A015237 -id:A015237 - cp's OEIS frontend

A277636 Number of 3 X 3 matrices having all elements in {0,...,n} with determinant = permanent.

1, 343, 6859, 50653, 226981, 753571, 2048383, 4826809, 10218313, 19902511, 36264691, 62570773, 103161709, 163667323, 251239591, 374805361, 545338513, 776151559, 1083206683, 1485446221, 2005142581, 2668267603, 3504881359, 4549540393, 5841725401, 7426288351

Offset: 0

Indranil Ghosh, Jan 02 2017

a(n) is a perfect cube.

Indranil Ghosh, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A059976 (Number of 3 X 3 singular matrices with all elements in {0,...,n})
Cf. A015237 (Number of 2 X 2 matrices with all elements in {0,...,n} with determinant = permanent )
Cf. A003215.

Vec((1 + 336*x + 4479*x^2 + 9808*x^3 + 4479*x^4 + 336*x^5 + x^6) / (1 - x)^7 + O(x^30)) \\ Colin Barker, Jan 02 2017

def a(n):
    return 27*n**6-81*n**5+108*n**4-81*n**3+36*n**2-9*n+1

a(n) = A003215(n-1)^3.
a(n) = (3*n^2 - 3*n + 1)^3.
G.f.: (1 + 336*x + 4479*x^2 + 9808*x^3 + 4479*x^4 + 336*x^5 + x^6) / (1 - x)^7. - Colin Barker, Jan 02 2017

A280058 Number of 2 X 2 matrices with entries in {0,1,...,n} with determinant = permanent with no entries repeated.

Original entry on oeis.org

0, 0, 0, 12, 48, 120, 240, 420, 672, 1008, 1440, 1980, 2640, 3432, 4368, 5460, 6720, 8160, 9792, 11628, 13680, 15960, 18480, 21252, 24288, 27600, 31200, 35100, 39312, 43848, 48720, 53940, 59520, 65472, 71808, 78540, 85680, 93240, 101232, 109668, 118560

Offset: 0

Indranil Ghosh, Dec 25 2016

Consider all Pythagorean triples (X,Y,Z=Y+2) ordered by increasing Z; A005843, A005563, A002522 and A007531 give the X, Y, Z and area A values of related triangles; for n >= 2 altitude h(n) = a(n+1)/A002522(n) or h(n)/2 is irreducible fraction in Q\Z. - Ralf Steiner, Mar 29 2020

Indranil Ghosh, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A015237 (where the entries can be repeated), A005843, A005563, A002522, A016742, A099761, A007531.

Table[2*n*(n-1)*(n-2), {n, 0, 50}] (* G. C. Greubel, Dec 25 2016 *)

for(n=0, 50, print1(2*n*(n-1)*(n-2), ", ")) \\ G. C. Greubel, Dec 25 2016

a(n)=12*binomial(n,3) \\ Charles R Greathouse IV, Dec 25 2016

def t(n):
    s=0
    for a in range(0,n+1):
        for b in range(0,n+1):
            if a!=b:
                for c in range(0,n+1):
                    if a!=c and b!=c:
                        for d in range(0,n+1):
                            if d!=a and d!=b and d!=c:
                                if (a*d-b*c)==(a*d+b*c):
                                    s+=1
    return s
for i in range(0,201):
    print(str(i)+" "+str(t(i)))

a = lambda n: 2*n*(n-1)*(n-2) # David Radcliffe, Jun 14 2025

a(n) = 2*((n+1)^3 - 6*(n+1)^2 + 11*(n+1) - 6), for n>0.
a(n) = 2*n*(n-1)*(n-2). - David Radcliffe, Jun 14 2025
a(n) == 0 (mod 12).
From G. C. Greubel, Dec 25 2016: (Start)
G.f.: (12*x^3)/(1 - x)^4.
E.g.f.: 2*x^3*exp(x).
a(n) = 2*n*(n-1)*(n-2).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
a(n) = 12 * A000292(n-2) for n>1. - Alois P. Heinz, Jan 30 2017
a(n+1) = sqrt(A016742(n)*A099761(n-1)) for n>=2. - Ralf Steiner, Mar 29 2020
From Amiram Eldar, Jun 30 2025: (Start)
Sum_{n>=3} 1/a(n) = 1/8.
Sum_{n>=3} (-1)^(n+1)/a(n) = log(2) - 5/8. (End)

A280407 Number of 2 X 2 matrices with all elements in {-n,..,0,..,n} with permanent = determinant * n.

Original entry on oeis.org

1, 45, 81, 233, 289, 601, 625, 1113, 1153, 1785, 1681, 2761, 2401, 3577, 3505, 4665, 4225, 6185, 5329, 7673, 6945, 8601, 7921, 11033, 9665, 12265, 11793, 14089, 12769, 18073, 14641, 19945, 17281, 20121, 20593, 23961, 21025, 25417, 24177, 29177, 25921, 35449, 28561, 36233

Offset: 0

Indranil Ghosh, Jan 06 2017

nonn

			For n = 2, few of the possible matrices are [-2,-2,0,0], [-2,-1,0,0], [-2,0,-2,0], [-2,0,-1,0], [-2,0,0,0], [-2,0,1,0], [-2,0,2,0], [1,0,0,0], [1,0,1,0], [1,0,2,0], [1,1,0,0], [1,2,0,0], [2,-2,0,0], [2,-1,0,0], [2,0,-2,0], .... There are 81 possibilities. Here each of the matrices is defined as M = [a,b,c,d] where a = M[1][1], b = M[1][2], c = M[2][1], d = M[2][2]. So for n = 2, a(2)=81.

David Radcliffe, Table of n, a(n) for n = 0..10000 (terms n = 0..155 from Indranil Ghosh).

Number of 2 X 2 matrices with all elements in {0,..,n}: A280391 (permanent = determinant * n), A280321 (determinant = permanent * n), A015237 (determinant = permanent) and A016754 (determinant = 2* permanent).

def t(n):
    s=0
    for a in range(-n,n+1):
        for b in range(-n,n+1):
            for c in range(-n,n+1):
                for d in range(-n,n+1):
                    if (a*d-b*c)*n==(a*d+b*c):
                        s+=1
    return s
for i in range(0,156):
    print(t(i))

import numpy as np
def a280417(N):
    if N > 0: yield 1
    if N > 1: yield 45
    if N <= 2: return
    prods = np.zeros(N * N, dtype=np.int32)
    prods[1] = 1 # prods[k] counts integer solutions to x*y = k with 1 <= x,y <= n
    for n in range(2, N):
        n_sq = n * n
        prods[n: n_sq: n] += 2
        prods[n_sq] += 1
        dx = (n + 1) // 2 if n % 2 else n + 1
        dy = (n - 1) // 2 if n % 2 else n - 1
        ad = prods[dx : n_sq : dx]
        bc = prods[dy : dy * ad.shape[0] + 1 : dy]
        yield (4 * n + 1) ** 2 + 8 * int(ad @ bc)
        # (4*n+1)**2 = solutions to a*d = b*c = 0 with -n <= a,b <= n.
        # ad @ bc = solutions to (n-1)*a*d = (n+1)*b*c > 0 with 1 <= a,b <= n.
        # Multiply by 8 to account for all consistent sign changes of a,b,c,d.
print(list(a280417(44))) # David Radcliffe, May 22 2025

From David Radcliffe, May 22 2025: (Start)
a(n) = (4*n+1)^2 iff n=0 or n+1 is an odd prime, otherwise a(n) > (4*n+1)^2.
a(n) = 8 * A280391(n) - 8*(2*n+1)^2 + (4*n+1)^2 for n>1. (End)

A357042 The sum of the numbers of the central diamond of the multiplication table [1..k] X [1..k] for k=2*n-1.

Original entry on oeis.org

1, 20, 117, 400, 1025, 2196, 4165, 7232, 11745, 18100, 26741, 38160, 52897, 71540, 94725, 123136, 157505, 198612, 247285, 304400, 370881, 447700, 535877, 636480, 750625, 879476, 1024245, 1186192, 1366625, 1566900, 1788421, 2032640, 2301057, 2595220, 2916725, 3267216, 3648385

Offset: 1

Nicolay Avilov, Sep 18 2022

a(n) is the sum of the elements of the multiplication table, forming the maximum diamond in its center.

			In the multiplication table [1..3] X [1..3]: a(2) = 2+2+4+6+6 = 20;
In the multiplication table [1..5] X [1..5]: a(3) = 3+4+3+6+6+8+9+8+12+12+15+16+15 = 117.
For n=3, the multiplication table [1..5] X [1..5] and the terms summed are
  *   1  2  3  4  5
   -----------------
  1|        3
  2|     4  6  8
  3|  3  6  9 12 15
  4|     8 12 16
  5|       15

Paolo Xausa, Table of n, a(n) for n = 1..10000
Nicolay Avilov, Drawing for a(1)-a(5)
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000290, A001844, A003991.

Cf. A000583, A015237.

A357042[n_] := n^2*(2*(n-1)*n + 1); Array[A357042, 50] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 20, 117, 400, 1025}, 50] (* Paolo Xausa, Oct 03 2024 *)

a(n) = n^2*(2*n^2 - 2*n + 1).
a(n) = 2*A000583(n) - A015237(n).
From Stefano Spezia, Sep 19 2022: (Start)
G.f.: x*(1 + 15*x + 27*x^2 + 5*x^3)/(1 - x)^5.
a(n) = A000290(n)*A001844(n-1). (End)

A192418 Molecular topological indices of the complete bipartite graphs K_{n,n}.

Original entry on oeis.org

4, 48, 180, 448, 900, 1584, 2548, 3840, 5508, 7600, 10164, 13248, 16900, 21168, 26100, 31744, 38148, 45360, 53428, 62400, 72324, 83248, 95220, 108288, 122500, 137904, 154548, 172480, 191748, 212400

Offset: 1

Eric W. Weisstein, Jul 10 2011

Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Molecular Topological Index
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Table[4n^2(2n-1),{n,30}] (* or *) LinearRecurrence[{4,-6,4,-1},{4,48,180,448},30] (* Harvey P. Dale, Apr 08 2018 *)

a(n) = 4*n^2*(2*n-1).
a(n) = 4*A015237(n).
G.f.: 4*x*(3*x^2+8*x+1)/(x-1)^4. - Colin Barker, Nov 04 2012
a(n) = 2*n * A002939(n). - Bruce J. Nicholson, Oct 14 2019
E.g.f.: 4*exp(x)*x*(1 + 5*x + 2*x^2). - Stefano Spezia, Oct 15 2019

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A277636 Number of 3 X 3 matrices having all elements in {0,...,n} with determinant = permanent.

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A280058 Number of 2 X 2 matrices with entries in {0,1,...,n} with determinant = permanent with no entries repeated.

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A280407 Number of 2 X 2 matrices with all elements in {-n,..,0,..,n} with permanent = determinant * n.

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A357042 The sum of the numbers of the central diamond of the multiplication table [1..k] X [1..k] for k=2*n-1.

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A192418 Molecular topological indices of the complete bipartite graphs K_{n,n}.

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