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 A195038 #21 Dec 27 2024 20:21:52
%S A195038 0,41,123,246,410,615,861,1148,1476,1845,2255,2706,3198,3731,4305,
%T A195038 4920,5576,6273,7011,7790,8610,9471,10373,11316,12300,13325,14391,
%U A195038 15498,16646,17835,19065,20336,21648,23001,24395,25830,27306,28823,30381,31980
%N A195038 41 times triangular numbers.
%C A195038 Related to the primitive Pythagorean triple [21, 20, 29].
%C A195038 Sequence found by reading the line from 0, in the direction 0, 41, ..., and the same line from 0, in the direction 0, 123, ..., in the Pythagorean spiral whose edges have length A195033 and whose vertices are the numbers A195034. This is the main diagonal of the square spiral.
%H A195038 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A195038 a(n) = (41*n^2 + 41*n)/2 = 41*n*(n+1)/2 = 41*A000217(n).
%F A195038 G.f.: -41*x/(x-1)^3. - _R. J. Mathar_, Oct 04 2014
%F A195038 From _Elmo R. Oliveira_, Dec 27 2024: (Start)
%F A195038 E.g.f.: 41*exp(x)*x*(2 + x)/2.
%F A195038 a(n) = A195034(2*n).
%F A195038 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%t A195038 41*Accumulate[Range[0,40]] (* or *) LinearRecurrence[{3,-3,1},{0,41,123},40] (* _Harvey P. Dale_, Nov 20 2015 *)
%o A195038 (PARI) a(n)=41*n*(n+1)/2 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A195038 Bisection of A195034.
%Y A195038 Cf. A000217, A195033.
%K A195038 nonn,easy
%O A195038 0,2
%A A195038 _Omar E. Pol_, Sep 12 2011