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A052905 a(n) = (n^2 + 7*n + 2)/2. 20
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 (list; graph; refs; listen; history; text; internal format)
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( a(i), i=0..n-1 ) 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, Lara Pudwell, Pattern Avoidance in Double Lists, 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

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

Cf. A002522, A131899, A134199.

Sequence in context: A313939 A313940 A212455 * A306351 A215341 A194275

Adjacent sequences:  A052902 A052903 A052904 * A052906 A052907 A052908

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

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Last modified October 16 20:55 EDT 2019. Contains 328103 sequences. (Running on oeis4.)