A052905 a(n) = (n^2 + 7*n + 2)/2.

1, 5, 10, 16, 23, 31, 40, 50, 61, 73, 86, 100, 115, 131, 148, 166, 185, 205, 226, 248, 271, 295, 320, 346, 373, 401, 430, 460, 491, 523, 556, 590, 625, 661, 698, 736, 775, 815, 856, 898, 941, 985, 1030, 1076, 1123, 1171, 1220, 1270, 1321, 1373, 1426, 1480

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• nonn
• easy

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

Examples

			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]
		

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).

Crossrefs

Cf. A002522, A131899, A134199, A159920.

Programs

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 *)

PARI

a(n)=n*(n+7)/2+1 \\ Charles R Greathouse IV, Nov 20 2011

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

Extensions

More terms from James Sellers, Jun 08 2000

Edited by Charles R Greathouse IV, Jul 25 2010