A034262 -id:A034262 - cp's OEIS frontend

A265129 Triangle read by rows, formed as the sum of the two versions of the natural numbers filling an equilateral triangle.

2, 5, 5, 10, 10, 10, 17, 17, 17, 17, 26, 26, 26, 26, 26, 37, 37, 37, 37, 37, 37, 50, 50, 50, 50, 50, 50, 50, 65, 65, 65, 65, 65, 65, 65, 65, 82, 82, 82, 82, 82, 82, 82, 82, 82, 101, 101, 101, 101, 101, 101, 101, 101, 101, 101

Offset: 1

Craig Knecht, Dec 02 2015

The natural numbers can sequentially fill a right- or left-handed equilateral triangle. Componentwise addition of the values of these two triangles produces the present triangle.
The row sums for this triangle give A034262.
The difference between the right- and left-handed triangles produces A049581.

			Displayed as a triangle:
   2;
   5  5;
  10 10 10;
  17 17 17 17;
  26 26 26 26 26;
  37 37 37 37 37 37;
  ...

Craig Knecht, Sum of right and left triangles.
Craig Knecht, Difference between right and left triangles A049581.

Column k=1 gives A002522.
Cf. A049581 (difference of triangles), A034262 (row sum of triangle), A069894 (center column).
Cf. A071237.

seq(seq(n^2+1,k=1..n),n=1..10); # Georg Fischer, Oct 01 2021

T(n,k) = n^2 + 1 for k = 1..n and n >= 1. - Georg Fischer, Oct 01 2021

Sum_{k=1..n} k * T(n,k) = A071237(n). - Alois P. Heinz, Oct 01 2021

Row 6 with T(6,k)=37 inserted by Georg Fischer, Oct 01 2021

A303944 Number of partitions of n into at most 1^2 copy of 1, 2^2 copies of 2, 3^2 copies of 3, ... .

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 11, 15, 19, 25, 34, 43, 55, 71, 90, 113, 143, 178, 222, 276, 340, 418, 515, 628, 765, 931, 1128, 1362, 1643, 1974, 2369, 2836, 3385, 4033, 4800, 5694, 6745, 7978, 9418, 11096, 13057, 15334, 17985, 21062, 24626, 28753, 33534, 39045, 45408, 52744, 61187

Offset: 0

Seiichi Manyama, May 03 2018

nonn

			  n |                                | a(n)
----+--------------------------------+------
  1 | 1                              |  1
  2 | 2                              |  1
  3 | 3, 2+1                         |  2
  4 | 4, 3+1, 2+2                    |  3
  5 | 5, 4+1, 3+2, 2+2+1             |  4
  6 | 6, 5+1, 4+2, 3+3, 3+2+1, 2+2+2 |  6

Cf. A034262, A052335, A303942.

G.f.: Product_{k>=1} (1-x^(k*(k^2+1)))/(1-x^k).

A318765 a(n) = (n + 2)*(n^2 + n - 1).

Original entry on oeis.org

-2, 3, 20, 55, 114, 203, 328, 495, 710, 979, 1308, 1703, 2170, 2715, 3344, 4063, 4878, 5795, 6820, 7959, 9218, 10603, 12120, 13775, 15574, 17523, 19628, 21895, 24330, 26939, 29728, 32703, 35870, 39235, 42804, 46583, 50578, 54795, 59240, 63919, 68838, 74003, 79420, 85095

Offset: 0

Bruno Berselli, Sep 04 2018

First differences are in A004538.

a(n) is divisible by 11 for n = 3, 7, 9, 14, 18, 20, 25, 29, 31, 36, 40, ... with formula (1/3)*(11*m + (1 + (m mod 3))*(-1)^((m-1) mod 3) + 8), m >= 0.

Colin Barker, Table of n, a(n) for n = 0..1000
Bruno Berselli, Table of similar sequences (row k=3, m>1).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A004538.
Subsequence of A047216.
Similar sequences (see Table in Links section): A011379, A027444, A033445, A034262, A045991, A069778.

GAP
```
List([0..50], n -> (n+2)*(n^2+n-1));
```

[(n+2)*(n^2+n-1) for n in 0:50] |> println

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

seq((n+2)*(n^2+n-1),n=0..43); # Paolo P. Lava, Sep 04 2018

Table[(n + 2) (n^2 + n - 1), {n, 0, 50}]

Maxima
```
makelist((n+2)*(n^2+n-1), n, 0, 50);
```
PARI
```
vector(50, n, n--; (n+2)*(n^2+n-1))
```
Python
```
[(n+2)*(n**2+n-1) for n in range(50)]
```
Sage
```
[(n+2)*(n^2+n-1) for n in (0..50)]
```

O.g.f.: (-2 + 11*x - 4*x^2 + x^3)/(1 - x)^4.
E.g.f.: (-2 + 5*x + 6*x^2 + x^3)*exp(x).
a(n) = -A033445(-n-1).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 5. - Wesley Ivan Hurt, Dec 18 2020

A373878 Row products of A317617.

Original entry on oeis.org

1, 1, 25, 3360, 1336336, 1158917760, 1870414552161, 5024209998931200, 20882706457600000000, 126806525870641017811200, 1078732544346879404306640625, 12413512011036114072165367296000, 188031682201497672618081000000000000, 3661653518107730704646938085472357120000

Offset: 0

Stefano Spezia, Jun 20 2024

nonn

Stefano Spezia, Table of n, a(n) for n = 0..150

Cf. A000079, A001093, A006003, A034262, A185826, A317617.

a[0]=1; a[n_]:=If[EvenQ[n], (n+n^3)^n/2^n, Pochhammer[(1+n^3)/2, n]]; Array[a,14,0]

a(n) = A185826(n) = A006003(n)^n for even n.

a(n) = Pochhammer((1 + n^3)/2, n) for odd n.

A055378 Table read by antidiagonals: T(n,k) = n^trinv(k)+n^(k-((trinv(k)(trinv(k)-1))/2)) where trinv (k) = floor((1+sqrt(1+8k))/2) and with 0^0 = 1.

Original entry on oeis.org

2, 1, 2, 0, 2, 2, 1, 2, 3, 2, 0, 2, 4, 4, 2, 0, 2, 5, 6, 5, 2, 1, 2, 6, 10, 8, 6, 2, 0, 2, 8, 12, 17, 10, 7, 2, 0, 2, 9, 18, 20, 26, 12, 8, 2, 0, 2, 10, 28, 32, 30, 37, 14, 9, 2, 1, 2, 12, 30, 65, 50, 42, 50, 16, 10, 2, 0, 2, 16, 36, 68, 126, 72, 56, 65, 18, 11, 2, 0, 2, 17, 54, 80, 130

Offset: 0

Henry Bottomley, Jun 22 2000

			a(50) = T(5,4) = 5^2+5^1 = 30

Rows include A010054 (apart from initial term), A007395 and A048645 (offset). Subsequent rows are sums of two powers of a given number and also rewritings of A052216 from a particular base to base 10. Columns include A007395, A000027, A005843, A002522, A002378, A001105, A001093, A034262, A011379, A033431, A002523.

T(n, k) = n^A025581(k)+n^A002262(k)

A119536 a(n) = 3n^3 + 3n.

Original entry on oeis.org

0, 6, 30, 90, 204, 390, 666, 1050, 1560, 2214, 3030, 4026, 5220, 6630, 8274, 10170, 12336, 14790, 17550, 20634, 24060, 27846, 32010, 36570, 41544, 46950, 52806, 59130, 65940, 73254, 81090, 89466, 98400, 107910, 118014, 128730, 140076, 152070

Offset: 0

Zerinvary Lajos, May 28 2006

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Maple
```
[seq (3*n^3+3*n,n=0..60)];
```

Table[3n^3+3n,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,6,30,90},40] (* Harvey P. Dale, Mar 19 2013 *)

a(n) = 3*A034262(n) = 6*A006003(n). - J. M. Bergot, Apr 16 2012

a(0)=0, a(1)=6, a(2)=30, a(3)=90, a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Mar 19 2013

A188542 Number of primes between n^3-n and n^3+n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 1, 2, 3, 2, 3, 2, 4, 4, 4, 3, 5, 5, 4, 4, 7, 4, 4, 8, 7, 6, 5, 6, 5, 4, 6, 6, 5, 6, 9, 5, 5, 5, 7, 9, 7, 5, 7, 12, 8, 8, 12, 7, 8, 13, 10, 10, 9, 10, 7, 11, 11, 11, 6, 5, 11, 13, 12, 8, 8, 14, 9, 5, 7, 5, 14, 12, 13, 8, 11, 15, 15, 12, 11, 5, 11, 17, 14, 15, 15, 12, 15, 14, 16, 14, 17, 13, 15, 15, 12, 6, 17, 16, 14

Offset: 1

Franklin T. Adams-Watters, Apr 07 2011

nonn

We include the end points in the range; this affects the value only for n=1.

Conjecture: the sequence contains no 0's. Verified up to n = 100000.

Cf. A110835, A000578, A007531, A034262.

vector(100,m,sum(k=m^3-m,m^3+m,isprime(k)))

cp's OEIS Frontend

A265129 Triangle read by rows, formed as the sum of the two versions of the natural numbers filling an equilateral triangle.

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A303944 Number of partitions of n into at most 1^2 copy of 1, 2^2 copies of 2, 3^2 copies of 3, ... .

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Formula

A318765 a(n) = (n + 2)*(n^2 + n - 1).

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GAP

Julia

Magma

Maple

Mathematica

Maxima

PARI

Python

Sage

Formula

A373878 Row products of A317617.

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Mathematica

Formula

A055378 Table read by antidiagonals: T(n,k) = n^trinv(k)+n^(k-((trinv(k)*(trinv(k)-1))/2)) where trinv (k) = floor((1+sqrt(1+8*k))/2) and with 0^0 = 1.

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Examples

Crossrefs

Formula

A119536 a(n) = 3*n^3 + 3*n.

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Programs

Maple

Mathematica

Formula

A188542 Number of primes between n^3-n and n^3+n.

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Author

Keywords

Comments

Crossrefs

Programs

PARI

A055378 Table read by antidiagonals: T(n,k) = n^trinv(k)+n^(k-((trinv(k)(trinv(k)-1))/2)) where trinv (k) = floor((1+sqrt(1+8k))/2) and with 0^0 = 1.

A119536 a(n) = 3n^3 + 3n.