A140091 -id:A140091 - cp's OEIS frontend

A067728 a(n) = 2n^2 + 8n.

10, 24, 42, 64, 90, 120, 154, 192, 234, 280, 330, 384, 442, 504, 570, 640, 714, 792, 874, 960, 1050, 1144, 1242, 1344, 1450, 1560, 1674, 1792, 1914, 2040, 2170, 2304, 2442, 2584, 2730, 2880, 3034, 3192, 3354, 3520, 3690, 3864, 4042, 4224, 4410, 4600, 4794

Offset: 1

Robert G. Wilson v, Feb 05 2002

Positive numbers k such that 8*(8 + 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. 7: A067727, 6: A067726, 5: A067724, 3: A067725.

Cf. A000217, A005563, A140091, A140681, A212331.

[2*n*(n+4): n in [1..50]] // Vincenzo Librandi, Jul 08 2012

Select[ Range[10000], IntegerQ[ Sqrt[ 8(8 + # )]] & ]
CoefficientList[Series[2*(5-3*x)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 08 2012 *)

a(n)=2*n*(n+4) \\ Charles R Greathouse IV, Dec 07 2011

def a(n): return (2*n + 8)*n
print([a(n) for n in range(1, 48)]) # Michael S. Branicky, Oct 24 2021

a(n+1) = 2*n*n + 12*n + 10. - Frank Ellermann
a(n) = Sum_{k=0..n} Sum_{j=4..n} (j - k), n >= 4. - Zerinvary Lajos, May 11 2007
From Vincenzo Librandi, Jul 08 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 2*x*(5-3*x)/(1-x)^3. (End)
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 25/96.
Sum_{n>=1} (-1)^(n+1)/a(n) = 7/96. (End)
E.g.f.: 2*exp(x)*x*(5 + x). - Stefano Spezia, Oct 01 2023

A141419 Triangle read by rows: T(n, k) = A000217(n) - A000217(n - k) with 1 <= k <= n.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 4, 7, 9, 10, 5, 9, 12, 14, 15, 6, 11, 15, 18, 20, 21, 7, 13, 18, 22, 25, 27, 28, 8, 15, 21, 26, 30, 33, 35, 36, 9, 17, 24, 30, 35, 39, 42, 44, 45, 10, 19, 27, 34, 40, 45, 49, 52, 54, 55

Offset: 1

Roger L. Bagula, Aug 05 2008

As a rectangle, the accumulation array of A051340.
From Clark Kimberling, Feb 05 2011: (Start)
Here all the weights are divided by two where they aren't in Cahn.
As a rectangle, A141419 is in the accumulation chain
... < A051340 < A141419 < A185874 < A185875 < A185876 < ...
(See A144112 for the definition of accumulation array.)
row 1: A000027
col 1: A000217
diag (1,5,...): A000326 (pentagonal numbers)
diag (2,7,...): A005449 (second pentagonal numbers)
diag (3,9,...): A045943 (triangular matchstick numbers)
diag (4,11,...): A115067
diag (5,13,...): A140090
diag (6,15,...): A140091
diag (7,17,...): A059845
diag (8,19,...): A140672
(End)
Let N=2*n+1 and k=1,2,...,n. Let A_{N,n-1} = [0,...,0,1; 0,...,0,1,1; ...; 0,1,...,1; 1,...,1], an n X n unit-primitive matrix (see [Jeffery]). Let M_n=[A_{N,n-1}]^4. Then t(n,k)=[M_n](1,k), that is, the n-th row of the triangle is given by the first row of M_n. - _L. Edson Jeffery, Nov 20 2011
Conjecture. Let N=2*n+1 and k=1,...,n. Let A_{N,0}, A_{N,1}, ..., A_{N,n-1} be the n X n unit-primitive matrices (again see [Jeffery]) associated with N, and define the Chebyshev polynomials of the second kind by the recurrence U_0(x) = 1, U_1(x) = 2*x and U_r(x) = 2*x*U_(r-1)(x) - U_(r-2)(x)  (r>1). Define the column vectors V_(k-1) = (U_(k-1)(cos(Pi/N)), U_(k-1)(cos(3*Pi/N)), ..., U_(k-1)(cos((2*n-1)*Pi/N)))^T, where T denotes matrix transpose. Let S_N = [V_0, V_1, ..., V_(n-1)] be the n X n matrix formed by taking V_(k-1) as column k-1. Let X_N = [S_N]^T*S_N, and let [X_N](i,j) denote the entry in row i and column j of X_N, i,j in {0,...,n-1}. Then t(n,k) = [X_N](k-1,k-1), and row n of the triangle is given by the main diagonal entries of X_N. Remarks: Hence t(n,k) is the sum of squares t(n,k) = sum[m=1,...,n (U_(k-1)(cos((2*m-1)*Pi/N)))^2]. Finally, this sequence is related to A057059, since X_N = [sum_{m=1,...,n} A057059(n,m)*A_{N,m-1}] is also an integral linear combination of unit-primitive matrices from the N-th set. - L. Edson Jeffery, Jan 20 2012
Row sums: n*(n+1)*(2*n+1)/6. - L. Edson Jeffery, Jan 25 2013
n-th row = partial sums of n-th row of A004736. - Reinhard Zumkeller, Aug 04 2014
T(n,k) is the number of distinct sums made by at most k elements in {1, 2, ... n}, for 1 <= k <= n, e.g., T(6,2) = the number of distinct sums made by at most 2 elements in {1,2,3,4,5,6}. The sums range from 1, to 5+6=11. So there are 11 distinct sums. - Derek Orr, Nov 26 2014
A number n occurs in this sequence A001227(n) times, the number of odd divisors of n, see A209260. - Hartmut F. W. Hoft, Apr 14 2016
Conjecture: 2*n + 1 is composite if and only if gcd(t(n,m),m) != 1, for some m. - L. Edson Jeffery, Jan 30 2018
From Peter Munn, Aug 21 2019 in respect of the sequence read as a triangle: (Start)
A number m can be found in column k if and only if A286013(m, k) is nonzero, in which case m occurs in column k on row A286013(m, k).
The first occurrence of m is in row A212652(m) column A109814(m), which is the rightmost column in which m occurs. This occurrence determines where m appears in A209260. The last occurrence of m is in row m column 1.
Viewed as a sequence of rows, consider the subsequences (of rows) that contain every positive integer. The lexicographically latest of these subsequences consists of the rows with row numbers in A270877; this is the only one that contains its own row numbers only once.
(End)

			As a triangle:
   1,
   2,  3,
   3,  5,  6,
   4,  7,  9, 10,
   5,  9, 12, 14, 15,
   6, 11, 15, 18, 20, 21,
   7, 13, 18, 22, 25, 27, 28,
   8, 15, 21, 26, 30, 33, 35, 36,
   9, 17, 24, 30, 35, 39, 42, 44, 45,
  10, 19, 27, 34, 40, 45, 49, 52, 54, 55;
As a rectangle:
   1   2   3   4   5   6   7   8   9  10
   3   5   7   9  11  13  15  17  19  21
   6   9  12  15  18  21  24  27  30  33
  10  14  18  22  26  30  34  38  42  46
  15  20  25  30  35  40  45  50  55  60
  21  27  33  39  45  51  57  63  69  75
  28  35  42  49  56  63  70  77  84  91
  36  44  52  60  68  76  84  92 100 108
  45  54  63  72  81  90  99 108 117 126
  55  65  75  85  95 105 115 125 135 145
Since the odd divisors of 15 are 1, 3, 5 and 15, number 15 appears four times in the triangle at t(3+(5-1)/2, 5) in column 5 since 5+1 <= 2*3, t(5+(3-1)/2, 3), t(1+(15-1)/2, 2*1) in column 2 since 15+1 > 2*1, and t(15+(1-1)/2, 1). - _Hartmut F. W. Hoft_, Apr 14 2016

R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 24.
Carlton Gamer, David W. Roeder, and John J. Watkins, Trapezoidal Numbers, Mathematics Magazine 58:2 (1985), pp. 108-110.
L. E. Jeffery, Unit-primitive matrices
M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.

Cf. A000330 (row sums), A004736, A057059, A070543.
A144112, A051340, A141419, A185874, A185875, A185876 are accumulation chain related.
A141418 is a variant.
Cf. A001227, A209260. - Hartmut F. W. Hoft, Apr 14 2016
A109814, A212652, A270877, A286013 relate to where each natural number appears in this sequence.
A000027, A000217, A000326, A005449, A045943, A059845, A115067, A140090, A140091, A140672 are rows, columns or diagonals - refer to comments.

a141419 n k =  k * (2 * n - k + 1) `div` 2
a141419_row n = a141419_tabl !! (n-1)
a141419_tabl = map (scanl1 (+)) a004736_tabl
-- Reinhard Zumkeller, Aug 04 2014

a:=(n,k)->k*n-binomial(k,2): seq(seq(a(n,k),k=1..n),n=1..12); # Muniru A Asiru, Oct 14 2018

T[n_, m_] = m*(2*n - m + 1)/2; a = Table[Table[T[n, m], {m, 1, n}], {n, 1, 10}]; Flatten[a]

t(n,m) = m*(2*n - m + 1)/2.
t(n,m) = A000217(n) - A000217(n-m). - L. Edson Jeffery, Jan 16 2013
Let v = d*h with h odd be an integer factorization, then v = t(d+(h-1)/2, h) if h+1 <= 2*d, and v = t(d+(h-1)/2, 2*d) if h+1 > 2*d; see A209260. - Hartmut F. W. Hoft, Apr 14 2016
G.f.: y*(-x + y)/((-1 + x)^2*(-1 + y)^3). - Stefano Spezia, Oct 14 2018
T(n, 2) = A060747(n) for n > 1. T(n, 3) = A008585(n - 1) for n > 2. T(n, 4) = A016825(n - 2) for n > 3. T(n, 5) = A008587(n - 2) for n > 4. T(n, 6) = A016945(n - 3) for n > 5. T(n, 7) = A008589(n - 3) for n > 6. T(n, 8) = A017113(n - 4) for n > 7.r n > 5. T(n, 7) = A008589(n - 3) for n > 6. T(n, 8) = A017113(n - 4) for n > 7. T(n, 9) = A008591(n - 4) for n > 8. T(n, 10) = A017329(n - 5) for n > 9. T(n, 11) = A008593(n - 5) for n > 10.  T(n, 12) = A017593(n - 6) for n > 11. T(n, 13) = A008595(n - 6) for n > 12. T(n, 14) = A147587(n - 7) for n > 13. T(n, 15) = A008597(n - 7) for n > 14. T(n, 16) = A051062(n - 8) for n > 15. T(n, 17) = A008599(n - 8) for n > 16. - Stefano Spezia, Oct 14 2018
T(2*n-k, k) = A070543(n, k). - Peter Munn, Aug 21 2019

Simpler name by Stefano Spezia, Oct 14 2018

A140672 a(n) = n(3n + 13)/2.

Original entry on oeis.org

0, 8, 19, 33, 50, 70, 93, 119, 148, 180, 215, 253, 294, 338, 385, 435, 488, 544, 603, 665, 730, 798, 869, 943, 1020, 1100, 1183, 1269, 1358, 1450, 1545, 1643, 1744, 1848, 1955, 2065, 2178, 2294, 2413, 2535, 2660, 2788, 2919, 3053

Offset: 0

Omar E. Pol, May 22 2008

G. C. Greubel, Table of n, a(n) for n = 0..5000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 5.
Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.

[(3*n^2 + 13*n)/2 : n in [0..80]]; // Wesley Ivan Hurt, Dec 27 2023

Table[n (3 n + 13)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 8, 19}, 50] (* Harvey P. Dale, Dec 16 2011 *)

a(n)=n*(3*n+13)/2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = (3*n^2 + 13*n)/2.
a(n) = 3*n + a(n-1) + 5 for n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=8, a(2)=19; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Dec 16 2011
G.f.: x*(8 - 5*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
E.g.f.: (1/2)*(3*x^2 +16*x)*exp(x). - G. C. Greubel, Jul 17 2017

A049777 Triangular array read by rows: T(m,n) = n + n+1 + ... + m = (m+n)(m-n+1)/2.

Original entry on oeis.org

1, 3, 2, 6, 5, 3, 10, 9, 7, 4, 15, 14, 12, 9, 5, 21, 20, 18, 15, 11, 6, 28, 27, 25, 22, 18, 13, 7, 36, 35, 33, 30, 26, 21, 15, 8, 45, 44, 42, 39, 35, 30, 24, 17, 9, 55, 54, 52, 49, 45, 40, 34, 27, 19, 10, 66, 65, 63, 60, 56, 51, 45, 38, 30, 21, 11, 78, 77, 75, 72, 68, 63, 57, 50

Offset: 1

Clark Kimberling

Triangle read by rows, T(n,k) = A000217(n) - A000217(k), 0 <= k < n. - Philippe Deléham, Mar 07 2013
Subtriangle of triangle in A049780. - Philippe Deléham, Mar 07 2013
No primes and all composite numbers (except 2^x) are generated after the first two columns of the square array for this sequence. In other words, no primes and all composites except 2^x are generated when m-n >= 2. - Bob Selcoe, Jun 18 2013
Diagonal sums in the square array equal partial sums of squares (A000330). - Bob Selcoe, Feb 14 2014
From Bob Selcoe, Oct 27 2014: (Start)
The following apply to the triangle as a square array read by rows unless otherwise specified (see Table link);
Conjecture: There is at least one prime in interval [T(n,k), T(n,k+1)]. Since T(n,k+1)/T(n,k) decreases to (k+1)/k as n increases, this is true for k=1 ("Bertrand's Postulate", first proved by P. Chebyshev), k=2 (proved by El Bachraoui) and k=3 (proved by Loo).
Starting with T(1,1), The falling diagonal of the first 2 numbers in each column (read by column) are the generalized pentagonal numbers (A001318). That is, the coefficients of T(1,1), T(2,1), T(2,2), T(3,2), T(3,3), T(4,3), T(4,4) etc. are the generalized pentagonal numbers. These are A000326 and A005449 (Pentagonal and Second pentagonal numbers: n*(3*n+1)/2, respectively), interweaved.
Let D(n,k) denote falling diagonals starting with T(n,k):
Treating n as constant: pentagonal numbers of the form n*k + 3*k*(k-1)/2 are D(n,1); sequences A000326, 005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542 are formed by n = 1 through 12, respectively.
Treating k as constant: D(1,k) are (3*n^2 + (4k-5)*n + (k-1)*(k-2))/2. When k = 2(mod3), D(1,k), is same as D(k+1,1) omitting the first (k-2)/3 numbers in the sequences. So D(1,2) is same as D(3,1); D(1,5) is same as D(6,1) omitting the 6; D(1,8) is same as D(9,1) omitting the 9 and 21; etc.
D(1,3) and D(1,4) are sequences A095794 and A140229, respectively.
(End)

			Rows: {1}; {3,2}; {6,5,3}; ...
Triangle begins:
   1;
   3,  2;
   6,  5,  3;
  10,  9,  7,  4;
  15, 14, 12,  9,  5;
  21, 20, 18, 15, 11,  6;
  28, 27, 25, 22, 18, 13,  7;
  36, 35, 33, 30, 26, 21, 15,  8;
  45, 44, 42, 39, 35, 30, 24, 17,  9;
  55, 54, 52, 49, 45, 40, 34, 27, 19, 10; ...

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
M. El Bachraoui, Primes in the interval [2n,3n], Int. J. Contemp. Math. Sciences 1:13 (2006), pp. 617-621.
A. Loo, On the primes in the interval [3n,4n], Int. J. Contemp. Math. Sciences 6 (2011), no. 38, 1871-1882.
S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv, arXiv:1212.2785 [math.NT], 2012.

Row sums = A000330.
Cf. A001318 (generalized pentagonal numbers).
Cf. A000217, A002260, A049780, A094728, A095794, A140229.
Cf. A000326, 005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542 (pentagonal numbers of form n*k + 3*k*(k-1)/2).

/* As triangle */ [[(m+n)*(m-n+1) div 2: n in [1..m]]: m in [1.. 15]]; // Vincenzo Librandi, Oct 27 2014

Flatten[Table[(n+k) (n-k+1)/2,{n,15},{k,n}]] (* Harvey P. Dale, Feb 27 2012 *)

{T(n,k) = if( k<1 || nMichael Somos, Oct 06 2007 */

Partial sums of A002260 row terms, starting from the right; e.g., row 3 of A002260 = (1, 2, 3), giving (6, 5, 3). - Gary W. Adamson, Oct 23 2007
Sum_{k=0..n-1} (-1)^k*(2*k+1)*A000203(T(n,k)) = (-1)^(n-1)*A000330(n). - Philippe Deléham, Mar 07 2013
Read as a square array: T(n,k) = k*(k+2n-1)/2. - Bob Selcoe, Oct 27 2014

A140675 a(n) = n(3n + 19)/2.

Original entry on oeis.org

0, 11, 25, 42, 62, 85, 111, 140, 172, 207, 245, 286, 330, 377, 427, 480, 536, 595, 657, 722, 790, 861, 935, 1012, 1092, 1175, 1261, 1350, 1442, 1537, 1635, 1736, 1840, 1947, 2057, 2170, 2286, 2405, 2527, 2652, 2780, 2911, 3045, 3182

Offset: 0

Omar E. Pol, May 22 2008

G. C. Greubel, Table of n, a(n) for n = 0..5000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 5.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.

Table[(n(3n+19))/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,11,25},50] (* Harvey P. Dale, Apr 26 2018 *)

a(n)=n*(3*n+19)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (3*n^2 + 19*n)/2.
a(n) = 3*n + a(n-1) + 8 for n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: x*(11 - 8*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
E.g.f.: (1/2)*(3*x^2 + 22*x)*exp(x). - G. C. Greubel, Jul 17 2017

A151542 Generalized pentagonal numbers: a(n) = 12n + 3n*(n-1)/2.

Original entry on oeis.org

0, 12, 27, 45, 66, 90, 117, 147, 180, 216, 255, 297, 342, 390, 441, 495, 552, 612, 675, 741, 810, 882, 957, 1035, 1116, 1200, 1287, 1377, 1470, 1566, 1665, 1767, 1872, 1980, 2091, 2205, 2322, 2442, 2565, 2691, 2820, 2952, 3087, 3225, 3366, 3510, 3657, 3807, 3960

Offset: 0

N. J. A. Sloane, May 15 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 5.
Leo Tavares, Illustration: Star Triangles
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

The generalized pentagonal numbers b*n + 3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.

Cf. A003154, A060544.

s=0;lst={};Do[AppendTo[lst,s+=n],{n,12,6!,3}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 05 2010 *)
LinearRecurrence[{3,-3,1}, {0,12,27}, 50] (* or *) With[{nn = 50}, CoefficientList[Series[(3/2)*(8*x + x^2)*Exp[x], {x, 0, nn}], x] Range[0, nn]!] (* G. C. Greubel, May 26 2017 *)

x='x+O('x^50); concat([0], Vec(serlaplace((3/2)*(8*x + x^2)*exp(x)))) \\ G. C. Greubel, May 26 2017

a(n)=(3*n^2+21*n)/2 \\ Charles R Greathouse IV, Jun 16 2017

a(n) = a(n-1) + 3*n + 9 (with a(0)=0). - Vincenzo Librandi, Nov 26 2010
G.f.: 3*x*(4 - 3*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
From G. C. Greubel, May 26 2017: (Start)
a(n) = 3*n*(n+7)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (3/2)*(8*x + x^2)*exp(x). (End)
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 121/490.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/21 - 319/4410. (End)
a(n) = A003154(n+1) - A060544(n). - Leo Tavares, Mar 26 2022

A140673 a(n) = 3n(n + 5)/2.

Original entry on oeis.org

0, 9, 21, 36, 54, 75, 99, 126, 156, 189, 225, 264, 306, 351, 399, 450, 504, 561, 621, 684, 750, 819, 891, 966, 1044, 1125, 1209, 1296, 1386, 1479, 1575, 1674, 1776, 1881, 1989, 2100, 2214, 2331, 2451, 2574, 2700, 2829, 2961, 3096

Offset: 0

Omar E. Pol, May 22 2008

a(n) equals the number of vertices of the A256666(n)-th graph (see Illustration of initial terms in A256666 Links). - Ivan N. Ianakiev, Apr 20 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 5.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A055998.

The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.

Table[Sum[i + n - 3, {i, 6, n}], {n, 5, 52}] (* Zerinvary Lajos, Jul 11 2009 *)
Table[3 n (n + 5)/2, {n, 0, 50}] (* Bruno Berselli, Sep 05 2018 *)
LinearRecurrence[{3,-3,1},{0,9,21},50] (* Harvey P. Dale, Jul 20 2023 *)

concat(0, Vec(3*x*(3 - 2*x)/(1 - x)^3 + O(x^100))) \\ Michel Marcus, Apr 20 2015

a(n) = 3*n*(n+5)/2; \\ Altug Alkan, Sep 05 2018

a(n) = A055998(n)*3 = (3*n^2 + 15*n)/2 = n*(3*n + 15)/2.
a(n) = 3*n + a(n-1) + 6 for n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: 3*x*(3 - 2*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
E.g.f.: (1/2)*(3*x^2 + 18*x)*exp(x). - G. C. Greubel, Jul 17 2017
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 137/450.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/15 - 47/450. (End)

A140674 a(n) = n(3n + 17)/2.

Original entry on oeis.org

0, 10, 23, 39, 58, 80, 105, 133, 164, 198, 235, 275, 318, 364, 413, 465, 520, 578, 639, 703, 770, 840, 913, 989, 1068, 1150, 1235, 1323, 1414, 1508, 1605, 1705, 1808, 1914, 2023, 2135, 2250, 2368, 2489, 2613, 2740, 2870, 3003, 3139

Offset: 0

Omar E. Pol, May 22 2008

G. C. Greubel, Table of n, a(n) for n = 0..5000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 5.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.

[n*(3*n + 17)/2: n in [0..70]]; // Wesley Ivan Hurt, Apr 21 2021

Table[(3*n^2 + 17*n)/2, {n, 0, 50}] (* G. C. Greubel, Jul 17 2017 *)

a(n)=n*(3*n+17)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (3*n^2 + 17*n)/2.
a(n) = 7*n + 3*A000217(n). - Reinhard Zumkeller, May 28 2008
a(n) = 3*n + a(n-1) + 7 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
G.f.: x*(10 - 7*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
E.g.f.: (1/2)*(3*x^2 + 20*x)*exp(x). - G. C. Greubel, Jul 17 2017

A140681 a(n) = 3n(n+6).

Original entry on oeis.org

0, 21, 48, 81, 120, 165, 216, 273, 336, 405, 480, 561, 648, 741, 840, 945, 1056, 1173, 1296, 1425, 1560, 1701, 1848, 2001, 2160, 2325, 2496, 2673, 2856, 3045, 3240, 3441, 3648, 3861, 4080, 4305, 4536, 4773, 5016, 5265, 5520, 5781

Offset: 0

Omar E. Pol, May 22 2008

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A028560, A033428, A045944, A140676, A067725, A140677, A140678, A067707, A140679, A140680, A140689, A000217, A005563, A067728, A140091, A212331.

A140681:=n->3*n*(n+6); seq(A140681(n), n=0..100); # Wesley Ivan Hurt, Dec 10 2013

Table[3n(n+6), {n,0,100}] (* Wesley Ivan Hurt, Dec 10 2013 *)
LinearRecurrence[{3,-3,1},{0,21,48},50] (* Harvey P. Dale, May 03 2023 *)

a(n)=3*n*(n+6) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A028560(n)*3 = 3*n^2 + 18*n = n*(3*n+18).
a(n) = 6*n + a(n-1) + 15 with n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
from G. C. Greubel, Jul 20 2017: (Start)
G.f.: 3*x*(7 - 5*x)/(1-x)^3.
E.g.f.: 3*x*(x+7)*exp(x). (End)
From Amiram Eldar, Feb 26 2022: (Start)
Sum_{n>=1} 1/a(n) = 49/360.
Sum_{n>=1} (-1)^(n+1)/a(n) = 37/1080. (End)

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

cp's OEIS Frontend

A067728 a(n) = 2*n^2 + 8*n.

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A141419 Triangle read by rows: T(n, k) = A000217(n) - A000217(n - k) with 1 <= k <= n.

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A140672 a(n) = n*(3*n + 13)/2.

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A049777 Triangular array read by rows: T(m,n) = n + n+1 + ... + m = (m+n)(m-n+1)/2.

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A140675 a(n) = n*(3*n + 19)/2.

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A151542 Generalized pentagonal numbers: a(n) = 12*n + 3*n*(n-1)/2.

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A140673 a(n) = 3*n*(n + 5)/2.

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A067728 a(n) = 2n^2 + 8n.

A140672 a(n) = n(3n + 13)/2.

A140675 a(n) = n(3n + 19)/2.

A151542 Generalized pentagonal numbers: a(n) = 12n + 3n*(n-1)/2.

A140673 a(n) = 3n(n + 5)/2.

A140674 a(n) = n(3n + 17)/2.

A140681 a(n) = 3n(n+6).

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