This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A033577 #48 Sep 08 2022 08:44:51
%S A033577 1,20,63,130,221,336,475,638,825,1036,1271,1530,1813,2120,2451,2806,
%T A033577 3185,3588,4015,4466,4941,5440,5963,6510,7081,7676,8295,8938,9605,
%U A033577 10296,11011,11750,12513,13300,14111,14946,15805,16688,17595,18526,19481,20460,21463
%N A033577 a(n) = (3*n+1) * (4*n+1).
%C A033577 Also the 120ยบ spoke (or ray) of a hexagonal spiral of Ulam. - _Robert G. Wilson v_, Jul 06 2014
%C A033577 If two independent real random variables x and y are distributed according to the same exponential distribution with pdf(x) = lambda * exp(-lambda * x) for some lambda > 0, then the probability that 3 <= x/(n*y) < 4 is given by n/a(n) for n>1. - _Andres Cicuttin_, Dec 11 2016
%H A033577 Colin Barker, <a href="/A033577/b033577.txt">Table of n, a(n) for n = 0..1000</a>
%H A033577 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A033577 From _Colin Barker_, Dec 12 2016: (Start)
%F A033577 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2.
%F A033577 G.f.: (1 + 17*x + 6*x^2)/(1-x)^3. (End)
%F A033577 E.g.f.: (1 + 19*x + 12*x^2)*exp(x). - _G. C. Greubel_, Oct 12 2019
%e A033577 See A056105 example section for hexagonal spiral of Ulam diagram. - _Robert G. Wilson v_, Jul 06 2014
%p A033577 A033577:=n->(3*n+1)*(4*n+1): seq(A033577(n), n=0..50); # _Wesley Ivan Hurt_, Jul 06 2014
%t A033577 f[n_] := (3n + 1)(4n + 1); Array[f, 50, 0] (* _Robert G. Wilson v_, Jul 06 2014 *)
%t A033577 LinearRecurrence[{3,-3,1},{1,20,63},50] (* _Harvey P. Dale_, Jul 16 2020 *)
%o A033577 (PARI) vector(50, m, 12*m^2 - 17*m + 6) \\ _Michel Marcus_, Jul 06 2014
%o A033577 (PARI) Vec((1 + 17*x + 6*x^2) / (1 - x)^3 + O(x^50)) \\ _Colin Barker_, Dec 12 2016
%o A033577 (Magma) [(3*n+1)*(4*n+1) : n in [0..50]]; // _Wesley Ivan Hurt_, Jul 06 2014
%o A033577 (Sage) [(3*n+1)*(4*n+1) for n in range(50)] # _G. C. Greubel_, Oct 12 2019
%o A033577 (GAP) List([0..50], n-> (3*n+1)*(4*n+1)); # _G. C. Greubel_, Oct 12 2019
%Y A033577 Subsequence of A281333.
%Y A033577 Cf. A003215, A056105, A056106, A056107, A056108, A056109, A244802, A244803, A244804, A244805, A244806.
%K A033577 nonn,easy
%O A033577 0,2
%A A033577 _N. J. A. Sloane_
%E A033577 More terms from _Wesley Ivan Hurt_, Jul 06 2014