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
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).
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
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved