A056000 a(n) = n*(n+9)/2.

0, 5, 11, 18, 26, 35, 45, 56, 68, 81, 95, 110, 126, 143, 161, 180, 200, 221, 243, 266, 290, 315, 341, 368, 396, 425, 455, 486, 518, 551, 585, 620, 656, 693, 731, 770, 810, 851, 893, 936, 980, 1025, 1071, 1118, 1166, 1215, 1265, 1316, 1368, 1421, 1475

Offset: 0

Author

Barry E. Williams, Jun 16 2000

Keywords

• easy
• nonn

Comments

Numbers m >= 0 such that 8m+81 is a square. - Bruce J. Nicholson, Jul 29 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
• Leo Tavares, Illustration: Triangular Pairs.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000096, A055998, A055999, A001477.

Column m=2 of (1, 5)-Pascal triangle A096940.

Cf. numbers of the form n*(d*n+10-d)/2 indexed in A140090.

Cf. A000217, A008585.

Programs

Mathematica

Table[n (n + 9)/2, {n, 0, 50}] (* or *)
FoldList[#1 + #2 + 4 &, Range[0, 50]] (* or *)
Table[PolygonalNumber[n + 4] - 10, {n, 0, 50}] (* or *)
CoefficientList[Series[x (5 - 4 x)/(1 - x)^3, {x, 0, 50}], x] (* Michael De Vlieger, Jul 30 2017 *)

PARI

a(n)=n*(n+9)/2 \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = A000217(n+4) - 10.

G.f.: x(5-4x)/(1-x)^3.

From Zerinvary Lajos, Oct 01 2006: (Start)

a(n) = A000096(n) + 3*n.

a(n) = A055999(n) + n.

a(n) = A056115(n) - n.

(End)

a(n) = binomial(n,2) - 4*n, n >= 9. - Zerinvary Lajos, Nov 25 2006

a(n) = A126890(n,4) for n > 3. - Reinhard Zumkeller, Dec 30 2006

a(n) = A028569(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-1,5), for n >= 1. - Milan Janjic, Dec 20 2008

a(n) = n + a(n-1) + 4. - Vincenzo Librandi, Aug 07 2010

a(n) = Sum_{k=1..n} (k+4). - Gary Detlefs, Aug 10 2010

Sum_{n>=1} 1/a(n) = 7129/11340. - R. J. Mathar, Jul 14 2012

a(n) = 5n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013

E.g.f.: (1/2)*(x^2 + 10*x)*exp(x). - G. C. Greubel, Jul 17 2017

Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/9 - 1879/11340. - Amiram Eldar, Jul 03 2020

a(n) = A000217(n+1) + A008585(n) - 1. - Leo Tavares, Sep 22 2022

From Amiram Eldar, Feb 12 2024: (Start)

Product_{n>=1} (1 - 1/a(n)) = -567*cos(sqrt(89)*Pi/2)/(220*Pi).

Product_{n>=1} (1 + 1/a(n)) = 35*cos(sqrt(73)*Pi/2)/(4*Pi). (End)

Extensions

More terms from James Sellers, Jul 04 2000