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 8m+41 is a square. - Bruce J. Nicholson, Jul 28 2017
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.
Milan Janjic, Two Enumerative Functions.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 884.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
G.f.: (-2*x+2*x^2-1)/(-1+x)^3.
Recurrence: {a(0)=1, a(1)=5, a(2)=10, -2*a(n)+n^2+7*n+2}.
a(n) = n+a(n-1)+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
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
[Bruno Berselli, Feb 28 2014]
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
a:=n->sum((n-4)/2, j=0..n): seq(a(n)-2, n=5..56); # Zerinvary Lajos, Apr 30 2007
with (combinat):seq((fibonacci(3, n)+n-11)/2, n=3..54); # Zerinvary Lajos, Jun 07 2008
a:=n->sum(k, k=0..n):seq(a(n)/2+sum(k, k=5..n)/2, n=3..54); # Zerinvary Lajos, Jun 10 2008
MATHEMATICA
i=4; s=1; lst={s}; Do[s+=n+i; If[s>=0, AppendTo[lst, s]], {n, 0, 6!, 1}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 30 2008 *)
k = 3; NestList[(k++; # + k) &, 1, 45] (* Robert G. Wilson v, Feb 03 2011 *)
Table[(n^2 + 7n + 2)/2, {n, 0, 49}] (* Alonso del Arte, Feb 03 2011 *)
LinearRecurrence[{3, -3, 1}, {1, 5, 10}, 60] (* Harvey P. Dale, Sep 15 2018 *)
PROG
(PARI) a(n)=n*(n+7)/2+1 \\ Charles R Greathouse IV, Nov 20 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
More terms from James A. Sellers, Jun 08 2000
Edited by Charles R Greathouse IV, Jul 25 2010
STATUS
approved