A067728 -id:A067728 - cp's OEIS frontend

A067726 a(n) = 6n^2 + 12n.

18, 48, 90, 144, 210, 288, 378, 480, 594, 720, 858, 1008, 1170, 1344, 1530, 1728, 1938, 2160, 2394, 2640, 2898, 3168, 3450, 3744, 4050, 4368, 4698, 5040, 5394, 5760, 6138, 6528, 6930, 7344, 7770, 8208, 8658, 9120, 9594, 10080, 10578, 11088, 11610

Offset: 1

Robert G. Wilson v, Feb 05 2002

Positive numbers k such that 6*(6 + k) is a perfect square.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Leo Tavares, Illustration: Star Cut Hexagons
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. numbers k such that k*(k + m) is a perfect square: A028560 (k=9), A067728 (k=8), A067727 (k=7), A067724 (k=5), A028347 (k=4), A067725 (k=3), A054000 (k=2), A005563 (k=1).

Cf. A003154, A003215.

List([1..45], n-> 6*n*(n+2)); # G. C. Greubel, Sep 01 2019

[6*n*(n+2): n in [1..50]]; // Vincenzo Librandi, Jul 08 2012

seq(6*n*(n+2), n=1..45); # G. C. Greubel, Sep 01 2019

Select[ Range[15000], IntegerQ[ Sqrt[ 6(6 + # )]] & ]
CoefficientList[Series[6*(3-x)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 08 2012 *)
6*(Range[2, 45]^2 -1) (* G. C. Greubel, Sep 01 2019 *)
LinearRecurrence[{3,-3,1},{18,48,90},60] (* Harvey P. Dale, May 10 2022 *)

a(n)=6*n*(n+2) \\ Charles R Greathouse IV, Dec 07 2011

[6*n*(n+2) for n in (1..45)] # G. C. Greubel, Sep 01 2019

G.f.: 6*x*(3 - x)/(1 - x)^3. - Vincenzo Librandi, Jul 08 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 08 2012
E.g.f.: 6*x*(3 + x)*exp(x). - G. C. Greubel, Sep 01 2019
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/8.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/24. (End)
a(n) = A003215(2*n) - A003154(n). - Leo Tavares, May 20 2023
a(n) = 6*A005563(n). - Hugo Pfoertner, May 24 2023

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

Original entry on oeis.org

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

A067705 a(n) = 11n^2 + 22n.

Original entry on oeis.org

33, 88, 165, 264, 385, 528, 693, 880, 1089, 1320, 1573, 1848, 2145, 2464, 2805, 3168, 3553, 3960, 4389, 4840, 5313, 5808, 6325, 6864, 7425, 8008, 8613, 9240, 9889, 10560, 11253, 11968, 12705, 13464, 14245, 15048, 15873, 16720, 17589, 18480, 19393, 20328, 21285

Offset: 1

Robert G. Wilson v, Feb 05 2002

Numbers k such that 11*(11 + k) is a perfect square.

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

Cf. A067724, A067725, A067726, A067727, A067728 (if 11 is replaced by 3, 5, 6, 7, 8 respectively), A067707 (12).

Cf. A005563.

[11*n*(n+2): n in [1..50]]; // Vincenzo Librandi, Jul 07 2012

Select[ Range[20000], IntegerQ[ Sqrt[ 11(11 + # )]] & ]
CoefficientList[Series[11 (3 - x)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Jul 07 2012 *)

a(n)=11*n*(n+2) \\ Charles R Greathouse IV, Dec 07 2011

From Vincenzo Librandi, Jul 07 2012: (Start)
G.f.: 11*x*(3-x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Elmo R. Oliveira, Jan 28 2025: (Start)
E.g.f.: 11*exp(x)*x*(3 + x).
a(n) = 11*A005563(n). (End)

A185732 Accumulation array of the polygonal number array (A086270), by antidiagonals.

Original entry on oeis.org

1, 4, 2, 10, 9, 3, 20, 24, 15, 4, 35, 50, 42, 22, 5, 56, 90, 90, 64, 30, 6, 84, 147, 165, 140, 90, 39, 7, 120, 224, 273, 260, 200, 120, 49, 8, 165, 324, 420, 434, 375, 270, 154, 60, 9, 220, 450, 612, 672, 630, 510, 350, 192, 72, 10, 286, 605, 855, 984, 980, 861, 665, 440, 234, 85, 11, 364, 792, 1155, 1380, 1440, 1344, 1127, 840, 540, 280, 99, 12, 455, 1014, 1518, 1870, 2025, 1980, 1764, 1428, 1035, 650, 330, 114, 13, 560, 1274, 1950, 2464, 2750, 2790, 2604, 2240

Offset: 1

Clark Kimberling, Feb 01 2011

This is the (first) accumulation array of A086270; the second is A185733. See A144112 for the definition of accumulation array.

			Northwest corner:
1....4....10...20...35
2....9....24...50...90
3....15...42...90...165
4....22...64...140..260
5....30...90...200..375

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Rows 1 to 5: A000292, A006002, A059270, A177814, 5*A002411.
Columns 1 to 4: A000027, A055999, A067728, 10*A000096.
Cf. A144112, A086270, A185732.

f[n_,k_]:=k+n*k(k-1)/2;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]  (* Array A086270 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten  (* A086270 *)
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* acc. arr. of {f(n,k)} *)
Factor[s[n,k]]  (* formula for A185732 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* acc. arr. A185732 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten (* A185732 *)

T(n,k) = k*(k+1)*n*(n+1)*(k*n-n+k+5)/12.

A269457 a(n) = 5(n + 1)(n + 4)/2.

Original entry on oeis.org

10, 25, 45, 70, 100, 135, 175, 220, 270, 325, 385, 450, 520, 595, 675, 760, 850, 945, 1045, 1150, 1260, 1375, 1495, 1620, 1750, 1885, 2025, 2170, 2320, 2475, 2635, 2800, 2970, 3145, 3325, 3510, 3700, 3895, 4095, 4300, 4510, 4725, 4945, 5170, 5400, 5635

Offset: 0

Ilya Gutkovskiy, Feb 27 2016

More generally, the ordinary generating function for the sequences of the form k*(n + 1)*(n - 1 + k)/2 is (k*(k - 1)/2 + (k*(3 - k)/2)*x)/(1 - x)^3 (see links section).

			a(0) = 0 + 1 + 2 + 3 + 4 = 10;
a(1) = 0 + 1 + 2 + 3 + 4 + 1 + 2 + 3 + 4 + 5 = 25;
a(2) = 0 + 1 + 2 + 3 + 4 + 1 + 2 + 3 + 4 + 5 + 2 + 3 + 4 + 5 + 6 = 45, etc.

Ilya Gutkovskiy, Sequences of the form k*(n + 1)*(n - 1 + k)/2.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000096, A008587, A028895, A040002, A045943, A054000, A055998, A067728.

[5*(n+1)*(n+4)/2: n in [0..50]]; // Vincenzo Librandi, Feb 28 2016

Table[5 (n + 1) ((n + 4)/2), {n, 0, 45}]
Table[Sum[5 (k + 2), {k, 0, n}], {n, 0, 45}]
LinearRecurrence[{3, -3, 1}, {10, 25, 45}, 46]

a(n) = 5*(n + 1)*(n + 4)/2; \\ Michel Marcus, Feb 29 2016

Vec(5*(2-x)/(1-x)^3 + O(x^100)) \\ Altug Alkan, Mar 04 2016

G.f.: 5*(2 - x)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = Sum_{k=0..n} 5*(k + 2) = Sum_{k=0..n} A008587(k + 2).
Sum_{n>=0} 1/a(n) = 11/45 = 0.24444444444... = A040002.
a(n) = 5*A000096(n+1).
a(n) = A055998(2*n+2) + A055998(n+1). - Bruno Berselli, Sep 23 2016
E.g.f.: 5*exp(x)*(4 + 6*x + x^2)/2. - Elmo R. Oliveira, Dec 24 2024

A158942 Nonsquares coprime to 10.

Original entry on oeis.org

3, 7, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 83, 87, 89, 91, 93, 97, 99, 101, 103, 107, 109, 111, 113, 117, 119, 123, 127, 129, 131, 133, 137, 139, 141, 143, 147, 149, 151, 153, 157, 159, 161, 163

Offset: 1

Eric Desbiaux, Mar 31 2009

Odd primes + odd nonprime integers that have an odd numbers of proper divisors A082686, are the result of a suppression of integers satisfying: 2n (A005843); n^2 (A000290); n^2+n (A002378). Of these, we can suppress the multiples of 5 (A008587).
Decimal expansion of 1/10^(n^2+n) + 1/10^(n^2) + 1/10^(5*n) + 1/10^(2*n) gives a 0 for these integers.
2n + n(n+1) + n^2 = 2n^2 + 3n = A014106.
2n^2 + 3n + 5n = 2n^2 + 8n = 2n(n+4) = A067728 8(8+n) is a perfect square.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Intersection of A000037 and A045572.

Cf. A082868, A002378, A000290, A005843, A008587, A014106, A067728.

Select[Range@ 163, ! IntegerQ@ Sqrt@ # && CoprimeQ[#, 10] &] (* Michael De Vlieger, Dec 11 2015 *)

isok(n) = (n % 2) && (n % 5) && (isprime(n) || (numdiv(n) % 2 == 0)); \\ Michel Marcus, Aug 27 2013

is(n)=gcd(n,10)==1 && !issquare(n) \\ Charles R Greathouse IV, Sep 05 2013

New name from Charles R Greathouse IV, Sep 05 2013

A302576 Numbers k such that k/10 + 1 is a square.

Original entry on oeis.org

-10, 0, 30, 80, 150, 240, 350, 480, 630, 800, 990, 1200, 1430, 1680, 1950, 2240, 2550, 2880, 3230, 3600, 3990, 4400, 4830, 5280, 5750, 6240, 6750, 7280, 7830, 8400, 8990, 9600, 10230, 10880, 11550, 12240, 12950, 13680, 14430, 15200, 15990, 16800, 17630, 18480, 19350, 20240

Offset: 1

Bruno Berselli, Apr 10 2018

Equivalently, numbers k such that (k + 10)*10 is a square.

The positive terms belong to the fourth column of the array in A185781.

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

Cf. A000290, A008592, A033583, A185781.
After -10, subsequence of A174133 because a(n) = ((n-1)^2-1)*(3^2+1).
Similar lists of k for which k/j + 1 is a square: A067998 (j=1), A054000 (j=2), A067725 (j=3), A134582 (j=4), A067724 (j=5), A067726 (j=6), A067727 (j=7), second bisection of A067728 (j=8), A147651 (j=9), this sequence (j=10), A067705 (j=11), second bisection of A067707 (j=12).

GAP
```
List([1..50], n -> 10*n*(n-2));
```
Julia
```
[10*n*(n-2) for n in 1:50] |> println
```
Magma
```
[10*n*(n-2): n in [1..50]];
```
Mathematica
```
Table[10 n (n - 2), {n, 1, 50}]
```
Maxima
```
makelist(10*n*(n-2), n, 1, 50);
```
PARI
```
vector(50, n, nn; 10*n*(n-2))
```
Python
```
[10*n*(n-2) for n in range(1, 50)]
```
Sage
```
[10*n*(n-2) for n in (1..50)]
```

O.g.f.: -10*x*(1 - 3*x)/(1 - x)^3.
E.g.f.: -10*x*(1 - x)*exp(x).
a(n) = a(2-n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 10*n*(n - 2) = 10*A067998(n).
a(n) = A033583(n-1) - 10. - Altug Alkan, Apr 10 2018

cp's OEIS Frontend

A067726 a(n) = 6*n^2 + 12*n.

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A212331 a(n) = 5*n*(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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A067705 a(n) = 11*n^2 + 22*n.

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A185732 Accumulation array of the polygonal number array (A086270), by antidiagonals.

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Mathematica

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A269457 a(n) = 5*(n + 1)*(n + 4)/2.

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A067726 a(n) = 6n^2 + 12n.

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

A067705 a(n) = 11n^2 + 22n.

A269457 a(n) = 5(n + 1)(n + 4)/2.