A061925 a(n) = ceiling(n^2/2) + 1.

1, 2, 3, 6, 9, 14, 19, 26, 33, 42, 51, 62, 73, 86, 99, 114, 129, 146, 163, 182, 201, 222, 243, 266, 289, 314, 339, 366, 393, 422, 451, 482, 513, 546, 579, 614, 649, 686, 723, 762, 801, 842, 883, 926, 969, 1014, 1059, 1106, 1153, 1202, 1251, 1302, 1353, 1406

Offset: 0

Author

Henry Bottomley, May 17 2001

Keywords

• nonn
• easy

Comments

a(n+1) gives index of the first occurrence of n in A100795. - Amarnath Murthy, Dec 05 2004

First term in each group in A074148. - Amarnath Murthy, Aug 28 2002

From Christian Barrientos, Jan 01 2021: (Start)

For n >= 3, a(n) is the number of square polyominoes with at least 2n - 2 cells whose bounding box has size 2 X n.

For n = 3, there are 6 square polyominoes with a bounding box of size 2 X 3:

_ _ _ _ _

|||_| |||_| |||_| |||_| |||_| |||_

|||_| ||| || || || || |||

(End)

Except for a(2), a(n) agrees with the lower matching number of the (n+1) X (n+1) bishop graph up to at least n = 13. - Eric W. Weisstein, Dec 23 2024

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000
• Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, Cyclic permutations avoiding patterns in both one-line and cycle forms, arXiv:2312.05145 [math.CO], 2023. See p. 2.
• Eric Weisstein's World of Mathematics, Bishop Graph.
• Eric Weisstein's World of Mathematics, Lower Matching Number.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A100795, A074147-A074149.

Programs

Maple

seq(floor((n^2+3)/2),n=0..25); # Gary Detlefs, Feb 13 2010

Mathematica

Table[Ceiling[n^2/2]+1,{n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011 *)
LinearRecurrence[{2,0,-2,1},{1,2,3,6},60] (* Harvey P. Dale, Jan 03 2024 *)

PARI

a(n) = { ceil(n^2/2) + 1 } \\ Harry J. Smith, Jul 29 2009

Formula

a(n) = a(n-1) + 2*floor((n-1)/2) + 1 = A061926(3, k) = 2*A002620(n+1) - (n-1) = A000982(n) + 1.

a(2*n) = a(2*n-1) + 2*n - 1 = 2*n^2 + 1 = A058331(n).

a(2*n+1) = a(2*n) + 2*n + 1 = 2*(n^2 + n + 1) = A051890(n+1).

a(n) = floor((n^2+3)/2). - Gary Detlefs, Feb 13 2010

From R. J. Mathar, Feb 19 2010: (Start)

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

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

a(n) = (2*n^2 - (-1)^n + 5)/4. - Bruno Berselli, Sep 29 2011

a(n) = A007590(n+1) - n + 1. - Wesley Ivan Hurt, Jul 15 2013

a(n) + a(n+1) = A027688(n). a(n+1) - a(n) = A109613(n). - R. J. Mathar, Jul 20 2013

E.g.f.: ((2 + x + x^2)*cosh(x) + (3 + x + x^2)*sinh(x))/2. - Stefano Spezia, May 07 2021

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 09 2007