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A183575 a(n) = n - 1 + ceiling((n^2-2)/2); complement of A183574. 4
2, 6, 10, 16, 22, 30, 38, 48, 58, 70, 82, 96, 110, 126, 142, 160, 178, 198, 218, 240, 262, 286, 310, 336, 362, 390, 418, 448, 478, 510, 542, 576, 610, 646, 682, 720, 758, 798, 838, 880, 922, 966, 1010, 1056, 1102, 1150, 1198, 1248, 1298, 1350, 1402, 1456, 1510, 1566, 1622, 1680, 1738, 1798, 1858, 1920, 1982, 2046, 2110, 2176, 2242, 2310, 2378, 2448, 2518 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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 centre. - 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).

a(n) = A000217(n) + A047215(n). - Wesley Ivan Hurt, Jul 15 2013

From Colin Barker, Dec 07 2017: (Start)

G.f.: 2*x*(1 + x - x^2) / ((1 - x)^3*(1 + x)).

a(n) = n*(n + 4)/2 for n even.

a(n) = (n^2 + 4*n - 1)/2 for n odd.

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 4.

(End)

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}, {2, 6, 10, 16}, 80] (* Harvey P. Dale, Feb 19 2021 *)

PROG

(PARI) Vec(2*x*(1 + x - x^2) / ((1 - x)^3*(1 + x)) + O(x^60)) \\ Colin Barker, Dec 07 2017

CROSSREFS

Cf. A183574 (complement).

Sequence in context: A220453 A195957 A294013 * A096184 A254829 A030511

Adjacent sequences:  A183572 A183573 A183574 * A183576 A183577 A183578

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jan 05 2011

EXTENSIONS

Description corrected by Eric W. Weisstein, May 18 2017

STATUS

approved

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Last modified June 19 07:26 EDT 2021. Contains 345126 sequences. (Running on oeis4.)