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A000969
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Expansion of g.f. (1 + x + 2*x^2)/((1 - x)^2*(1 - x^3)).
(Formerly M2630 N1042)
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28
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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
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OFFSET
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0,2
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COMMENTS
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Terms that are on the x-axis of the following spiral (without 0):
28--29--29--30--31--31--32
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27 13--14--15--15--16--17
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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
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24--23--23--22--21--21--20 (End)
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REFERENCES
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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).
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LINKS
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FORMULA
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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(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
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
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MAPLE
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
(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
(Magma) [Floor(Binomial(2*n+3, 2)/3): n in [0..60]]; // G. C. Greubel, Apr 18 2023
(SageMath) [(binomial(2*n+3, 2)//3) for n in range(61)] # G. C. Greubel, Apr 18 2023
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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