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

0,2

Comments

Apart from initial terms, same as A008611. - Anton Chupin, Oct 24 2009

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).

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)

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 *)

Prog

(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
(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
(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

Crossrefs

Cf. A008611, A049604, A057078, A194960, A330396.

Sequence in context: A260450 A036583 A351557 * A324782 A256427 A120441

Adjacent sequences: A047875 A047876 A047877 * A047879 A047880 A047881

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved