OFFSET
1,2
COMMENTS
Agrees with the circumference of the n X n stacked book graph for n = 2 up to at least n = 8. - Eric W. Weisstein, Dec 05 2017
It seems that a(n-1) is the maximal length of an optimal solution path required to solve any n X n maze. Here the maze has a single start point, a single end point, and any number of walls that cannot be traversed. The maze is 4-connected, so the allowed moves are: up, down, left and right. For odd n, the hardest mazes have walls located in a spiral, start point in one corner and end point in the center. - Dmitry Kamenetsky, Mar 06 2018
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Graph Circumference.
Eric Weisstein's World of Mathematics, Stacked Book Graph.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
FORMULA
a(n) = n - 1 + ceiling(n^2/2-1).
From Colin Barker, Dec 07 2017: (Start)
G.f.: 2*x^2*(1 + x - x^2) / ((1 - x)^3*(1 + x)).
a(n) = (n^2 + 2*n - 4)/2 for n even.
a(n) = (n^2 + 2*n - 3)/2 for n odd.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 4.
(End)
Sum_{n>=2} 1/a(n) = 7/8 + tan(sqrt(5)*Pi/2)*Pi/(2*sqrt(5)). - Amiram Eldar, Sep 16 2022
E.g.f.: (4 + (x^2 + 3*x - 4)*cosh(x) + (x^2 + 3*x - 3)*sinh(x))/2. - Stefano Spezia, Sep 05 2023
MATHEMATICA
Table[Ceiling[n^2/2 - 1] + n - 1, {n, 20}] (* Eric W. Weisstein, May 18 2017 *)
Table[(2 n (n + 2) - 7 - (-1)^n)/4, {n, 20}] (* Eric W. Weisstein, May 18 2017 *)
Table[If[Mod[n, 2] == 0, n^2 + 2 n - 4, (n + 3) (n - 1)]/2, {n, 20}] (* Eric W. Weisstein, May 18 2017 *)
LinearRecurrence[{2, 0, -2, 1}, {0, 2, 6, 10}, 80] (* Harvey P. Dale, Feb 19 2021 *)
PROG
(PARI) concat(0, Vec(2*x*(1 + x - x^2) / ((1 - x)^3*(1 + x)) + O(x^60))) \\ Colin Barker, Dec 07 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jan 05 2011
EXTENSIONS
Description corrected by Eric W. Weisstein, May 18 2017
a(1)=0 inserted by Amiram Eldar, Sep 16 2022
STATUS
approved