

A047878


a(n) is the least number of knight's moves from corner (0,0) to nth diagonal of unbounded chessboard.


3



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
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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,ni).
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(n1) + a(n3)  a(n4) for n>5.
G.f.: x*(3xx^22*x^3+2*x^4) / ((1x)^2*(1+x+x^2)). (End)
From Guenther Schrack, Nov 19 2020: (Start)
a(n) = a(n3) + 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(n2) 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^42*x^3x^2x+3)/((x1)^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(n1) +Self(n3) Self(n4): 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



