OFFSET
0,2
COMMENTS
The knight starts at (0,0) and we count the least number of steps. Row 1 of the array at A065775. - Clark Kimberling, Dec 20 2010
Apparently also the minimum number of steps of the (1,3)-leaper to reach (n,n) starting at (0,0). - R. J. Mathar, Jan 05 2018
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Francis N. Castro, Oscar E. González and Luis A. Medina, Generalized exponential sums and the power of computers, Involve, Vol. 11 (2018), Issue 1, pp. 127-142. Also, authors' copy.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
FORMULA
a(n) = 2[ (n+2)/4 ] if n even, 2[ (n+1)/4 ]+1 if n odd (n >= 8).
G.f.: x*(3-x+x^2-x^3-2*x^4+2*x^5)/((1-x)^2*(1+x)*(1+x^2)). a(n)=A083219(n), n<>1. - R. J. Mathar, Dec 15 2008
T(0,0)=0, T(1,0)=3, and for m>=1, T(4m-2,0)=2m, T(4m-1,0)=2m+1, T(4m,0)=2m, T(4m+1,0)=2m+1 where T(.,.) = A065775(.,.). - Clark Kimberling, Dec 20 2010
Sum_{n>=1} (-1)^n/a(n) = 5/3 - 2*log(2). - Amiram Eldar, Sep 10 2023
EXAMPLE
a(1)=3 counts these moves: (0,0) to (2,1) to (0,2) to (1,0). - Clark Kimberling, Dec 20 2010
MATHEMATICA
CoefficientList[Series[x (3 - x + x^2 - x^3 - 2 x^4 + 2 x^5)/((1-x)^2 (1+x) (1+x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Jan 06 2018 *)
Array[Which[#==1, 3, True, (#+Mod[#, 4])/2]&, 100, 0] (* Elisha Hollander, Aug 05 2021 *)
PROG
(PARI) concat([0], Vec( x*(3-x+x^2-x^3-2*x^4+2*x^5)/((1-x)^2*(1+x)*(1+x^2)) + O(x^166) ) ) \\ Joerg Arndt, Sep 10 2014
(Python) def a(n): return 3 if n == 1 else (n + n % 4) // 2 # Elisha Hollander, Aug 05 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved