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

%I A175287 #60 Mar 08 2025 14:25:58
%S A175287 0,1,2,5,9,16,25,38,54,75,100,131,167,210,259,316,380,453,534,625,725,
%T A175287 836,957,1090,1234,1391,1560,1743,1939,2150,2375,2616,2872,3145,3434,
%U A175287 3741,4065,4408,4769,5150,5550,5971,6412,6875,7359,7866,8395,8948,9524,10125,10750
%N A175287 Partial sums of ceiling(n^2/4).
%C A175287 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
%C A175287 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
%H A175287 Vincenzo Librandi, <a href="/A175287/b175287.txt">Table of n, a(n) for n = 0..10000</a>
%H A175287 L. Colmenarejo, <a href="http://arxiv.org/abs/1604.00803">Combinatorics on several families of Kronecker coefficients related to plane partitions</a>, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.
%H A175287 Juan B. Gil and Jessica A. Tomasko, <a href="https://arxiv.org/abs/2207.12617">Pattern-avoiding even and odd Grassmannian permutations</a>, arXiv:2207.12617 [math.CO], 2022.
%H A175287 Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Merca/merca3.html">Inequalities and Identities Involving Sums of Integer Functions</a> J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
%H A175287 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).
%F A175287 a(n) = round((2*n+1)*(2*n^2+2*n+9)/48).
%F A175287 a(n) = floor((n+1)*(2*n^2+n+9)/24).
%F A175287 a(n) = ceiling((2*n^3+3*n^2+10*n)/24).
%F A175287 a(n) = round((2*n^3+3*n^2+10*n)/24).
%F A175287 a(n) = a(n-4)+n^2-3*n+5 , n>3.
%F A175287 G.f.: x*(1-x+x^2) / ( (1+x)*(x-1)^4 ).
%F A175287 a(n) = (2*n*(2*n^2+3*n+10)-9*(-1)^n+9)/48. - _Bruno Berselli_, Dec 03 2010
%F A175287 a(n)+a(n+1) = A004006(n+1). - _R. J. Mathar_, Mar 08 2025
%e A175287 a(4) = ceil(0/4)+ceil(1/4)+ceil(4/4)+ceil(9/4)+ceil(16/4) = 0+1+1+3+4=9.
%p A175287 a:= n-> round((2*n^(3)+3*n^(2)+10*n)/24): seq(a(n), n=0..20);
%t A175287 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 *)
%t A175287 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 *)
%t A175287 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 *)
%o A175287 (Magma) [Floor((n+1)*(2*n^2+n+9)/24): n in [0..60]]; // _Vincenzo Librandi_, Jun 22 2011
%o A175287 (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
%Y A175287 Partial sums of A004652.
%Y A175287 Cf. A361272.
%K A175287 nonn,easy
%O A175287 0,3
%A A175287 _Mircea Merca_, Dec 03 2010