OFFSET
0,3
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
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).
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
EXAMPLE
a(4) = ceil(0/4)+ceil(1/4)+ceil(4/4)+ceil(9/4)+ceil(16/4) = 0+1+1+3+4=9.
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 *)
PROG
(Magma) [Floor((n+1)*(2*n^2+n+9)/24): n in [0..60]]; // Vincenzo Librandi, Jun 22 2011
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Mircea Merca, Dec 03 2010
STATUS
approved