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 A195319 #66 Mar 14 2023 10:16:01
%S A195319 0,9,30,63,108,165,234,315,408,513,630,759,900,1053,1218,1395,1584,
%T A195319 1785,1998,2223,2460,2709,2970,3243,3528,3825,4134,4455,4788,5133,
%U A195319 5490,5859,6240,6633,7038,7455,7884,8325,8778,9243,9720,10209,10710,11223
%N A195319 Three times second hexagonal numbers: 3*n*(2*n+1).
%C A195319 Sequence found by reading the line from 0, in the direction 0, 9, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Semi-axis opposite to A094159 in the same spiral.
%C A195319 Sum of the numbers from 2*n to 4*n. - _Wesley Ivan Hurt_, Nov 27 2015
%C A195319 From _Peter M. Chema_, Jan 21 2017: (Start)
%C A195319 Also 0 together with the partial sums of A017629.
%C A195319 Digit root is 0 together with period 3: repeat [9,3,9].
%C A195319 Final digits cycle a length period 10: repeat [0,9,0,3,8,5,4,5,8,3]. (End)
%C A195319 Sequence found by reading the line from 0, in the direction 0, 9, ..., in the triangle spiral. - _Hans G. Oberlack_, Dec 08 2018
%H A195319 Vincenzo Librandi, <a href="/A195319/b195319.txt">Table of n, a(n) for n = 0..10000</a>
%H A195319 Hans G. Oberlack, <a href="/A195319/a195319.png">Triangle spiral</a>.
%H A195319 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A195319 a(n) = 6*n^2 + 3*n = 3*A014105(n).
%F A195319 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - _Harvey P. Dale_, Oct 13 2013
%F A195319 G.f.: 3*x*(3+x) / (1-x)^3. - _Wesley Ivan Hurt_, Nov 27 2015
%F A195319 a(n) = A000217(3*n) + 3*A000217(n). - _Bruno Berselli_, Aug 31 2017
%F A195319 E.g.f.: 3*x*(2*x+3)*exp(x). - _G. C. Greubel_, Dec 07 2018
%F A195319 From _Amiram Eldar_, Feb 27 2022: (Start)
%F A195319 Sum_{n>=1} 1/a(n) = 2*(1 - log(2))/3.
%F A195319 Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/2 + log(2) - 2)/3. (End)
%p A195319 A195319:=n->6*n^2 + 3*n: seq(A195319(n), n=0..50); # _Wesley Ivan Hurt_, Nov 27 2015
%t A195319 Table[6n^2+3n,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,9,30},50] (* _Harvey P. Dale_, Oct 13 2013 *)
%t A195319 CoefficientList[Series[3 x (3 + x)/(1 - x)^3, {x, 0, 50}], x] (* _Wesley Ivan Hurt_, Nov 27 2015 *)
%o A195319 (Magma) [3*n*(2*n+1): n in [0..50]]; // _Vincenzo Librandi_, Sep 20 2011
%o A195319 (PARI) a(n)=3*n*(2*n+1) \\ _Charles R Greathouse IV_, Oct 16 2015
%o A195319 (Sage) [3*n*(2*n+1) for n in range(50)] # _G. C. Greubel_, Dec 07 2018
%o A195319 (GAP) List([0..30], n -> 3*n*(2*n+1)); # _G. C. Greubel_, Dec 07 2018
%Y A195319 Bisection of A045943.
%Y A195319 Cf. A000217, A001318, A014105, A094159, A017629.
%K A195319 nonn,easy
%O A195319 0,2
%A A195319 _Omar E. Pol_, Sep 17 2011