A212331 a(n) = 5*n*(n+5)/2.

0, 15, 35, 60, 90, 125, 165, 210, 260, 315, 375, 440, 510, 585, 665, 750, 840, 935, 1035, 1140, 1250, 1365, 1485, 1610, 1740, 1875, 2015, 2160, 2310, 2465, 2625, 2790, 2960, 3135, 3315, 3500, 3690, 3885, 4085, 4290, 4500, 4715, 4935, 5160, 5390, 5625, 5865

Offset: 0

Author

Bruno Berselli, May 30 2012

Keywords

• nonn
• easy

Comments

Numbers of the form n*t(n+5,h)-(n+5)*t(n,h), where t(k,h) = k*(k+2*h+1)/2 for any h. Likewise:

A000217(n) = n*t(n+1,h)-(n+1)*t(n,h),

A005563(n) = n*t(n+2,h)-(n+2)*t(n,h),

A140091(n) = n*t(n+3,h)-(n+3)*t(n,h),

A067728(n) = n*t(n+4,h)-(n+4)*t(n,h) (n>0),

A140681(n) = n*t(n+6,h)-(n+6)*t(n,h).

This is the case r=7 in the formula:

u(r,n) = (P(r, P(n+r, r+6)) - P(n+r, P(r, r+6))) / ((r+5)*(r+6)/2)^2, where P(s, m) is the m-th s-gonal number.

Also, a(k) is a square for k = (5/2)*(A078986(n)-1).

Sum of reciprocals of a(n), for n>0: 137/750.

Also, numbers h such that 8*h/5+25 is a square.

The table given below as example gives the dimensions D(h, n) of the irreducible SU(3) multiplets (h,n). See the triangle A098737 with offset 0, and the comments there, also with a link and the Coleman reference. - Wolfdieter Lang, Dec 18 2020

Examples

			From the first and second comment derives the following table:
----------------------------------------------------------------
h \ n | 0   1    2    3    4    5    6    7    8    9    10
------|---------------------------------------------------------
0     | 0,  1,   3,   6,  10,  15,  21,  28,  36,  45,   55, ...  (A000217)
1     | 0,  3,   8,  15,  24,  35,  48,  63,  80,  99,  120, ...  (A005563)
2     | 0,  6,  15,  27,  42,  60,  81, 105, 132, 162,  195, ...  (A140091)
3     | 0, 10,  24,  42,  64,  90, 120, 154, 192, 234,  280, ...  (A067728)
4     | 0, 15,  35,  60,  90, 125, 165, 210, 260, 315,  375, ...  (A212331)
5     | 0, 21,  48,  81, 120, 165, 216, 273, 336, 405,  480, ...  (A140681)
6     | 0, 28,  63, 105, 154, 210, 273, 343, 420, 504,  595, ...
7     | 0, 36,  80, 132, 192, 260, 336, 420, 512, 612,  720, ...
8     | 0, 45,  99, 162, 234, 315, 405, 504, 612, 729,  855, ...
9     | 0, 55, 120, 195, 280, 375, 480, 595, 720, 855, 1000, ...
with the formula n*(h+1)*(h+n+1)/2. See also A098737.
		

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A000537, A005563, A055998, A067728, A098737, A140091, A140681.

Programs

Magma

[5*n*(n+5)/2: n in [0..46]];

Mathematica

Table[(5/2) n (n + 5), {n, 0, 46}]

PARI

a(n)=5*n*(n+5)/2 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: 5*x*(3-2*x)/(1-x)^3.

a(n) = a(-n-5) = 5*A055998(n).

E.g.f.: (5/2)*x*(x + 6)*exp(x). - G. C. Greubel, Jul 21 2017

Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/25 - 47/750. - Amiram Eldar, Feb 26 2022

Extensions

Extended by Bruno Berselli, Aug 05 2015