A055999 a(n) = n*(n + 7)/2.

0, 4, 9, 15, 22, 30, 39, 49, 60, 72, 85, 99, 114, 130, 147, 165, 184, 204, 225, 247, 270, 294, 319, 345, 372, 400, 429, 459, 490, 522, 555, 589, 624, 660, 697, 735, 774, 814, 855, 897, 940, 984, 1029, 1075, 1122, 1170, 1219, 1269, 1320, 1372, 1425, 1479

Offset: 0

Author

Barry E. Williams, Jun 16 2000

Keywords

• easy
• nonn

Comments

If X is an n-set and Y a fixed (n-4)-subset of X then a(n-3) is equal to the number of 2-subsets of X intersecting Y. - Milan Janjic, Aug 15 2007

Numbers m >= 0 such that 8m+49 is a square. - Bruce J. Nicholson, Jul 28 2017

References

• Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 193.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Milan Janjic, Two Enumerative Functions.
• Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 3.
• Leo Tavares, Diamond illustration.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Equals A000217(n+3) - 6.

Cf. A000096, A055998, A074171, A056000, A001477, A002522, A028563, A126890.

Third column (m=2) of (1, 4)-Pascal triangle A095666.

Cf. A000290.

Programs

Mathematica

Table[n*(n + 7)/2, {n, 0, 50}] (* G. C. Greubel, Jul 13 2017 *)

PARI

a(n)=n*(n+7)/2 \\ Charles R Greathouse IV, Sep 24 2015

Formula

G.f.: x*(4-3*x)/(1-x)^3.

a(n) = A126890(n,3) for n>2. - Reinhard Zumkeller, Dec 30 2006

a(n) = A028563(n)/2. - Zerinvary Lajos, Feb 12 2007

If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then a(n) = -f(n,n-1,4), for n>=1. - Milan Janjic, Dec 20 2008

a(n) = n + a(n-1) + 3 (with a(0)=0). - Vincenzo Librandi, Aug 07 2010

a(n) = Sum_{k=1..n} (k+3). - Gary Detlefs, Aug 10 2010

Sum_{n>=1} 1/a(n) = 363/490. - R. J. Mathar, Jul 14 2012

a(n) = 4n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013

a(n) = Sum_{i=4..n+3} i. - Wesley Ivan Hurt, Jun 28 2013

E.g.f.: (1/2)*x*(x+8)*exp(x). - G. C. Greubel, Jul 13 2017

Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/7 - 319/1470. - Amiram Eldar, Jan 10 2021

a(n) = A000290(n+1) - A000217(n-2). - Leo Tavares, Jan 28 2023

From Amiram Eldar, Feb 12 2024: (Start)

Product_{n>=1} (1 - 1/a(n)) = 15*cos(sqrt(57)*Pi/2)/(8*Pi).

Product_{n>=1} (1 + 1/a(n)) = -63*cos(sqrt(41)*Pi/2)/(8*Pi). (End)

Extensions

More terms from James Sellers, Jul 04 2000