A144216 -id:A144216 - cp's OEIS frontend

A144217 Weight array of A144216: a rectangular array by antidiagonals.

0, 1, 1, 2, 0, 2, 3, 0, 0, 3, 4, 0, 0, 0, 4, 5, 0, 0, 0, 0, 5, 6, 0, 0, 0, 0, 0, 6, 7, 0, 0, 0, 0, 0, 0, 7, 8, 0, 0, 0, 0, 0, 0, 0, 8, 9, 0, 0, 0, 0, 0, 0, 0, 0, 9, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Clark Kimberling, Sep 14 2008

			Northwest corner:
0 1 2 3 4 5
1 0 0 0 0 0
2 0 0 0 0 0
3 0 0 0 0 0
4 0 0 0 0 0

Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened).

Cf. A144217.

Array[Append[PadRight[{#},#,0],#]&,15,0] (* Paolo Xausa, Dec 21 2023 *)

Start with R(m,n)=(m(m-1)+n(n-1))/2 for m>=1,n>=1. Put R(m,n)=0 if m=0 or n=0. Define w(m,n)=R(m,n)-R(m-1,n-1)-R(m,n-1)-R(m-1,n) for m>=1, n>=1. Then the weight array W={w(m,n)} is A144217. (See A144112.)

A016061 a(n) = n(n+1)(4*n+5)/6.

Original entry on oeis.org

0, 3, 13, 34, 70, 125, 203, 308, 444, 615, 825, 1078, 1378, 1729, 2135, 2600, 3128, 3723, 4389, 5130, 5950, 6853, 7843, 8924, 10100, 11375, 12753, 14238, 15834, 17545, 19375, 21328, 23408, 25619, 27965, 30450, 33078, 35853, 38779, 41860

Offset: 0

Robert G. Wilson v

Number of ZnS molecules in cluster of n layers in zinc blende crystal.
(Zinc sulfide crystallizes in two different forms: wurtzite and zinc blende, the latter is also spelled zincblende.) - Jonathan Vos Post, Jan 22 2013
The Kn4 triangle sums of the Connell-Pol triangle A159797 lead to the sequence given above. For the definitions of the Kn4 and other triangle sums see A180662. - Johannes W. Meijer, May 20 2011
If one generated primitive Pythagorean triangles (2n+1, 2n+3) the collective sum of their perimeters for each n is four times the numbers listed in this sequence. - J. M. Bergot, Jul 18 2011
a(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and nA000292(n)+A000292(n+1)=n^3. - Clark Kimberling, Jun 04 2012
Degrees of the Hilbert polynomials for B_3 and C_3, per p. 13 of Gashi et al. - Jonathan Vos Post, Dec 14 2013
Number of solutions to a + b = c + d when 0 < a <= k, 0 <= b, c, d <= k, k = 0, 1, 2, 3.... Taken from Step 1 2007 problem #1(i) using 4 digit balanced numbers. - Bobby Milazzo, Mar 09 2013
From J. M. Bergot, Jun 18 2013: (Start)
Consider the lower half, including the main diagonal, of the array in A144216 as a triangle.  The rows begin:
 0;
 1, 2;
 3, 4, 6;
 6, 7, 9, 12, ...
The sum of the terms in row(n) is a(n). (End)
This sequence is related to A008865 by a(n) = n*A008865(n+1) - Sum_{i=1..n} A008865(i) for n>0. - Bruno Berselli, Aug 06 2015

P. Jena and S. N. Behera, Clusters and Nanostructured Materials, Nova Science Publishers, 1996.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
David N. Blauch, Crystal Structure of Zinc Blende.
Qëndrim R. Gashi, Travis Schedler and David E. Speyer, Looping of the numbers game and the alcoved hypercube, arXiv:0909.5324v1 [math.RT], 2009.
Johan Kok, Introduction to total chromatic vertex stress of graphs, Open J. Disc. Appl. Math. (ODAM, 2023) Vol. 6, No. 2, 32-38. See p. 34.
T. P. Martin, Shells of atoms, Phys. Reports, Vol. 273 (1996), pp. 199-241, see p. 233.
Gloria Olive, Problem #504, Factorizations and Sums, Two-Year College Math. Jnl., Vol. 25 (1994), pp. 244-245.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Bisection of A002623.
Cf. A000292, A000330, A002412, A008865, A014105, A144216, A159797, A180662.
Row sums of triangle A120070.

I:=[0,3,13,34]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 25 2013

A016061 := proc(n)
    n*(n+1)*(4*n+5)/6 ;
end proc: # R. J. Mathar, Sep 26 2013

CoefficientList[Series[x (3 + x) / (1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 25 2013 *)
Table[n(n+1)(4*n+5)/6, {n,0,100}] (* Wesley Ivan Hurt, Sep 25 2013 *)

v=vector(40,i,t(i)); s=0; forstep(i=2,40,2,s+=v[i]; print1(s","))

G.f.: x*(3+x)/(1-x)^4. - Paul Barry, Feb 27 2003
Partial sums of A014105. - Jon Perry, Jul 23 2003
a(n) = Sum_{i=0..n-1} 2*i^2 + i. - Jani Nurminen (slinky(AT)iki.fi), May 14 2006
a(n) = 2*n^3/3  +3*n^2/2 + 5*n/6. - Jonathan Vos Post, Dec 14 2013
a(n) = (4*n+5)/(2*n+1)*A000330(n). - Alexander R. Povolotsky, Mar 09 2013
a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - Bobby Milazzo, Mar 10 2013
Sum_{n>=1} 1/a(n) = 12*Pi/5 + 72*log(2)/5 - 426/25. - Amiram Eldar, Jan 04 2022
E.g.f.: exp(x)*x*(18 + 21*x + 4*x^2)/6. - Stefano Spezia, Jul 31 2022

A367964 Triangle of 2-parameter triangular numbers, read by rows. T(n, k) = (n(n + 1) + k(k + 1)) / 2.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 6, 7, 9, 12, 10, 11, 13, 16, 20, 15, 16, 18, 21, 25, 30, 21, 22, 24, 27, 31, 36, 42, 28, 29, 31, 34, 38, 43, 49, 56, 36, 37, 39, 42, 46, 51, 57, 64, 72, 45, 46, 48, 51, 55, 60, 66, 73, 81, 90, 55, 56, 58, 61, 65, 70, 76, 83, 91, 100, 110

Offset: 0

Peter Luschny, Dec 07 2023

If the rows of the triangle are extended for k > n, the array A144216 is created, which is symmetrical to the main diagonal and therefore contains no new information compared to this triangle.

			Triangle T(n, k) starts:
  0 |  0;
  1 |  1,  2;
  2 |  3,  4,  6;
  3 |  6,  7,  9, 12;
  4 | 10, 11, 13, 16, 20;
  5 | 15, 16, 18, 21, 25, 30;
  6 | 21, 22, 24, 27, 31, 36, 42;
  7 | 28, 29, 31, 34, 38, 43, 49, 56;
  8 | 36, 37, 39, 42, 46, 51, 57, 64, 72;
  9 | 45, 46, 48, 51, 55, 60, 66, 73, 81,  90;
 10 | 55, 56, 58, 61, 65, 70, 76, 83, 91, 100, 110;
.
Start at row 0, column 0 with 0. Go down by adding the column index in step n. At row n, restart the counting and go n steps right by adding the row index in step n, then change direction and go down again by adding the column index. After 3*n steps on this path you are at T(2*n, n) which is 2*triangular(n) + (triangular(2*n) - triangular(n)) = (5*n^2 + 3*n)/2. These are the sliced heptagonal numbers A147875 (see the illustration of Leo Tavares).
.
The equation T(n, k) = (n*(n + 1) + k*(k + 1))/2 can be extended to all n, k in ZZ.
  [n\k] ... -6  -5  -4  -3  -2  -1   0   1   2   3   4   5  ...
  -------------------------------------------------------------
  [-5] ..., 25, 20, 16, 13, 11, 10, 10, 11, 13, 16, 20, 25, ...
  [-4] ..., 21, 16, 12,  9,  7,  6,  6,  7,  9, 12, 16, 21, ...
  [-3] ..., 18, 13,  9,  6,  4,  3,  3,  4,  6,  9, 13, 18, ...
  [-2] ..., 16, 11,  7,  4,  2,  1,  1,  2,  4,  7, 11, 16, ...
  [-1] ..., 15, 10,  6,  3,  1,  0,  0,  1,  3,  6, 10, 15, ...
  [ 0] ..., 15, 10,  6,  3,  1,  0,  0,  1,  3,  6, 10, 15, ...
  [ 1] ..., 16, 11,  7,  4,  2,  1,  1,  2,  4,  7, 11, 16, ...
  [ 2] ..., 18, 13,  9,  6,  4,  3,  3,  4,  6,  9, 13, 18, ...
  [ 3] ..., 21, 16, 12,  9,  7,  6,  6,  7,  9, 12, 16, 21, ...
  [ 4] ..., 25, 20, 16, 13, 11, 10, 10, 11, 13, 16, 20, 25, ...

Columns: A000217, A000124, A152950, A166136, A121263.
Diagonals: A002378, A000290, A002061, A059100, A027689.
Cf. A147875 (T(2*n, n)), A016061 (row sums), A367965 (alternating row sums), A143216 (the multiplicative equivalent), A144216 (extended array).

T := (n, k) -> (n*(n + 1) + k*(k + 1)) / 2:
for n from 0 to 10 do seq(T(n, k), k = 0..n) od;

Module[{n=1},NestList[Append[#+n,n*++n]&,{0},10]] (* or *)
Table[(n(n+1)+k(k+1))/2,{n,0,10},{k,0,n}] (* Paolo Xausa, Dec 07 2023 *)

# A purely additive construction:
from functools import cache
@cache
def a_row(n: int) -> list[int]:
    if n == 0: return [0]
    row = a_row(n - 1) + [0]
    for k in range(n): row[k] += n
    row[n] = row[n - 1] + n
    return row

Recurrence: T(n, n) = n + T(n, n-1) starting with T(0, 0) = 0.
For k <> n: T(n, k) = n + T(n-1, k).
T(n, k) = t(n) + t(k), where t(n) are the triangular numbers A000217.
G.f.: (x + x*(2 - 5*x + x^2)*y + x^4*y^2)/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Dec 07 2023

A357503 a(n) is the hafnian of the 2n X 2n symmetric matrix whose element (i,j) equals abs(i-j).

Original entry on oeis.org

1, 1, 8, 174, 7360, 512720, 53245824, 7713320944, 1486382446592, 367691598791424, 113570289012090880

Offset: 0

Stefano Spezia, Oct 01 2022

			a(2) = M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 8 is the hafnian of
    0, 1, 2, 3;
    1, 0, 1, 2;
    2, 1, 0, 1;
    3, 2, 1, 0.

Wikipedia, Hafnian
Wikipedia, Symmetric matrix

Cf. A049581, A085750 (determinant of M(n)), A085807 (permanent of M(n)), A094053 (super- and subdiagonal sums of M(n) in reversed order), A144216 (row- and column sums of M(n)), A338456.

M[i_, j_, n_]:=Part[Part[Table[Abs[r-c], {r, n}, {c, n}], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]

tm(n) = matrix(n, n, i, j, abs(i-j));
a(n) = my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]); ); s/(n!*2^n); \\ Michel Marcus, May 02 2023

a(6) from Michel Marcus, May 02 2023

a(7)-a(10) from Pontus von Brömssen, Oct 15 2023

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A144217 Weight array of A144216: a rectangular array by antidiagonals.

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A016061 a(n) = n*(n+1)*(4*n+5)/6.

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A367964 Triangle of 2-parameter triangular numbers, read by rows. T(n, k) = (n*(n + 1) + k*(k + 1)) / 2.

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Formula

A357503 a(n) is the hafnian of the 2n X 2n symmetric matrix whose element (i,j) equals abs(i-j).

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A016061 a(n) = n(n+1)(4*n+5)/6.

A367964 Triangle of 2-parameter triangular numbers, read by rows. T(n, k) = (n(n + 1) + k(k + 1)) / 2.