A047878 a(n) is the least number of knight's moves from corner (0,0) to n-th diagonal of unbounded chessboard.
0, 3, 2, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 9, 8, 9, 10, 9, 10, 11, 10, 11, 12, 11, 12, 13, 12, 13, 14, 13, 14, 15, 14, 15, 16, 15, 16, 17, 16, 17, 18, 17, 18, 19, 18, 19, 20, 19, 20, 21, 20, 21, 22, 21, 22, 23, 22, 23, 24, 23, 24, 25
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..5000
- Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
Programs
-
Magma
I:=[2, 1, 2, 3]; [0,3] cat [n le 4 select I[n] else Self(n-1) +Self(n-3) -Self(n-4): n in [1..81]]; // G. C. Greubel, Oct 22 2022
-
Mathematica
LinearRecurrence[{1,0,1,-1},{0,3,2,1,2,3},80] (* Harvey P. Dale, Sep 01 2018 *) Join[{0,3}, Table[(n+2 -2*ChebyshevU[2*n, 1/2])/3, {n,2,75}]] (* G. C. Greubel, Oct 22 2022 *) -
PARI
concat(0, Vec(x*(2*x^4-2*x^3-x^2-x+3)/((x-1)^2*(x^2+x+1)) + O(x^100))) \\ Colin Barker, May 04 2014
-
SageMath
(Sage) [0,3]+[(n+2 - 2*chebyshev_U(2*n, 1/2))/3 for n in (2..75)] # G. C. Greubel, Oct 22 2022
Formula
a(n) = Min_{i=0..n} A049604(i,n-i).
a(3n) = n, a(3n+1) = n+1, a(3n+2) = n+2 for n >= 1.
From Colin Barker, May 04 2014: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>5.
G.f.: x*(3-x-x^2-2*x^3+2*x^4) / ((1-x)^2*(1+x+x^2)). (End)
From Guenther Schrack, Nov 19 2020: (Start)
a(n) = a(n-3) + 1, for n > 4 with a(0) = 0, a(1) = 3, a(2) = 2, a(3) = 1, a(4) = 2;
a(n) = (3*n + 6 - 2*(w^(2*n)*(2 + w) + w^n*(1 - w)))/9, for n > 1 with a(0) = 0, a(1) = 3, where w = (-1 + sqrt(-3))/2, a primitive third root of unity;
a(n) = (n + 2 - 2*A057078(n))/3 for n > 1;
a(n) = A194960(n-2) for n > 2;
a(n) = (2*n + 2 - A330396(n))/3 for n > 1. (End)
Comments