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A212331
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a(n) = 5*n*(n+5)/2.
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7
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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FORMULA
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G.f.: 5*x*(3-2*x)/(1-x)^3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/25 - 47/750. - Amiram Eldar, Feb 26 2022
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EXAMPLE
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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.
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MATHEMATICA
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Table[(5/2) n (n + 5), {n, 0, 46}]
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PROG
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(Magma) [5*n*(n+5)/2: n in [0..46]];
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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