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

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

Author

Daniël Karssen, Jul 15 2018

Keywords

• nonn
• easy

Comments

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 .

.

Links

• 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).

Crossrefs

Cf. A316588, A316668, A316669, A316670.

Programs

Mathematica

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 *)

PARI

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

Formula

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)