login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000969 Expansion of g.f. (1 + x + 2*x^2)/((1 - x)^2*(1 - x^3)).
(Formerly M2630 N1042)
28
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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
| | | | |
26 12 3 0---1 7 18
| | | | | |
25 11 3---2---1 7 19
| | | |
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
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.
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
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 *)
PROG
(Haskell)
a000969 = flip div 3 . a014105 . (+ 1) -- Reinhard Zumkeller, Jun 23 2015
(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
CROSSREFS
Cf. A004773 (first differences), A092498 (partial sums).
Sequence in context: A257941 A257944 A005228 * A194117 A122250 A169679
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 16:28 EDT 2024. Contains 371254 sequences. (Running on oeis4.)