A316669 -id:A316669 - cp's OEIS frontend

A316964 Same as A316669, except numbering of the squares starts at 0 rather than 1.

0, 1, 2, 4, 3, 5, 8, 6, 7, 9, 13, 10, 11, 12, 14, 19, 15, 16, 17, 18, 20, 26, 21, 22, 23, 24, 25, 27, 34, 28, 29, 30, 31, 32, 33, 35, 43, 36, 37, 38, 39, 40, 41, 42, 44, 53, 45, 46, 47, 48, 49, 50, 51, 52, 54, 64, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 76

Offset: 0

Daniël Karssen, Jul 17 2018

nonn

a(n) = A316669(n+1)-1.

See A316669 for further information.

Daniël Karssen, Table of n, a(n) for n = 0..999
Daniël Karssen, Figure showing the first 22 steps of the sequence

Cf. A316669.

A316671 Squares visited by moving diagonally one square on a diagonally numbered board and moving to the lowest available unvisited square at each step.

Original entry on oeis.org

1, 5, 4, 12, 11, 23, 22, 38, 37, 57, 56, 80, 79, 107, 106, 138, 137, 173, 172, 212, 211, 255, 254, 302, 301, 353, 352, 408, 407, 467, 466, 530, 529, 597, 596, 668, 667, 743, 742, 822, 821, 905, 904, 992, 991, 1083, 1082, 1178, 1177, 1277, 1276, 1380, 1379

Offset: 1

Daniël Karssen, Jul 15 2018

Board is numbered as follows:
2  4  7 11 16  .
5  8 12 17  .
9 13 18  .
14 19  .
20  .
.
   .

Daniël Karssen, Table of n, a(n) for n = 1..10000
Daniël Karssen, Figure showing the first 6 steps of the sequence
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A316588, A316668, A316669, A316670.

CoefficientList[ Series[-(2x^4 - 3x^2 + 4x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 52}], x] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {1, 5, 4, 12, 11}, 53] (* Robert G. Wilson v, Jul 18 2018 *)

Vec(x*(1 + 4*x - 3*x^2 + 2*x^4) / ((1 - x)^3*(1 + x)^2) + O(x^40)) \\ Colin Barker, Jul 18 2018

From Colin Barker, Jul 18 2018: (Start)
G.f.: x*(1 + 4*x - 3*x^2 + 2*x^4) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5.
a(n) = (n^2 + n + 4)/2 for n even.
a(n) = (n^2 - n + 2)/2 for n odd.
(End)

A316668 Squares visited by king moves on a diagonally numbered board and moving to the lowest available unvisited square at each step.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 8, 9, 6, 10, 14, 13, 12, 11, 16, 17, 18, 19, 20, 15, 21, 27, 26, 25, 24, 23, 22, 29, 30, 31, 32, 33, 34, 35, 28, 36, 44, 43, 42, 41, 40, 39, 38, 37, 46, 47, 48, 49, 50, 51, 52, 53, 54, 45, 55, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 67, 68

Offset: 1

Daniël Karssen, Jul 15 2018

nonn

Board is numbered as follows:
   1  2  4  7 11 16  .
   3  5  8 12 17  .
   6  9 13 18  .
  10 14 19  .
  15 20  .
  21  .
   .
Same as A316588 but with king move instead of knight move.

Daniël Karssen, Table of n, a(n) for n = 1..100000
Daniël Karssen, Figure showing the first 26 steps of the sequence

Cf. A316588, A316669, A316670, A316671.

A316670 Squares visited by bishop moves on a diagonally numbered board and moving to the lowest available unvisited square at each step.

Original entry on oeis.org

1, 5, 4, 6, 14, 11, 12, 13, 15, 27, 22, 23, 24, 25, 26, 28, 44, 37, 38, 39, 40, 41, 42, 43, 45, 65, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 90, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 119, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117

Offset: 1

Daniël Karssen, Jul 15 2018

nonn

Board is numbered as follows:
   1  2  4  7 11 16  .
   3  5  8 12 17  .
   6  9 13 18  .
  10 14 19  .
  15 20  .
  21  .
   .
Same as A316588 but with bishop move instead of knight move.

Daniël Karssen, Table of n, a(n) for n = 1..10000
Daniël Karssen, Figure showing the first 10 steps of the sequence

Cf. A316588, A316668, A316669, A316671.

cp's OEIS Frontend

A316964 Same as A316669, except numbering of the squares starts at 0 rather than 1.

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A316671 Squares visited by moving diagonally one square on a diagonally numbered board and moving to the lowest available unvisited square at each step.

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Programs

Mathematica

PARI

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A316668 Squares visited by king moves on a diagonally numbered board and moving to the lowest available unvisited square at each step.

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A316670 Squares visited by bishop moves on a diagonally numbered board and moving to the lowest available unvisited square at each step.

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Author

Keywords

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