OFFSET
0,2
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
LINKS
Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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
EXAMPLE
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.
MATHEMATICA
Table[(5/2) n (n + 5), {n, 0, 46}]
PROG
(Magma) [5*n*(n+5)/2: n in [0..46]];
(PARI) a(n)=5*n*(n+5)/2 \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, May 30 2012
EXTENSIONS
Extended by Bruno Berselli, Aug 05 2015
STATUS
approved