A120413 Largest even number strictly less than n^2.

0, 2, 8, 14, 24, 34, 48, 62, 80, 98, 120, 142, 168, 194, 224, 254, 288, 322, 360, 398, 440, 482, 528, 574, 624, 674, 728, 782, 840, 898, 960, 1022, 1088, 1154, 1224, 1294, 1368, 1442, 1520, 1598, 1680, 1762, 1848, 1934, 2024, 2114, 2208, 2302, 2400, 2498, 2600

Offset: 1

Author

Henry Bottomley, Jul 06 2006

Keywords

• easy
• nonn

Comments

Longest non-intersecting route from (0, 0) to (n - 1, n - 1) staying in an (n - 1) X (n - 1) box (shortest route is length 2n A005843).

Links

• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Programs

Maple

seq(2*ceil(n^2/2)-2,n=1..50);

Mathematica

Flatten[Table[{(2n - 1)^2 - 1, 4n^2 - 2}, {n, 25}]] (* Alonso del Arte, Apr 15 2016 *)

PARI

lista(nn) = for(n=0, nn, print1((-1+(-1)^n+4*n+2*n^2)/2, ", ")); \\ Altug Alkan, Apr 15 2016

Formula

a(n) = 2*ceiling[n^2/2] - 2 = 2*A074148(n) = A085046(n) - 1.

From Colin Barker, Jul 29 2012: (Start)

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

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

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

a(n) = n^2 - 2 for even n; a(n) = n^2 - 1 for odd n. -Dennis P. Walsh, Apr 15 2016

Extensions

Offset corrected by N. J. A. Sloane, Apr 15 2016