A175287 Partial sums of ceiling(n^2/4).

0, 1, 2, 5, 9, 16, 25, 38, 54, 75, 100, 131, 167, 210, 259, 316, 380, 453, 534, 625, 725, 836, 957, 1090, 1234, 1391, 1560, 1743, 1939, 2150, 2375, 2616, 2872, 3145, 3434, 3741, 4065, 4408, 4769, 5150, 5550, 5971, 6412, 6875, 7359, 7866, 8395, 8948, 9524, 10125, 10750

Offset: 0

Author

Mircea Merca, Dec 03 2010

Keywords

• nonn
• easy

Comments

a(n) is the number of 1243-avoiding odd Grassmannian permutations of size n+1. Avoiding any of the patterns 2134, 2341, or 4123, gives the same sequence. - Juan B. Gil, Mar 09 2023

Conjecture: a(n) is the number of perimeter-magic (hollow) triangles of order 3 with magic sum n+2. Order 3 means each of the 3 edges has 3 elements >=1; the triangle has 6 elements. The elements do not need to be distinct, and triangles obtained by rotations are counted only once. The triangle (read ccw) for magic sum 3 has elements 1 1 1 1 1 1. The 2 triangles with magic sum 4 are 1 1 2 1 1 2 and 1 2 1 2 1 2. - R. J. Mathar, Mar 08 2025

Examples

			a(4) = ceil(0/4)+ceil(1/4)+ceil(4/4)+ceil(9/4)+ceil(16/4) = 0+1+1+3+4=9.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• L. Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.
• Juan B. Gil and Jessica A. Tomasko, Pattern-avoiding even and odd Grassmannian permutations, arXiv:2207.12617 [math.CO], 2022.
• Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Crossrefs

Partial sums of A004652.

Cf. A361272.

Programs

Magma

[Floor((n+1)*(2*n^2+n+9)/24): n in [0..60]]; // Vincenzo Librandi, Jun 22 2011

Maple

a:= n-> round((2*n^(3)+3*n^(2)+10*n)/24): seq(a(n), n=0..20);

Mathematica

Table[Sum[Ceiling[i^2/4], {i, 0, n}], {n, 0, 49}] (* or *) Table[(2n(2n^2 + 3n + 10) -9(-1)^n + 9)/48, {n, 0, 49}] (* Alonso del Arte, Dec 03 2010 *)
CoefficientList[Series[(x^3 - x^2 + x)/(x^5 - 3 x^4 + 2 x^3 + 2 x^2 - 3 x + 1), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2014 *)
Accumulate[Ceiling[Range[0,50]^2/4]] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{0,1,2,5,9},60] (* Harvey P. Dale, Nov 19 2014 *)

PARI

x='x+O('x^99); concat(0, Vec((x^3-x^2+x)/ (x^5-3*x^4+2*x^3+2*x^2-3*x+1))) \\ Altug Alkan, Apr 05 2016

Formula

a(n) = round((2*n+1)*(2*n^2+2*n+9)/48).

a(n) = floor((n+1)*(2*n^2+n+9)/24).

a(n) = ceiling((2*n^3+3*n^2+10*n)/24).

a(n) = round((2*n^3+3*n^2+10*n)/24).

a(n) = a(n-4)+n^2-3*n+5 , n>3.

G.f.: x*(1-x+x^2) / ( (1+x)*(x-1)^4 ).

a(n) = (2*n*(2*n^2+3*n+10)-9*(-1)^n+9)/48. - Bruno Berselli, Dec 03 2010

a(n)+a(n+1) = A004006(n+1). - R. J. Mathar, Mar 08 2025