A049604 -id:A049604 - cp's OEIS frontend

A047879 a(n)=Sum{T(n,i): i=0,1,...,n}+Sum{T(i,n): i=0,1,...,n-1}, array T counting least number of knight's moves as in A049604.

0, 10, 10, 18, 24, 38, 50, 70, 82, 108, 126, 156, 176, 212, 236, 278, 304, 350, 382, 434, 466, 524, 562, 624, 664, 732, 776, 850, 896, 974, 1026, 1110, 1162, 1252, 1310, 1404, 1464, 1564, 1628, 1734, 1800, 1910, 1982, 2098, 2170

Offset: 0

Clark Kimberling

nonn

Cf. A049604.

Empirical g.f.: 2*x*(x^10+3*x^9-x^8-4*x^7-5*x^6-4*x^5-2*x^3-4*x^2-5*x-5) / ((x-1)^3*(x+1)^2*(x^2+1)*(x^2+x+1)). - Colin Barker, May 04 2014

A047881 a(n) = Sum_{h=0..n, k=0..n} T(h,k), array T counting knights' moves as in A049604.

Original entry on oeis.org

0, 10, 20, 38, 62, 100, 150, 220, 302, 410, 536, 692, 868, 1080, 1316, 1594, 1898, 2248, 2630, 3064, 3530, 4054, 4616, 5240, 5904, 6636, 7412, 8262, 9158, 10132, 11158, 12268, 13430, 14682, 15992, 17396, 18860, 20424, 22052

Offset: 0

Clark Kimberling

nonn

Partial sums of A047879.

Conjecture: a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10) for n > 11. - Chai Wah Wu, May 25 2016

A047876 a(n) = Sum{T(i,n-i): i=0,1,...,n}, array T as in A049604.

Original entry on oeis.org

0, 6, 8, 8, 12, 18, 22, 28, 36, 42, 52, 64, 68, 82, 98, 104, 118, 138, 146, 164, 184, 194, 216, 240, 248, 274, 302, 312, 338, 370, 382, 412, 444, 458, 492, 528, 540, 578, 618, 632, 670, 714, 730, 772, 816, 834, 880, 928, 944, 994, 1046

Offset: 0

Clark Kimberling

nonn

From Chai Wah Wu, Jun 11 2016: (Start)
a(n) = 2*a(n-3) + a(n-4) - a(n-6) - 2*a(n-7) + a(n-10) for n > 14 (conjectured).
G.f.: 2*x*(x^13 + x^11 - x^9 - 2*x^8 - 4*x^7 - x^6 + x^5 + 2*x^4 - 4*x^2 - 4*x - 3)/(x^10 - 2*x^7 - x^6 + x^4 + 2*x^3 - 1) (conjectured). (End)

A047877 a(n) = (1/2)*Sum_{i=0..n} T(i,n-i), array T as in A049604.

Original entry on oeis.org

0, 3, 4, 4, 6, 9, 11, 14, 18, 21, 26, 32, 34, 41, 49, 52, 59, 69, 73, 82, 92, 97, 108, 120, 124, 137, 151, 156, 169, 185, 191, 206, 222, 229, 246, 264, 270, 289, 309, 316, 335, 357, 365, 386, 408, 417, 440, 464, 472, 497, 523

Offset: 0

Clark Kimberling

nonn

From Chai Wah Wu, Jun 11 2016: (Start)
a(n) = 2*a(n-3) + a(n-4) - a(n-6) - 2*a(n-7) + a(n-10) for n > 14 (conjectured).
G.f.: x*(x^13 + x^11 - x^9 - 2*x^8 - 4*x^7 - x^6 + x^5 + 2*x^4 - 4*x^2 - 4*x - 3)/(x^10 - 2*x^7 - x^6 + x^4 + 2*x^3 - 1) (conjectured). (End)

A065775 Array T read by diagonals: T(i,j)=least number of knight's moves on a chessboard (infinite in all directions) needed to move from (0,0) to (i,j).

Original entry on oeis.org

0, 3, 3, 2, 2, 2, 3, 1, 1, 3, 2, 2, 4, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 2, 2, 2, 4, 4, 5, 3, 3, 3, 3, 3, 3, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 3, 3, 3, 3, 5, 5, 5, 6, 6, 4, 4, 4, 4, 4, 4, 4, 6, 6, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 6, 6, 6, 6, 4, 4, 4, 4, 4, 6, 6, 6, 6, 7, 7, 7, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7

Offset: 0

Stewart Gordon, Dec 05 2001

			From _Clark Kimberling_, Dec 20 2010: (Start)
T(i,j) for -2<=i<=2 and -2<=j<=2:
  4 1 2 1 4=T(2,2)
  1 2 3 2 1=T(2,1)
  2 3 0 3 2=T(2,0)
  1 2 3 2 1=T(2,-1)
  4 1 2 1 4=T(2,-2)
Corner of the array, T(i,j) for i>=0, j>=0: [Corrected Oct 14 2016]
  0 3 2 3 2 3 4...
  3 2 1 2 3 4 3...
  2 1 4 3 2 3 4...
  3 2 3 2 3 4 2... (End)

Identical to A049604 except for T(1, 1).

For number of knight's moves to various subsets of the chessboard, see A018837, A183041-A183053.

From Clark Kimberling, Dec 20 2010: (Start)
T(i,j) is given in cases:
Case 1:  row 0
  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.
Case 2:  row 1
  T(0,1)=3, T(1,1)=2, and for m>=1,
  T(4m-2,1)=2m-1, T(4m-1,1)=2m, T(4m,1)=2m+1,
  T(4m+1,1)=2m+2.
Case 3:  columns 0 and 1
  (column 0 = row 0); (column 1 = row 1).
Case 4:  For i>=2 and j>=2,
   T(i,j)=1+min{T(i-2,j-1),T(i-1,j-2)}.
Cases 1-4 determine T in the 1st quadrant;
all other T(i,j) are easily obtained by symmetry. (End)

A062103 Number of paths by which an unpromoted knight (keima) of Shogi can move to various squares on infinite board, if it starts from its origin square, the second leftmost square of the back rank.

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 14

Offset: 1

Antti Karttunen, May 30 2001

Table formatted as a square array shows the top-left corner of the infinite board. This is an aerated and sligthly skewed variant of Catalan's triangle A009766.

Hans L. Bodlaender, The Chess Variant Pages
Fairbairn, Leggett et al., Information about Shogi (Japanese chess)

A009766, A049604, A062104, trinv given at A054425.

[seq(ShoogiKnightSeq(j),j=1..120)]; ShoogiKnightSeq := n -> ShoogiKnightTriangle(trinv(n-1)-1,(n-((trinv(n-1)*(trinv(n-1)-1))/2))-1);
ShoogiKnightTriangle := proc(r,m) option remember; if(m < 0) then RETURN(0); fi; if(r < 0) then RETURN(0); fi; if(m > r) then RETURN(0); fi; if((1 = r) and (0 = m)) then RETURN(1); fi; RETURN(ShoogiKnightTriangle(r-3,m-2) + ShoogiKnightTriangle(r-1,m-2)); end;

trinv[n_] := Floor[(1 + Sqrt[8 n + 1])/2];
ShoogiKnightSeq[n_] := ShoogiKnightTriangle[trinv[n - 1] - 1, (n - ((trinv[n - 1]*(trinv[n - 1] - 1))/2)) - 1];
ShoogiKnightTriangle[r_, m_] := ShoogiKnightTriangle[r, m] = Which[m < 0, 0, r < 0, 0, m > r, 0, r == 1 && m == 0, 1, True, ShoogiKnightTriangle[r - 3, m - 2] + ShoogiKnightTriangle[r - 1, m - 2]];
Array[ShoogiKnightSeq, 120] (* Jean-François Alcover, Mar 06 2016, adapted from Maple *)

A047878 a(n) is the least number of knight's moves from corner (0,0) to n-th diagonal of unbounded chessboard.

Original entry on oeis.org

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

Clark Kimberling

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

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).

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

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

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 *)

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

(Sage) [0,3]+[(n+2 - 2*chebyshev_U(2*n, 1/2))/3 for n in (2..75)] # G. C. Greubel, Oct 22 2022

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)

A279605 Triangle T(n, k) read by rows: minimal number of knight moves to reach the central square on a (2n+1) X (2n+1) board starting from the k-th outermost square counted from middle of first rank for k = 1..n+1, or -1 if reaching the central square is impossible.

Original entry on oeis.org

0, -1, -1, 4, 1, 2, 2, 3, 2, 3, 4, 3, 2, 3, 2, 4, 3, 4, 3, 4, 3, 4, 5, 4, 3, 4, 3, 4, 6, 5, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 4, 5, 4, 5, 4, 6, 7, 6, 5, 6, 5, 6, 5, 6, 5, 8, 7, 6, 7, 6, 5, 6, 5, 6, 5, 6, 8, 7, 8, 7, 6, 7, 6, 7, 6, 7, 6, 7, 8, 9, 8, 7, 8, 7, 6, 7, 6, 7, 6, 7, 6

Offset: 0

Felix Fröhlich, Dec 15 2016

			Triangle starts
   0;
  -1, -1;
   4,  1,  2;
   2,  3,  2,  3;
   4,  3,  2,  3,  2;
   4,  3,  4,  3,  4,  3;
   4,  5,  4,  3,  4,  3,  4;
   6,  5,  4,  5,  4,  5,  4,  5;
   6,  5,  6,  5,  4,  5,  4,  5,  4;
   6,  7,  6,  5,  6,  5,  6,  5,  6,  5;
   ...
T(0, 1) = 0, because the board has just 1 square where the knight must start.
T(1, 1) and T(1, 2) = -1, because reaching the central square with a knight is not possible on a 3 X 3 board.
T(2, 1) = 4, because at least 4 moves are necessary on a 5 X 5 board to reach the central square when starting from a corner square.
T(2, 3) = 2 because 2 moves are necessary on a 5 X 5 board to reach the central square when starting from the middle of one side. - _Andrew Howroyd_, Feb 28 2020

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows n = 0..50)
Wikipedia, Jeson Mor.

Cf. A035008, A049604, A065775.

T(n,k) = A049604(n, n-k) = A065775(n, n-k) for n > 1. - Andrew Howroyd, Feb 28 2020

a(5) corrected and terms a(15) and beyond from Andrew Howroyd, Feb 28 2020

cp's OEIS Frontend

A047879 a(n)=Sum{T(n,i): i=0,1,...,n}+Sum{T(i,n): i=0,1,...,n-1}, array T counting least number of knight's moves as in A049604.

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A047881 a(n) = Sum_{h=0..n, k=0..n} T(h,k), array T counting knights' moves as in A049604.

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A047876 a(n) = Sum{T(i,n-i): i=0,1,...,n}, array T as in A049604.

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Formula

A047877 a(n) = (1/2)*Sum_{i=0..n} T(i,n-i), array T as in A049604.

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A065775 Array T read by diagonals: T(i,j)=least number of knight's moves on a chessboard (infinite in all directions) needed to move from (0,0) to (i,j).

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A062103 Number of paths by which an unpromoted knight (keima) of Shogi can move to various squares on infinite board, if it starts from its origin square, the second leftmost square of the back rank.

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Maple

Mathematica

A047878 a(n) is the least number of knight's moves from corner (0,0) to n-th diagonal of unbounded chessboard.

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Magma

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A279605 Triangle T(n, k) read by rows: minimal number of knight moves to reach the central square on a (2*n+1) X (2*n+1) board starting from the k-th outermost square counted from middle of first rank for k = 1..n+1, or -1 if reaching the central square is impossible.

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A279605 Triangle T(n, k) read by rows: minimal number of knight moves to reach the central square on a (2n+1) X (2n+1) board starting from the k-th outermost square counted from middle of first rank for k = 1..n+1, or -1 if reaching the central square is impossible.