A317614 -id:A317614 - cp's OEIS frontend

A325657 a(n) = (1/2)(-1 + (-1)^n)(n-1) + n^2.

0, 1, 4, 7, 16, 21, 36, 43, 64, 73, 100, 111, 144, 157, 196, 211, 256, 273, 324, 343, 400, 421, 484, 507, 576, 601, 676, 703, 784, 813, 900, 931, 1024, 1057, 1156, 1191, 1296, 1333, 1444, 1483, 1600, 1641, 1764, 1807, 1936, 1981, 2116, 2163, 2304, 2353, 2500, 2551

Offset: 0

Stefano Spezia, May 13 2019

For n > 0, a(n) is the n-th element of the diagonal of the triangle A325655. Equivalently, a(n) is the element M_{n,1} of the matrix M(n) whose permanent is A322277(n).

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000290, A054569, A317614, A322277, A325655.

Flat(List([0..55], n->(1/2)*(- 1 + (- 1)^n)*(n - 1) + n^2));

[(1/2)*(- 1 + (- 1)^n)*(n - 1) + n^2: n in [0..55]];

a:=n->(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3): seq(a(n), n=0..55);

Table[(1/2)*(- 1+(-1)^n)*(n-1)+n^2,{n,0,55}]

a(n) = (1/2)*(- 1 + (- 1)^n)*(n - 1) + n^2;

O.g.f.: (-1 - 3*x - x^2 - 3*x^3)/((-1 + x)^3*(1+x)^2).
E.g.f.: (1/2)*exp(-x)*(-1 - x + exp(2*x)*(1 + x + 2*x^2)).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4
a(n) = n^2 if n is even.
a(n) = n^2 - n + 1 if n is odd.

A350236 a(n) is the sum of the entries in an n X n X n 3D matrix whose elements start at 1 in the corner cells and increase by 1 with each step towards the center.

Original entry on oeis.org

1, 8, 54, 160, 425, 864, 1666, 2816, 4617, 7000, 10406, 14688, 20449, 27440, 36450, 47104, 60401, 75816, 94582, 116000, 141561, 170368, 204194, 241920, 285625, 333944, 389286, 450016, 518897, 594000, 678466, 770048, 872289, 982600, 1104950, 1236384, 1381321

Offset: 1

Saeed Barari, Dec 21 2021

The 2D version of this problem is discussed in A317614.

			For n=3: we have the following 3D matrix: (sliced for each Z surface)
(z=1): 1 2 1
       2 3 2
       1 2 1
(z=2): 2 3 2
       3 4 3
       2 3 2
(z=3): 1 2 1
       2 3 2
       1 2 1
The sum of all elements is: (3/4)*n^2 * (n^2 - 2/3*n + (n mod 2)) = 54.

Saeed Barari, In depth (pdf)
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A317614.

a:=n->(3/4)*n^2 * (n^2 - (2/3)*n + modp(n, 2)): seq(a(n), n=1..50);

LinearRecurrence[{2,2,-6,0,6,-2,-2,1},{1,8,54,160,425,864,1666,2816},35] (* Stefano Spezia, May 19 2022 *)

for n in range(1, nmax):
  sum = round(3/4*n**2 * (n**2 - 2/3*n + n % 2))
  print(sum, end=', ')

a(n) = (3/4)*n^2 * (n^2 - 2/3*n + (n mod 2)).
From Stefano Spezia, May 19 2022: (Start)
O.g.f.: x*(1 + 6*x + 36*x^2 + 42*x^3 + 45*x^4 + 12*x^5 + 2*x^6)/((1 - x)^5*(1 + x)^3).
E.g.f.: x*((4 + 15*x + 16*x^2 + 3*x^3)*cosh(x) + (1 + 18*x + 16*x^2 + 3*x^3)*sinh(x))/4.
a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5)- 2*a(n-6) - 2*a(n-7) + a(n-8) for n > 8. (End)

Python program and a(23), a(34) corrected by Georg Fischer, Sep 30 2022

A382597 a(n) = Product_{i=1..n} 1 - i + n(n - i + 1) - (n - 2i + 1)*((n - i + 1) mod 2).

Original entry on oeis.org

1, 1, 8, 105, 3840, 181545, 15814656, 1635491025, 261144576000, 47396578806225, 12046266925056000, 3390530144534798265, 1256223498048110592000, 506594307608708171477625, 257699484814807738928332800, 140934799049120316306629726625, 94240120920042785192632469422080

Offset: 0

Stefano Spezia, Mar 31 2025

nonn

a(n) is the product of the elements of the main antidiagonal of the n X n square matrix M(n) formed by writing the numbers 1, ..., n^2 successively forward and backward along the rows in zig-zag pattern (see A317614).

Except for n = 0, 2, and 6, a(n) has trailing zeros iff n is even.

			a(4) = 3840:
   1,  2,  3,  4;
   8,  7,  6,  5;
   9, 10, 11, 12;
  16, 15, 14, 13.

Cf. A317614, A370753, A382532.

a[n_]:=Product[1-i+n(n-i+1)-(n-2i+1)Mod[n-i+1,2],{i,n}]; Array[a,17,0]

cp's OEIS Frontend

A325657 a(n) = (1/2)(-1 + (-1)^n)(n-1) + n^2.

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A350236 a(n) is the sum of the entries in an n X n X n 3D matrix whose elements start at 1 in the corner cells and increase by 1 with each step towards the center.

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A382597 a(n) = Product_{i=1..n} 1 - i + n(n - i + 1) - (n - 2i + 1)*((n - i + 1) mod 2).

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cp's OEIS Frontend

A325657 a(n) = (1/2)*(-1 + (-1)^n)*(n-1) + n^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

A350236 a(n) is the sum of the entries in an n X n X n 3D matrix whose elements start at 1 in the corner cells and increase by 1 with each step towards the center.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

Extensions

A382597 a(n) = Product_{i=1..n} 1 - i + n*(n - i + 1) - (n - 2*i + 1)*((n - i + 1) mod 2).

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Author

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Examples

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Programs

Mathematica

A325657 a(n) = (1/2)(-1 + (-1)^n)(n-1) + n^2.

A382597 a(n) = Product_{i=1..n} 1 - i + n(n - i + 1) - (n - 2i + 1)*((n - i + 1) mod 2).