OFFSET
0,2
COMMENTS
Starting 1, 5, 10, 16, 23, ... gives binomial transform of (1, 4, 1, 0, 0, 0, ...). Row sums of triangle A134199. - Gary W. Adamson, Jul 25 2007
If Y_i (i=1,2,3,4,5) are 2-blocks of an n-set X then, for n >= 10, a(n-4) is the number of (n-2)-subsets of X intersecting each Y_i (i=1,2,3,4,5). - Milan Janjic, Nov 09 2007
This sequence is related to A159920 by A159920(n+1) = n*a(n) - Sum_{i=0..n-1} a(i) for n > 0. - Bruno Berselli, Feb 28 2014
Numbers m > 0 such that 8*m + 41 is a square. - Bruce J. Nicholson, Jul 28 2017
For n >= 1, number of ways to distribute a mass equal to 2 under the repunit list of size n, in a quincunx grid that satisfies the constraint: the value of a cell is <= to the sum of the values of the two cells located just above it. - Luc Rousseau, May 15 2026
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000
Charles Cratty, Samuel Erickson, Frehiwet Negass, and Lara Pudwell, Pattern Avoidance in Double Lists, Involve, Vol. 10, No. 3 (2017), pp. 379-398; preprint, 2015.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 884.
Milan Janjic, Two Enumerative Functions.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
G.f.: (1 + 2*x - 2*x^2)/(1-x)^3.
Recurrence: a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = a(n-1) + n + 3, with n>0, a(0)=1. - Vincenzo Librandi, Aug 06 2010
E.g.f.: (1/2)*(x^2 + 8*x + 2)*exp(x). - G. C. Greubel, Jul 13 2017
Sum_{n>=0} 1/a(n) = 19/20 + 2*Pi*tan(sqrt(41)*Pi/2)/sqrt(41). - Amiram Eldar, Dec 13 2022
EXAMPLE
From Bruno Berselli, Feb 28 2014: (Start)
Illustration of initial terms:
o
o o
o o o o
o o o o o o
o o o o o o o o o
o o o o o o o o o o o o
o o o o o o o o o o . . . . . o
o o o o o o o . . . . o o . . . . . o
o o o o o . . . o o . . . . o o . . . . . o
o o o . . o o . . . o o . . . . o o . . . . . o
o o . o o . . o o . . . o o . . . . o o . . . . . o
o o o . o o . . o o . . . o o . . . . o o . . . . . o
o o o o o o o o o o o o o o o o o o o o o o o o o o o o
---------------------------------------------------------------------
1 5 10 16 23 31 40
(End)
From Luc Rousseau, May 15 2026: (Start)
a(2)=10 illustrated by means of the repartition of 2 under [1 1] in a quincunx grid:
-
[1 1] [1 1] [1 1] [1 1] [1 1]
1 1 . 1 . 1 . 1 1 1 . . 1 . .
. . . . . . . . . . . . 1 . . . . 1 . .
-
[1 1] [1 1] [1 1] [1 1] [1 1]
. 1 . . 1 . . . 1 . . 1 . 2 .
. 1 . . . . 1 . . . 1 . . . . 1 . . . .
-
(End)
MAPLE
spec := [S, {S=Prod(Sequence(Z), Sequence(Z), Union(Sequence(Z), Z, Z))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
seq(binomial(n, 2)-5, n=4..55); # Zerinvary Lajos, Jan 13 2007
MATHEMATICA
k = 3; NestList[(k++; # + k) &, 1, 45] (* Robert G. Wilson v, Feb 03 2011 *)
(* Alternative: *)
Table[(n^2 + 7n + 2)/2, {n, 0, 49}] (* Alonso del Arte, Feb 03 2011 *)
(* Alternative: *)
LinearRecurrence[{3, -3, 1}, {1, 5, 10}, 60] (* Harvey P. Dale, Sep 15 2018 *)
(* Alternative: *)
((2*Range[0, 60] +7)^2 -41)/8 (* G. C. Greubel, Dec 26 2025 *)
PROG
(PARI) a(n)=n*(n+7)/2+1 \\ Charles R Greathouse IV, Nov 20 2011
(Magma) A052905:= func< n | (n^2 +7*n +2)/2 >; // G. C. Greubel, Dec 26 2025
(SageMath)
def A052905(n): return (n**2 +7*n +2)//2 # G. C. Greubel, Dec 26 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
INRIA Encyclopedia of Combinatorial Structures, Jan 25 2000
EXTENSIONS
More terms from James Sellers, Jun 08 2000
Edited by Charles R Greathouse IV, Jul 25 2010
Edited by G. C. Greubel, Dec 26 2025
STATUS
approved