A140681 -id:A140681 - cp's OEIS frontend

A212331 a(n) = 5n(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

Bruno Berselli, May 30 2012

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

			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.

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

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

Magma
```
[5*n*(n+5)/2: n in [0..46]];
```
Mathematica
```
Table[(5/2) n (n + 5), {n, 0, 46}]
```

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

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

Extended by Bruno Berselli, Aug 05 2015

A098737 Triangle read by rows: number of triangles formed by lines from two vertices of a triangle to the opposite side that segment the opposite sides into m and n segments. Since f(m,n) = f(n,m), it suffices to give the results in a triangular table.

Original entry on oeis.org

1, 3, 8, 6, 15, 27, 10, 24, 42, 64, 15, 35, 60, 90, 125, 21, 48, 81, 120, 165, 216, 28, 63, 105, 154, 210, 273, 343, 36, 80, 132, 192, 260, 336, 420, 512, 45, 99, 162, 234, 315, 405, 504, 612, 729, 55, 120, 195, 280, 375, 480, 595, 720, 855, 1000, 66, 143, 231, 330

Offset: 1

Eugene McDonnell (eemcd(AT)mac.com), Oct 29 2004

Frank Buss gave this as a puzzle; K. L. Metlov solved it, submitting his result in the J language created by Kenneth Iverson. The program given below is only five tokens long. J defines a series of three functions to be a "fork" defined by x (f g h ) y = (x f y) g (f h y) - a generalization of the usual mathematical practice of writing (f + g) y to mean (f y) + (g y). J also has a primitive "half" and has a dummy function "cap" whose purpose is to permit more forks to be written. 3 (* * +) 5 is thus (3 * 5) * (3 + 5) or 120. cap half 3 (* * +) 5 is thus 60.
This sequence is the dimensions of the various irreducible representations of SU(3). In the language of physics, the integers m and n are one more than the numbers of quarks or antiquarks, respectively, that label the representation. - Alex Meiburg, Dec 13 2020 =
Comment on the previous one: D(n, m) = f(m+1, n+1) = (n+1)*(m+1)*(n+m+2), for 0 <= m <= n, (given as array D(n,m) as example in A212331) is the dimension of the irreducible SU(3) multiplet (n, m), denoted also by D(n, m). The multiplet (m, n) is denoted also by a bar over D(n, m). The irreducuble tensor t(n, m) is symmetric in n upper indices from {1,2,3}, symmetric in m lower indices, and traceless in every pair of an upper and a lower index. See the Coleman reference for a derivation. - Wolfdieter Lang, Dec 18 2020

			f(3, 5) is 60, from 1/2 * (3 * 5) * (3 + 5) or 1/2 * 15 * 8.
The triangle f(m, n) starts:
m\n     1   2   3   4   5   6   7   8   9   10   11 ...
1:      1
2:      3   8
3:      6  15  27
4:     10  24  42  64
5:     15  35  60  90 125
6:     21  48  81 120 165 216
7:     28  63 105 154 210 273 343
8:     36  80 132 192 260 336 420 512
9:     45  99 162 234 315 405 504 612 729
10:    55 120 195 280 375 480 595 720 855 1000
11:    66 143 231 330 440 561 693 836 990 1155 1331
... reformatted and extended by _Wolfdieter Lang_, Dec 18 2020

Sidney Coleman, Quantum Field Theory, Eds. Bryan Gin-ge Chen et al., World Scientific, 2019, eq. (37.8), p. 799.

Wikipedia, Clebsch-Gordan coefficients for SU(3)

Cf. A000217, A005563, A140091, A067728, A212331, A140681 (columns), A000578, A059270, A331433 (diagonals).
(diagonal).
See also A107985, A212331 (array as example).

J
```
cap half * * +
```

t[m_, n_] := (m*n)(m + n)/2; Flatten[ Table[ t[m, n], {m, 10}, {n, m}]] (* Robert G. Wilson v, Nov 04 2004 *)

f(m, n) = 1/2 * (m * n) * (m + n).

G.f.: x*y*(1 + 4*x*y + x^2*(y - 9)*y - 3*x^3*(y - 1)*y + 3*x^4*y^2)/((1 - x)^3*(1 - x*y)^4). - Stefano Spezia, Oct 01 2023

More terms from Robert G. Wilson v, Nov 04 2004

cp's OEIS Frontend

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

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A098737 Triangle read by rows: number of triangles formed by lines from two vertices of a triangle to the opposite side that segment the opposite sides into m and n segments. Since f(m,n) = f(n,m), it suffices to give the results in a triangular table.

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J

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A098737 Triangle read by rows: number of triangles formed by lines from two vertices of a triangle to the opposite side that segment the opposite sides into m and n segments. Since f(m,n) = f(n,m), it suffices to give the results in a triangular table.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

J

Mathematica

Formula

Extensions

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