964, p. 323.
%D A007533 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007533 Vincenzo Librandi, <a href="/A007533/b007533.txt">Table of n, a(n) for n = 0..1000</a>
%H A007533 W. Sierpiński, <a href="https://eudml.org/doc/219306">Elementary Theory of Numbers</a>, Warszawa 1964.
%H A007533 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A007533 From _Bruno Berselli_, Dec 11 2011: (Start)
%F A007533 a(n) = 25*n^2 + 14*n + 2.
%F A007533 G.f.: (2 + 35*x + 13*x^2)/(1-x)^3. (End)
%F A007533 From _Elmo R. Oliveira_, Oct 31 2024: (Start)
%F A007533 E.g.f.: (2 + 39*x + 25*x^2)*exp(x).
%F A007533 a(n) = A154355(n+1).
%F A007533 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%t A007533 Table[25n^2+14n+2,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{2,41,130},40] (* _Harvey P. Dale_, Dec 18 2013 *)
%o A007533 (Magma) [(5*n+1)^2 + 4*n+1: n in [0..40]]; // _Vincenzo Librandi_, May 02 2011
%o A007533 (PARI) a(n)=25*n^2 + 14*n + 2 \\ _Charles R Greathouse IV_, May 02 2011
%Y A007533 Cf. A154355.
%K A007533 nonn,easy
%O A007533 0,1
%A A007533 _N. J. A. Sloane_, _Robert G. Wilson v_