

A183575


a(n) = n  1 + ceiling((n^22)/2); complement of A183574.


4



0, 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
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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(n1) 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 4connected, 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^2/21).
a(n) = A000217(n2) + A047215(n1).  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^2 + 2*n  4)/2 for n even.
a(n) = (n^2 + 2*n  3)/2 for n odd.
a(n) = 2*a(n1)  2*a(n3) + a(n4) 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


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

Cf. A183574 (complement).
Sequence in context: A195957 A354425 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
a(1)=0 inserted by Amiram Eldar, Sep 16 2022


STATUS

approved



