A033431 -id:A033431 - cp's OEIS frontend

A366158 Number of distinct determinants of 3 X 3 matrices with entries from {0, 1, ..., n}.

1, 5, 25, 77, 179, 355, 609, 995, 1497, 2167, 2999, 4069, 5289, 6841, 8595, 10661, 13023, 15777, 18795, 22305, 26085, 30397, 35107, 40381, 45929, 52247, 58929, 66287, 74139, 82767, 91643, 101701, 112013, 123235

Offset: 0

Robert P. P. McKone, Oct 02 2023

These determinants a(n) equivalently represent the leading coefficient (coefficient of term with degree 0) of the characteristic polynomials for such matrices, thereby providing a direct measure and lower bound of the uniqueness of these polynomials within this matrix class.

The maximal determinant counted by a(n) is A033431(n) = 2*n^3.

Robert P. P. McKone, The distinct determinants for a(0)-a(18).

Cf. A058331 (distinct determinants for 2 X 2 matrices).
Cf. A272661, A272659.
Cf. A365926.
Cf. A272658, A272660, A272662, A272663.
Cf. A033431 (maximal determinant).
Cf. A097400 (distinct consecutive entries in 3 X 3 matrix).

mat[n_Integer?Positive] := mat[n] = Array[m, {n, n}]; flatMat[n_Integer?Positive] := flatMat[n] = Flatten[mat[n]]; detMat[n_Integer?Positive] := detMat[n] = Det[mat[n]] // FullSimplify; a[d_Integer?Positive, 0] = 1; a[d_Integer?Positive, n_Integer?Positive] := a[d, n] = Length[DeleteDuplicates[Flatten[ParallelTable[Evaluate[detMat[d]], ##] & @@ Table[{flatMat[d][[i]], 0, n}, {i, 1, d^2}]]]]; Table[a[3, n], {n, 0, 9}]

from itertools import product
def A366158(n): return len({a[0]*(a[4]*a[8] - a[5]*a[7]) - a[1]*(a[3]*a[8] - a[5]*a[6]) + a[2]*(a[3]*a[7] - a[4]*a[6]) for a in product(range(n+1),repeat=9)}) # Chai Wah Wu, Oct 06 2023

a(19)-a(26) from Robin Visser, May 08 2025

a(27)-a(33) from Robin Visser, Aug 26 2025

A304159 a(n) = 2n^3 - 4n^2 + 6*n - 2 (n>=1).

Original entry on oeis.org

2, 10, 34, 86, 178, 322, 530, 814, 1186, 1658, 2242, 2950, 3794, 4786, 5938, 7262, 8770, 10474, 12386, 14518, 16882, 19490, 22354, 25486, 28898, 32602, 36610, 40934, 45586, 50578, 55922, 61630, 67714, 74186, 81058, 88342, 96050, 104194, 112786, 121838, 131362, 141370, 151874, 162886, 174418, 186482, 199090

Offset: 1

Emeric Deutsch, May 09 2018

a(n) is the first Zagreb index of the Barbell graph B(n) (n>=3).
The Barbell graph B(n) is defined as two copies of the complete graph K(n) (n>=3), connected by a bridge.
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
The M-polynomial of the Barbell graph B(n) is M(B(n),x,y) = (n-1)*(n-2)*x^{n-1}*y^{n-1} + 2*(n-1)*x^{n-1}*y^n + x^n*y^n.

Colin Barker, Table of n, a(n) for n = 1..1000
Emeric Deutsch and Sandi Klavžar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
Eric Weisstein's World of Mathematics, Barbell Graph
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002943, A014105, A033431, A100705, A304160.

Maple
```
seq(2*n^3-4*n^2+6*n-2, n = 1 .. 40);
```

Table[2n^3-4n^2+6n-2 ,{n,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{2,10,34,86},50] (* Harvey P. Dale, Mar 05 2023 *)

Vec(2*x*(1 + x + 3*x^2 + x^3) / (1 - x)^4 + O(x^60)) \\ Colin Barker, May 09 2018

a(n) = 2*n^3-4*n^2+6*n-2; \\ Altug Alkan, May 09 2018

a(n) = 2 * A100705(n-1).
From Colin Barker, May 09 2018: (Start)
G.f.: 2*x*(1 + x + 3*x^2 + x^3) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4. (End)
a(n) = A033431(n) - A002943(n-1) = A033431(n) - 2*A014105(n-1). - Omar E. Pol, May 09 2018

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)

cp's OEIS Frontend

A366158 Number of distinct determinants of 3 X 3 matrices with entries from {0, 1, ..., n}.

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A304159 a(n) = 2n^3 - 4n^2 + 6*n - 2 (n>=1).

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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.

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cp's OEIS Frontend

A366158 Number of distinct determinants of 3 X 3 matrices with entries from {0, 1, ..., n}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A304159 a(n) = 2*n^3 - 4*n^2 + 6*n - 2 (n>=1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

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.

Views

Author

Keywords

Examples

Crossrefs

Formula

A304159 a(n) = 2n^3 - 4n^2 + 6*n - 2 (n>=1).

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.