A028896 -id:A028896 - cp's OEIS frontend

A227776 a(n) = 6*n^2 + 1.

1, 7, 25, 55, 97, 151, 217, 295, 385, 487, 601, 727, 865, 1015, 1177, 1351, 1537, 1735, 1945, 2167, 2401, 2647, 2905, 3175, 3457, 3751, 4057, 4375, 4705, 5047, 5401, 5767, 6145, 6535, 6937, 7351, 7777, 8215, 8665, 9127, 9601, 10087, 10585, 11095, 11617, 12151

Offset: 0

Clark Kimberling, Jul 30 2013

Least splitter is defined for x < y at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y. Conjecture: a(n) is the least splitter of s(n) and s(n+1), where s(n) = n*sin(1/n).

			The first eight least splitting rationals for {n*sin(1/n), n >=1 } are these fractions: 6/7, 24/25, 54/55, 96/97, 150/151, 216/217, 294/295, 384/385.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Leo Tavares, Illustration: Hexagonal Star Rays.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A227631, A287326.

Cf. A003154, A008458, A003215, A000217, A028896, A054554, A046092, A080855, A045943, A172043, A002378, A033581.

z = 40; r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; s[n_] := s[n] = n*Sin[1/n]; t = Table[r[s[n], s[n + 1]], {n, 1, z}] (* least splitting rationals *); fd = Denominator[t] (* Peter J. C. Moses, Jul 15 2013 *)
Array[6 #^2 + 1 &, 45] (* Michael De Vlieger, Nov 08 2017 *)
LinearRecurrence[{3,-3,1},{7,25,55},50] (* Harvey P. Dale, Dec 16 2017 *)

a(n)=6*n^2+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (1 + 4*x + 7*x^2)/(1 - x)^3.
a(n) = A287326(2n, n). - Kolosov Petro, Nov 06 2017
From Amiram Eldar, Jul 15 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(6))*coth(Pi/sqrt(6)))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(6))*csch(Pi/sqrt(6)))/2. (End)
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(6))*sinh(Pi/sqrt(3)).
Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(6))*csch(Pi/sqrt(6)).(End)
From Leo Tavares, Nov 20 2021: (Start)
a(n) = A003154(n+1) - A008458(n). See Hexagonal Star Rays illustration.
a(n) = A003215(n) + A028896(n-1).
a(n) = A054554(n+1) + A046092(n).
a(n) = A080855(n) + A045943(n).
a(n) = A172043(n) + A002378(n).
a(n) = A033581(n) + 1. (End)
E.g.f.: exp(x)*(1 + 6*x + 6*x^2). - Stefano Spezia, Sep 14 2024

a(0) = 1 prepended by Robert P. P. McKone, Oct 09 2023

A262221 a(n) = 25n(n + 1)/2 + 1.

Original entry on oeis.org

1, 26, 76, 151, 251, 376, 526, 701, 901, 1126, 1376, 1651, 1951, 2276, 2626, 3001, 3401, 3826, 4276, 4751, 5251, 5776, 6326, 6901, 7501, 8126, 8776, 9451, 10151, 10876, 11626, 12401, 13201, 14026, 14876, 15751, 16651, 17576, 18526, 19501, 20501, 21526, 22576, 23651

Offset: 0

Bruno Berselli, Sep 15 2015

Also centered 25-gonal (or icosipentagonal) numbers.
This is the case k=25 of the formula (k*n*(n+1) - (-1)^k + 1)/2. See table in Links section for similar sequences.
For k=2*n, the formula shown above gives A011379.
Primes in sequence: 151, 251, 701, 1951, 3001, 4751, 10151, 12401, ...

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 51 (23rd row of the table).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Table of numbers of the form (k*n*(n+1)-(-1)^k+1)/2.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A008607 (first differences), A011379, A034856, A054254, A080956, A123296, A186349, A255184.
Cf. centered polygonal numbers listed in A069190.
Similar sequences of the form (k*n*(n+1) - (-1)^k + 1)/2 with -1 <= k <= 26: A000004, A000124, A002378, A005448, A005891, A028896, A033996, A035008, A046092, A049598, A060544, A064200, A069099, A069125, A069126, A069128, A069130, A069132, A069174, A069178, A080956, A124080, A163756, A163758, A163761, A164136, A173307.

Magma
```
[25*n*(n+1)/2+1: n in [0..50]];
```

Table[25 n (n + 1)/2 + 1, {n, 0, 50}]
25*Accumulate[Range[0,50]]+1 (* or *) LinearRecurrence[{3,-3,1},{1,26,76},50] (* Harvey P. Dale, Jan 29 2023 *)

PARI
```
vector(50, n, n--; 25*n*(n+1)/2+1)
```
Sage
```
[25*n*(n+1)/2+1 for n in (0..50)]
```

G.f.: (1 + 23*x + x^2)/(1 - x)^3.
a(n) = a(-n-1) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A123296(n) + 1.
a(n) = A000217(5*n+2) - 2.
a(n) = A034856(5*n+1).
a(n) = A186349(10*n+1).
a(n) = A054254(5*n+2) with n>0, a(0)=1.
a(n) = A000217(n+1) + 23*A000217(n) + A000217(n-1) with A000217(-1)=0.
Sum_{i>=0} 1/a(i) = 1.078209111... = 2*Pi*tan(Pi*sqrt(17)/10)/(5*sqrt(17)).
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=0} a(n)/n! = 77*e/2.
Sum_{n>=0} (-1)^(n+1) * a(n)/n! = 23/(2*e). (End)
E.g.f.: exp(x)*(2 + 50*x + 25*x^2)/2. - Elmo R. Oliveira, Dec 24 2024

A307011 First coordinate in a redundant hexagonal coordinate system of the points of a counterclockwise spiral on an hexagonal grid. Second and third coordinates are given in A307012 and A345978.

Original entry on oeis.org

0, 1, 0, -1, -1, 0, 1, 2, 2, 1, 0, -1, -2, -2, -2, -1, 0, 1, 2, 3, 3, 3, 2, 1, 0, -1, -2, -3, -3, -3, -3, -2, -1, 0, 1, 2, 3, 4, 4, 4, 4, 3, 2, 1, 0, -1, -2, -3, -4, -4, -4, -4, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5

Offset: 0

Hugo Pfoertner, Mar 19 2019

From Peter Munn, Jul 22 2021: (Start)
The points of the spiral are equally the points of a hexagonal lattice, the points of an isometric (triangular) grid and the center points of the cells of a honeycomb (regular hexagonal tiling or grid). The coordinate system can be described using 3 axes that pass through spiral point 0 and one of points 1, 2 or 3. Along each axis, one of the coordinates is 0.
a(n) is the signed distance from spiral point n to the axis that passes through point 2. The distance is measured along either of the lines through point n that are parallel to one of the other 2 axes and the sign is such that point 1 has positive distance.
This coordinate can be paired with either of the other coordinates to form oblique coordinates as described in A307012. Alternatively, all 3 coordinates can be used together, symmetrically, as described in A345978.
There is a negated variant of the 3rd coordinate, which is the conventional sense of this coordinate for specifying (with the 2nd coordinate) the Eisenstein integers that can be the points of the spiral when it is embedded in the complex plane. See A307013.
(End)

Hugo Pfoertner, Table of n, a(n) for n = 0..10034
Margherita Barile, Oblique Coordinates, entry in Eric Weisstein's World of Mathematics.
HandWiki, Hexagonal Lattice.
Peter Munn, Illustration of signed distance of spiral points.
Hugo Pfoertner, Illustration of A307012 vs A307011, spiral.
Hugo Pfoertner, Illustration of A345978 vs A307011, spiral.
Wikipedia, Signed distance function.

Cf. A307012, A307013, A307014, A307016, A307017, A328818, A345978.
Numbers on the spokes of the spiral: A000567, A028896, A033428, A045944, A049450, A049451.
Positions on the spiral that correspond to Eisenstein primes: A345435.

r=-1;d=-1;print1(m=0,", ");for(k=0,8,for(j=1,r,print1(s,", "));if(k%2,,m++;r++);for(j=-m,m+1,if(d*j>=-m,print1(s=d*j,", ")));d=-d)

Name revised by Peter Munn, Jul 08 2021

A096948 Triangular table read by rows: T(n,m) = number of rectangles found in an n X m rectangle built from 1 X 1 squares, 1 <= m <= n.

Original entry on oeis.org

1, 3, 9, 6, 18, 36, 10, 30, 60, 100, 15, 45, 90, 150, 225, 21, 63, 126, 210, 315, 441, 28, 84, 168, 280, 420, 588, 784, 36, 108, 216, 360, 540, 756, 1008, 1296, 45, 135, 270, 450, 675, 945, 1260, 1620, 2025, 55, 165, 330, 550, 825, 1155, 1540, 1980, 2475, 3025

Offset: 1

Wolfdieter Lang, Jul 16 2004

Table of products of triangular numbers A000217.

Because of symmetry it is sufficient to consider n X m rectangles with n >= m. A square is a special rectangle.

			T(2,2) = 9 because in a 2 X 2 square there are four 1 X 1 squares, two 1 X 2 rectangles, two 2 X 1 rectangles and one 2 X 2 square: 4 + 2 + 2 + 1 =9.
T(3,2) = 18 = t(3)*t(2) because in a 3 X 2 rectangle there are six 1 X 1 squares, three 1 X 2 rectangles, four 2 X 1 rectangles, two 3 X 1 rectangles, two 2 X 2 squares and one 3 X 2 rectangle: 6 + 3 + 4 + 2 + 2 + 1 = 9 + 9 = 18.
Triangle begins:
   1,
   3,   9,
   6,  18,  36,
  10,  30,  60, 100,
  15,  45,  90, 150, 225,
  21,  63, 126, 210, 315, 441,
  28,  84, 168, 280, 420, 588, 784,
  36, 108, 216, 360, 540, 756,1008,1296,
  45, 135, 270, 450, 675, 945,1260,1620,2025,
  55, 165, 330, 550, 825,1155,1540,1980,2475,3025,
  (...)

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of triangle, flattened).
Steve Chow, Math for fun, how many rectangles?, blackpenredpen on YouTube, Apr 14 2018.
Wolfdieter Lang, First 10 rows.

Cf. A000217 (1st column), A045943 (2nd column), A028896 (3rd column), A124080 (4th col).

Table[n*(n+1)*m*(m+1)/4, {n, 10}, {m, n}] (* Paolo Xausa, Feb 25 2025 *)

T(n,m)=if(m>n,0,n*(n+1)*m*(m+1)/4) \\ Charles R Greathouse IV, Dec 14 2015

T(n, m) = t(n)*t(m) if n>=m else 0, with the triangular numbers t(n):= A000217(n), n>=1.

G.f. for column m (without leading zeros): t(m)*(x/(1-x)^3 - Sum_{k=0..m-1} t(k)*x^k)/x^m, m>=1.

Name edited by M. F. Hasler, Oct 22 2020

A320047 Consider coefficients U(m,l,k) defined by the identity Sum_{k=1..l} Sum_{j=0..m} A302971(m,j)/A304042(m,j) * k^j * (T-k)^j = Sum_{k=0..m} (-1)^(m-k) * U(m,l,k) * T^k that holds for all positive integers l,m,T. This sequence gives 2-column table read by rows, where n-th row lists coefficients U(1,n,k) for k = 0, 1 and n >= 1.

Original entry on oeis.org

5, 6, 28, 18, 81, 36, 176, 60, 325, 90, 540, 126, 833, 168, 1216, 216, 1701, 270, 2300, 330, 3025, 396, 3888, 468, 4901, 546, 6076, 630, 7425, 720, 8960, 816, 10693, 918, 12636, 1026, 14801, 1140, 17200, 1260, 19845, 1386, 22748, 1518, 25921, 1656

Offset: 1

Kolosov Petro, Oct 04 2018

For l=T, the identity takes the form T^(2m+1) = Sum_{k=0..m} (-1)^(m-k)*U(m,T,k)*T^k, which holds for all positive integers T and m.

			        column  column
   l      k=0     k=1
  ---   ------  ------
   1       5       6
   2      28      18
   3      81      36
   4     176      60
   5     325      90
   6     540     126
   7     833     168
   8    1216     216
   9    1701     270
  10    2300     330
  11    3025     396
  12    3888     468
  ...

Max Alekseyev, Derivation of the general formula for U(m,n,k), MathOverflow, 2018.
Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.
Petro Kolosov, More details on derivation of present sequence.
Petro Kolosov, Mathematica program, verifies the identity T^(2m+1) = Sum_{k=0..m} (-1)^(m-k)*U(m,T,k)*T^k for m=0,1,...,12.
Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.
Petro Kolosov, An efficient method of spline approximation for power function, arXiv:2503.07618 [math.GM], 2025.

The case m=2 is A316349.
The case m=3 is A316387.
Column k=0 is A275709.
Column k=1 is A028896.
Cf. A302971, A304042, A287326, A300656, A300785.

(* Define the R[n,k] := A302971(n,k)/A304042(n,k) *)
R[n_, k_] := 0
R[n_, k_] := (2 k + 1)*Binomial[2 k, k]*
   Sum[R[n, j]*Binomial[j, 2 k + 1]*(-1)^(j - 1)/(j - k)*
   BernoulliB[2 j - 2 k], {j, 2 k + 1, n}] /; 2 k + 1 <= n
R[n_, k_] := (2 n + 1)*Binomial[2 n, n] /; k == n;
(* Define the U(m,l,t) coefficients *)
U[m_, l_, t_] := (-1)^m Sum[Sum[Binomial[j, t] R[m,j] k^(2 j - t) (-1)^j, {j, t, m}], {k, 1, l}];
(* Define the value of the variable 'm' to be m = 1 for A320047 *)
m = 1;
(* Print first 10 rows of U(m,l,t) coefficients for 'm' defined above *)
Column[Table[U[m, l, t], {l, 1, 10}, {t, 0, m}]]

U(m,l,t) = (-1)^m * Sum_{k=1..l} Sum_{j=t..m} binomial(j,t) * R(m,j) * k^{2j-t} * (-1)^j, where m = 1, l >= 1 and R(m,j) = A302971(m,j)/A304042(m,j); after Max Alekseyev, see links.
Conjectures from Colin Barker, Aug 03 2019: (Start)
G.f.: x*(5 + 6*x + 8*x^2 - 6*x^3 - x^4) / ((1 - x)^4*(1 + x)^4).
a(n) = (4 - 4*(-1)^n - 3*(-5+(-1)^n)*n - 3*(-3+(-1)^n)*n^2 + (1+(-1)^(1+n))*n^3) / 8.
a(n) = 4*a(n-2) - 6*a(n-4) + 4*a(n-6) - a(n-8) for n>8.
(End)

A346721 Use the cells of a hexagonal grid to represent the algebraic integers in the integer ring of Q(sqrt(-7)), as explained in the comments. Number the cells along the counterclockwise hexagonal spiral that starts with cells 0 and 1 representing integers 0 and 1. List the cells that represent 0 or a prime in the ring.

Original entry on oeis.org

0, 2, 3, 5, 6, 11, 17, 19, 21, 23, 28, 30, 32, 45, 47, 57, 59, 61, 63, 65, 67, 69, 76, 78, 80, 82, 84, 103, 107, 121, 125, 127, 129, 131, 135, 137, 139, 148, 150, 152, 156, 158, 160, 187, 189, 211, 213

Offset: 1

Peter Munn, Jul 30 2021

In this entry we use "rational integers" to refer to integers in their usual sense as whole numbers - they form a subset of the algebraic integers that form the ring, which we denote "R".
The algebraic integers in R (the elements of R) are specifically quadratic integers of the form z = x + y*sqrt(-7) or z = (x+0.5) + (y+0.5)*sqrt(-7) where x and y are rational integers. Plotted as points on a plane, they can be joined in a grid of isosceles triangles or be seen as the center points of hexagonal regions. When the latter are adjusted to make them regular, it makes for appealing diagrams, which we will come to shortly.
(To be precise, we map each element, z, to the region of the complex plane containing the points that have z as their nearest ring element, then map these (hexagonal) regions continuously to the cells of a (regular) hexagonal grid.)
R is one of 9 related rings that are unique factorization domains, meaning their elements factorize into prime elements in a unique way, just as with rational integers and prime numbers. See the Wikipedia link or the Stark reference, for example.
This set of sequences is inspired by tilings: see the Wichmann link. Each tiling represents one of the 9 rings and shows the primes as distinctively colored squares or hexagons as appropriate.
General properties of the related hexagonal spiral sequences: (Start)
R is one of 7 rings where hexagons are appropriate. Each has elements of the form x + y*sqrt(-p) and (x+0.5) + (y+0.5)*sqrt(-p), where p is a (rational) prime congruent to 3 modulo 4.
When mapping the grid cells to quadratic integers, it is often convenient to write the latter as a + w*b, where w = 0.5*(1+sqrt(-p)). Cell m on the spiral represents A307011(m) + w*A307012(m).
We can find the primes without advanced mathematics, using multiplication formulas and a sieve as explained below.
w^2 = w - c, where c = (p+1)/4 (which is an integer as p == 3 (mod 4)). So, in general, the product of a_1 + w*b_1 and a_2 + w*b_2 is (a_1*a_2 - c*b_1*b_2) + w*(a_1*b_2 + a_2*b_1 + b_1*b_2). The norm (absolute square) of a + w*b is a^2 + a*b + c*b^2.
For k >= 1, the algebraic integers represented by cells numbered 3k*(k-1)+1 to 3k*(k+1) on the spiral (cells A003215(k-1) to A028896(k)) are positioned along a hexagon in the complex plane; they include rational integers k and -k, and have norms in the range [k^2*(4c-1)/4c, k^2*c] = [k^2*p/(p+1), k^2*c].
To determine the primes we may list the ring elements in an order such that they have nondecreasing norm, and use a sieve to remove the products of nonunits. So, we are only interested in elements with norm greater than 1 (i.e. nonzero, nonunit). At each round of sieving we note the first element, z, whose products we have not yet removed, and remove in turn the product of z and each element from z onwards in the list.
(End)

			Table showing derivation of initial terms.
The ring element, z, represented by spiral cell m is shown in the form A307011(m) + A307012(m)*w, where w = 0.5*(1+sqrt(-7)).
The column headed "(x,y)" gives x and y when z is written in the form z = x + y*sqrt(-7).
      A307011(m)
       |  A307012(m)
   m   |   |      z      (x,y)      status         n  a(n)=m
       |   |
   0   0   0      0   ( 0.0, 0.0)     zero         1    0
   1   1   0      1   ( 1.0, 0.0)     unit
   2   0   1      w   ( 0.5, 0.5)    prime         2    2
   3  -1   1   -1+w   (-0.5, 0.5)    prime         3    3
   4  -1   0     -1   (-1.0, 0.0)     unit
   5   0  -1     -w   (-0.5,-0.5)    prime         4    5
   6   1  -1    1-w   ( 0.5,-0.5)    prime         5    6
   7   2  -1    2-w   ( 1.5,-0.5)  =    -w*w
   8   2   0      2   ( 2.0, 0.0)  = (1-w)*w
   9   1   1    1+w   ( 1.5, 0.5)  = (1-w)*(w-1)
  10   0   2     2w   ( 1.0, 1.0)  =     2*w
  11  -1   2  -1+2w   ( 0.0, 1.0)    prime         6   11

L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910.
H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970; Theorem 8.22 on page 295 lists the nine UFDs of the form Q(sqrt(-d)), cf. A003173.

OEIS Wiki, Algebraic integers.
OEIS Wiki, Norm.
Eric Weisstein's World of Mathematics, Absolute Square, Complex Plane, Hexagonal Grid, Ring of Integers.
Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019. See Figure 5.
Wikipedia, Quadratic integer.
Wikipedia, Unique factorization domain.

Cf. A003173.
Norms of primes in R: A090348.
Sequences related to the geometry of the spiral: A003215, A028896, A307011, A307012, A345764.
Equivalent sequences for other Q(sqrt(D)): A345436 (D=-1), A345437 (D=-2), A345435 (D=-3), A346722 (D=-11), A346723 (D=-19), A346724 (D=-43), A346725 (D=-67), A346726 (D=-163).

m is a term if and only if A345764(m) is a term.

A292314 Numbers equal to the sum of three oblong numbers in arithmetic progression.

Original entry on oeis.org

18, 126, 168, 216, 468, 918, 1026, 1140, 1260, 1518, 1950, 2106, 2268, 2790, 3168, 3996, 4218, 5418, 5676, 5940, 6210, 6768, 7056, 7650, 8268, 8910, 9240, 9576, 9918, 10266, 10620, 11346, 11718, 13668, 14076, 15336, 15768, 16650, 17556, 18018, 18486, 18960, 20418, 21420, 22446

Offset: 1

Antonio Roldán, Sep 14 2017

nonn

Subsequence of A028896.

			126 = 3*4 + 6*7 + 8*9 = 12 + 42 + 72, with 72 - 42 = 42 - 12 = 30;
468 = 8*9 + 12*13 + 15*16 = 72 + 156 + 240, with 240 - 156 = 156 - 72 = 84.

Cf. A292309, A292310, A292313, A292316.

o[n_] := n(n+1); s[x_] := Reduce[ x+k == o[y] && x-k == o[z] && k>0 && z>0, {z, y, k}, Integers]; 3 Select[o@ Range@ 93, s[#] =!= False &] (* Giovanni Resta, Sep 18 2017 *)

t=2; k=2; while(t<=10^4, i=k; e=0; v=t+i; while(i>2&&e==0, if(issquare(4*v+1), m=3*t; e=1; print1(m,", ")); i+=-2; v+=i); k+=2; t+=k)

a(n) = 3*A292316(n).

A004538 a(n) = 3n^2 + 3n - 1.

Original entry on oeis.org

-1, 5, 17, 35, 59, 89, 125, 167, 215, 269, 329, 395, 467, 545, 629, 719, 815, 917, 1025, 1139, 1259, 1385, 1517, 1655, 1799, 1949, 2105, 2267, 2435, 2609, 2789, 2975, 3167, 3365, 3569, 3779, 3995, 4217, 4445

Offset: 0

N. J. A. Sloane, Eric T. Lane (ERICLANE(AT)UTCVM.UTC.EDU)

Numbers k such that (4*k + 7)/3 is a square. - Bruno Berselli, Sep 11 2018

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

First differences of A033445.

Cf. A028896, A003215, A077588, A000538, A000330.

[3*n^2 + 3*n -1: n in [0..50]]; // G. C. Greubel, Sep 10 2018

Table[5*Sum[k^4,{k,1,n}]/Sum[k^2,{k,1,n}], {n,1,20}] (* Alexander Adamchuk, Apr 12 2006 *)
Table[3n^2+3n-1,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{-1,5,17},40] (* Harvey P. Dale, Jan 18 2019 *)

a(n)=3*n^2+3*n-1 \\ Charles R Greathouse IV, Jun 17 2017

From Alexander Adamchuk, Apr 12 2006: (Start)
a(n) = 5 * Sum_{k=1..n} k^4 / Sum_{k=1..n} k^2, n > 0.
a(n) = 5 * A000538(n) / A000330(n), n > 0. (End)
a(n) = a(n-1) + 6*n with a(0)=-1. - Vincenzo Librandi, Nov 18 2010
From G. C. Greubel, Sep 10 2018: (Start)
G.f.: (-1 + 8*x - x^2)/(1 - x)^3.
E.g.f.: (-1 + 6*x + 3*x^2)*exp(x). (End)
Sum_{n>=0} 1/a(n) = ( psi(1/2+sqrt(21)/6) - psi(1/2-sqrt(21)/6)) /sqrt(21) = -0.6286929... R. J. Mathar, Apr 24 2024

A121757 Triangle read by rows: multiply Pascal's triangle by 1,2,6,24,120,720,... = A000142.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 6, 18, 24, 1, 8, 36, 96, 120, 1, 10, 60, 240, 600, 720, 1, 12, 90, 480, 1800, 4320, 5040, 1, 14, 126, 840, 4200, 15120, 35280, 40320, 1, 16, 168, 1344, 8400, 40320, 141120, 322560, 362880, 1, 18, 216, 2016, 15120, 90720, 423360, 1451520

Offset: 0

Alford Arnold, Aug 19 2006

Row sums are 1,3,11,49,261,1631,... = A001339

a(n,k) = D(n+1,k+1) Array D in A253938 is part of a conjectured formula for F(n,p,r) that relates Dyck path peaks and returns. a(n,k) was discovered prior to array D. - Roger Ford, May 19 2016

			Row 6 is 1*1 5*2 10*6 10*24 5*120 1*720.
From _Vincenzo Librandi_, Dec 16 2012: (Start)
Triangle begins:
1,
1, 2,
1, 4,  6,
1, 6,  18,  24,
1, 8,  36,  96,   120,
1, 10, 60,  240,  600,  720,
1, 12, 90,  480,  1800, 4320,  5040,
1, 14, 126, 840,  4200, 15120, 35280,  40320,
1, 16, 168, 1344, 8400, 40320, 141120, 322560, 362880 etc.
(End)

Vincenzo Librandi, Rows n = 0..100, flattened
J. Goldman, J. Haglund, Generalized rook polynomials, J. Combin. Theory A91 (2000), 509-530, 2-rook coefficients of k rooks on the 2xn board (all heights 2).
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007526 A000522, A005843 (2nd column), A028896 (3rd column).
Cf. A008279.
Cf. A008277, A132159 (mirrored).

a121757 n k = a121757_tabl !! n !! k
a121757_row n = a121757_tabl !! n
a121757_tabl = iterate
   (\xs -> zipWith (+) (xs ++ [0]) (zipWith (*) [1..] ([0] ++ xs))) [1]
-- Reinhard Zumkeller, Mar 06 2014

Flatten[Table[n!(k+1)/(n-k)!,{n,0,10},{k,0,n}]]  (* Harvey P. Dale, Apr 25 2011 *)

A000142(n)={ return(n!) ; } A007318(n,k)={ return(binomial(n,k)) ; } A121757(n,k)={ return(A007318(n,k)*A000142(k+1)) ; } { for(n=0,12, for(k=0,n, print1(A121757(n,k),",") ; ); ) ; } \\ R. J. Mathar, Sep 02 2006

a(n,k) = A007318(n,k)*A000142(k+1), k=0,1,..,n, n=0,1,2,3... - R. J. Mathar, Sep 02 2006

a(n,k) = A008279(n,k) * (k+1). a(n,k) = n!*(k+1)/(n-k)!. - Franklin T. Adams-Watters, Sep 20 2006

A131104 Rectangular array read by antidiagonals: a(n, k) is the number of ways to put k labeled objects into n labeled boxes so that there is one box with exactly one object (n, k >= 1).

Original entry on oeis.org

1, 2, 0, 3, 0, 0, 4, 0, 6, 0, 5, 0, 18, 8, 0, 6, 0, 36, 24, 10, 0, 7, 0, 60, 48, 120, 12, 0, 8, 0, 90, 80, 420, 396, 14, 0, 9, 0, 126, 120, 1000, 1512, 1092, 16, 0, 10, 0, 168, 168, 1950, 3720, 6804, 2736, 18, 0, 11, 0, 216, 224, 3360, 7380, 23240, 31008, 6480, 20, 0, 12, 0

Offset: 1

David Wasserman, Jun 14 2007, Jun 15 2007

Problem suggested by Brandon Zeidler. Columns 3 through 5 are A028896, A033996, 10*A007586.

			Array begins:
1 0 0 0 0 0 0
2 0 6 8 10 12 14
3 0 18 24 120 396 1092

Cf. A131103, A131105, A131106, A131107.

a(n, 1) = n. For k > 1, a(n, k) = sum_{j=1..min(floor((k-1)/2), n-1)} A008299(k-1, j)*n!*k*/(n-j-1)!.

cp's OEIS Frontend

A227776 a(n) = 6*n^2 + 1.

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A262221 a(n) = 25*n*(n + 1)/2 + 1.

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A307011 First coordinate in a redundant hexagonal coordinate system of the points of a counterclockwise spiral on an hexagonal grid. Second and third coordinates are given in A307012 and A345978.

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A096948 Triangular table read by rows: T(n,m) = number of rectangles found in an n X m rectangle built from 1 X 1 squares, 1 <= m <= n.

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A292314 Numbers equal to the sum of three oblong numbers in arithmetic progression.

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A262221 a(n) = 25n(n + 1)/2 + 1.

A004538 a(n) = 3n^2 + 3n - 1.