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A000969 Expansion of g.f. (1 + x + 2*x^2)/((1 - x)^2*(1 - x^3)).

%I A000969 M2630 N1042 #94 Apr 19 2023 09:04:10
%S A000969 1,3,7,12,18,26,35,45,57,70,84,100,117,135,155,176,198,222,247,273,
%T A000969 301,330,360,392,425,459,495,532,570,610,651,693,737,782,828,876,925,
%U A000969 975,1027,1080,1134,1190,1247,1305,1365,1426,1488,1552,1617,1683,1751,1820,1890
%N A000969 Expansion of g.f. (1 + x + 2*x^2)/((1 - x)^2*(1 - x^3)).
%C A000969 From _Paul Curtz_, Oct 07 2018: (Start)
%C A000969 Terms that are on the x-axis of the following spiral (without 0):
%C A000969      28--29--29--30--31--31--32
%C A000969       |
%C A000969      27  13--14--15--15--16--17
%C A000969       |   |                   |
%C A000969      27  13   4---5---5---6  17
%C A000969       |   |   |           |   |
%C A000969      26  12   3   0---1   7  18
%C A000969       |   |   |       |   |   |
%C A000969      25  11   3---2---1   7  19
%C A000969       |   |               |   |
%C A000969      25  11--10---9---9---8  19
%C A000969       |                       |
%C A000969      24--23--23--22--21--21--20  (End)
%C A000969 Diagonal 1, 4, 8, 13, 20, 28, ... (without 0) is A143978. - _Bruno Berselli_, Oct 08 2018
%D A000969 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000969 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000969 N. J. A. Sloane, <a href="/A000969/b000969.txt">Table of n, a(n) for n = 0..10000</a>
%H A000969 David Applegate, Omar E. Pol, and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
%H A000969 R. P. Loh, A. G. Shannon, and A. F. Horadam, <a href="/A000969/a000969.pdf">Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients</a>, Preprint, 1980.
%H A000969 P. A. Piza, <a href="http://www.jstor.org/stable/3029103">Fermat coefficients</a>, Math. Mag., 27 (1954), 141-146.
%H A000969 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A000969 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A000969 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>.
%H A000969 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F A000969 a(n) = floor( (2*n+3)*(n+1)/3 ). Or, a(n) = (2*n+3)*(n+1)/3 but subtract 1/3 if n == 1 mod 3. - _N. J. A. Sloane_, May 05 2010
%F A000969 a(2^k-2) = A139250(2^k-1), k >= 1. - _Omar E. Pol_, Feb 13 2010
%F A000969 a(n) = Sum_{i=0..n} floor(4*i/3). - _Enrique Pérez Herrero_, Apr 21 2012
%F A000969 a(n) = +2*a(n-1) -1*a(n-2) +1*a(n-3) -2*a(n-4) +1*a(n-5). - _Joerg Arndt_, Apr 22 2012
%F A000969 a(n) = A014105(n+1) = A258708(n+3,n). - _Reinhard Zumkeller_, Jun 23 2015
%F A000969 Sum_{n>=0} 1/a(n) = 6 - Pi/sqrt(3) - 10*log(2)/3. - _Amiram Eldar_, Oct 01 2022
%F A000969 E.g.f.: (exp(x)*(8 + 21*x + 6*x^2) + exp(-x/2)*(cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)))/9. - _Stefano Spezia_, Apr 05 2023
%p A000969 A000969:=-(1+z+2*z**2)/(z**2+z+1)/(z-1)**3; # _Simon Plouffe_ in his 1992 dissertation
%t A000969 f[x_, y_]:= Floor[Abs[y/x -x/y]]; Table[f[3, 2n^2+n+2], {n,53}] (* _Robert G. Wilson v_, Aug 11 2010 *)
%t A000969 CoefficientList[Series[(1+x+2*x^2)/((1-x)^2*(1-x^3)), {x, 0, 50}], x] (* _Stefano Spezia_, Oct 08 2018 *)
%o A000969 (Haskell)
%o A000969 a000969 = flip div 3 . a014105 . (+ 1)  -- _Reinhard Zumkeller_, Jun 23 2015
%o A000969 (PARI) a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,-2,1,-1,2]^n*[1;3;7;12;18])[1,1] \\ _Charles R Greathouse IV_, May 10 2016
%o A000969 (Magma) [Floor(Binomial(2*n+3,2)/3): n in [0..60]]; // _G. C. Greubel_, Apr 18 2023
%o A000969 (SageMath) [(binomial(2*n+3,2)//3) for n in range(61)] # _G. C. Greubel_, Apr 18 2023
%Y A000969 Cf. A004773 (first differences), A092498 (partial sums).
%Y A000969 Cf. A014105, A139250, A143978, A160165, A258708.
%K A000969 nonn,easy
%O A000969 0,2
%A A000969 _N. J. A. Sloane_