A008867 -id:A008867 - cp's OEIS frontend

A307783 The permanent of an n X n symmetric Toeplitz matrix M(n) whose first row consists of n, n-1, ..., 1.

1, 5, 62, 1472, 57228, 3300052, 264163120, 28004426240, 3796084024832, 640290996560896, 131495036625989504, 32300689159458652160, 9350873610168606862080, 3150550820854335942423808, 1222211647879605626853439488, 540858935979668390014623285248, 270804098518125729769134021574656

Offset: 1

Stefano Spezia, Apr 28 2019

nonn

The matrix M(n) differs from that of A204235 in using for the first row the positive integers 1, 2,..., n in decreasing order in place of in increasing order (see examples).
The trace of the matrix M(n) is A000290(n).
The determinant of the matrix M(n) is A001792(n-1).
The sum of the k-th row of the matrix M(n) is A008867(n,k).
For n > k, the sum of the k-diagonal of the matrix M(n) is A055461(n,k).

			For n = 1 the matrix M(1) is
  1
with permanent a(1) = 1.
For n = 2 the matrix M(2) is
  2, 1
  1, 2
with permanent a(2) = 5.
For n = 3 the matrix M(3) is
  3, 2, 1
  2, 3, 2
  1, 2, 3
with permanent a(3) = 62.

Vaclav Kotesovec, Table of n, a(n) for n = 1..35
Wikipedia, Toeplitz Matrix

Cf. A000290, A001792, A008867, A055461, A204235.

f:= proc(n) uses LinearAlgebra; Permanent(ToeplitzMatrix([i, i=n..1, -1)])) end proc: map(f, [$1..17]);

b[i_]:=i; a[n_]:=Permanent[ToeplitzMatrix[Reverse[Array[b, n]], Reverse[Array[b, n ]]]]; Array[a, 17]

{a(n) = matpermanent(matrix(n, n, i, j, n + 1 - max(i - j + 1, j - i + 1)))}
for(n=1, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Apr 29 2019

A008912 Truncated triangular numbers (of form n(n-3)/2 - k^2+kn+1 for 1<=k

Original entry on oeis.org

1, 3, 6, 7, 10, 12, 15, 18, 19, 21, 25, 27, 28, 33, 36, 37, 42, 45, 46, 48, 52, 55, 57, 60, 61, 63, 66, 69, 73, 75, 78, 82, 87, 88, 90, 91, 96, 102, 105, 106, 108, 111, 117, 118, 120, 123, 126, 127, 133, 135, 136, 141, 144, 145, 147, 150, 153, 160, 162, 165

Offset: 1

N. J. A. Sloane, R. K. Guy

nonn

J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).

Cf. A008867.

Numbers of the form (n^2+n+k^2-k+4kn)/2, n>=1, k>=0. - Bob Selcoe, Nov 14 2018

A319602 Numbers with at least two representations as truncated triangular numbers.

Original entry on oeis.org

36, 75, 91, 102, 127, 153, 168, 190, 192, 201, 213, 231, 267, 270, 300, 322, 333, 348, 351, 361, 388, 397, 420, 426, 432, 435, 465, 487, 498, 531, 543, 546, 558, 582, 586, 595, 621, 627, 630, 657, 663, 673, 685, 696, 712, 717, 738, 762, 768, 777, 811, 816, 817

Offset: 1

Allan C. Wechsler, Nov 15 2018

A truncated triangular number is a figurate number, the number of dots in a hexagonal diagram where the side lengths alternate between two values. Include a number in this list if there are two different side-length pairs that give the same count.
The underlying quadratic form is (4ab + a(a-3) + b(b-3) + 2)/2; n is in the list if n can be expressed in this form in two different ways, where a <= b. (That is, exchanging a and b is not considered different.)
A number occurs at least three times in A008867 if and only if it occurs in this sequence.

			75 is in the list because there are 75 dots in both the (2,10) hexagon and the (5,6) hexagon.
Table of solutions for the smallest 10 examples:
36: (1,8) (3,5)
75: (2,10) (5,6)
91: (1,13) (6,6)
102: (2,12) (4,9)
127: (3,12) (7,7)
153: (1,17) (4,12)
168: (2,16) (7,9)
190: (1,19) (7,10)
192: (4,14) (8,9)
201: (3,16) (5,13)

Cf. A008912 (all truncated triangular numbers), A008867 (see comments).

More terms from Alois P. Heinz, Nov 15 2018

A321740 Number of representations of n as a truncated triangular number.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0

Offset: 1

Allan C. Wechsler, Nov 17 2018

nonn

A truncated triangular number is a figurate number, the number of dots in a hexagonal diagram where the side lengths alternate between two values. This sequence gives the number of ways that a number can be represented in this form.
In a sense this sequence is a hexagonal analog of A038548, which asks the same question for rectangular numbers, and A001227 for trapezoidal numbers.
These sequences usually turn out to count divisors of a particular form, of a number simply related to n, but such a formulation is not yet known in this case.
Indices for which this sequence is nonzero are at A008912; this sequence is 2 or greater at the indices given in A319602.

			a(36) = 2 because 36 can be achieved with hexagons of sides (1,9,1,9,1,9) and (3,5,3,5,3,5).

Cf. A008912, A008867, A319602, A322491, A322492.

More terms from Hugo Pfoertner, Sep 18 2020

A322491 Numbers representable as truncated triangular numbers A008912 in more ways than any smaller number.

Original entry on oeis.org

1, 3, 36, 465, 1323, 9045, 38646, 117582, 334656, 1429893, 4350525, 18588606, 87223458, 265382010, 928000395, 4702917255, 5989820730, 12064005132, 56608024080

Offset: 1

Hugo Pfoertner, Dec 12 2018

The corresponding numbers of representations are provided as A322492.

			a(2) = 3 because 3 is the least number occurring more than 1 time in A008867.
a(3) = 36 because 36 is the first number occurring more often (A322492(2) = 4 times) than a(2) = 3 and all numbers < 36 in A008867.

Cf. A008867, A008912, A319602, A321740.

A322492 Records in the number of ways to represent a number as truncated triangular number A008912.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 108, 128, 144, 192

Offset: 1

Hugo Pfoertner, Dec 12 2018

The numbers where the record is first achieved are provided as A322491.

Cf. A008867, A008912, A319602, A321740, A322491.

v=vectorsmall(20000000);for(n=1,5100,for(k=1,n-1,my(t=n*(n-3)/2-k^2+k*n+1);v[t]++));vm=0;for(k=1,#v,if(v[k]>vm,print1(v[k],", ");vm=v[k])) \\ Hugo Pfoertner, Sep 18 2020

cp's OEIS Frontend

A307783 The permanent of an n X n symmetric Toeplitz matrix M(n) whose first row consists of n, n-1, ..., 1.

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A008912 Truncated triangular numbers (of form n*(n-3)/2 - k^2+k*n+1 for 1<=k

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A319602 Numbers with at least two representations as truncated triangular numbers.

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A321740 Number of representations of n as a truncated triangular number.

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A322491 Numbers representable as truncated triangular numbers A008912 in more ways than any smaller number.

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A322492 Records in the number of ways to represent a number as truncated triangular number A008912.

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PARI

A008912 Truncated triangular numbers (of form n(n-3)/2 - k^2+kn+1 for 1<=k