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

1, 3, 7, 12, 18, 26, 35, 45, 57, 70, 84, 100, 117, 135, 155, 176, 198, 222, 247, 273, 301, 330, 360, 392, 425, 459, 495, 532, 570, 610, 651, 693, 737, 782, 828, 876, 925, 975, 1027, 1080, 1134, 1190, 1247, 1305, 1365, 1426, 1488, 1552, 1617, 1683, 1751, 1820, 1890

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

From Paul Curtz, Oct 07 2018: (Start)

Terms that are on the x-axis of the following spiral (without 0):

28--29--29--30--31--31--32

|

27 13--14--15--15--16--17

| | |

27 13 4---5---5---6 17

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26 12 3 0---1 7 18

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25 11 3---2---1 7 19

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25 11--10---9---9---8 19

| |

24--23--23--22--21--21--20 (End)

Diagonal 1, 4, 8, 13, 20, 28, ... (without 0) is A143978. - Bruno Berselli, Oct 08 2018

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..10000
• David Applegate, Omar E. Pol, and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, 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.]
• R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
• P. A. Piza, Fermat coefficients, Math. Mag., 27 (1954), 141-146.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Crossrefs

Cf. A004773 (first differences), A092498 (partial sums).

Cf. A014105, A139250, A143978, A160165, A258708.

Programs

Haskell

a000969 = flip div 3 . a014105 . (+ 1)  -- Reinhard Zumkeller, Jun 23 2015

Magma

[Floor(Binomial(2*n+3,2)/3): n in [0..60]]; // G. C. Greubel, Apr 18 2023

Maple

A000969:=-(1+z+2*z**2)/(z**2+z+1)/(z-1)**3; # Simon Plouffe in his 1992 dissertation

Mathematica

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 *)
CoefficientList[Series[(1+x+2*x^2)/((1-x)^2*(1-x^3)), {x, 0, 50}], x] (* Stefano Spezia, Oct 08 2018 *)

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

SageMath

[(binomial(2*n+3,2)//3) for n in range(61)] # G. C. Greubel, Apr 18 2023

Formula

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

a(2^k-2) = A139250(2^k-1), k >= 1. - Omar E. Pol, Feb 13 2010

a(n) = Sum_{i=0..n} floor(4*i/3). - Enrique Pérez Herrero, Apr 21 2012

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

a(n) = A014105(n+1) = A258708(n+3,n). - Reinhard Zumkeller, Jun 23 2015

Sum_{n>=0} 1/a(n) = 6 - Pi/sqrt(3) - 10*log(2)/3. - Amiram Eldar, Oct 01 2022

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