A028566 -id:A028566 - cp's OEIS frontend

A091435 Array T(n,k) = n*(n+k), read by antidiagonals.

0, 1, 0, 4, 2, 0, 9, 6, 3, 0, 16, 12, 8, 4, 0, 25, 20, 15, 10, 5, 0, 36, 30, 24, 18, 12, 6, 0, 49, 42, 35, 28, 21, 14, 7, 0, 64, 56, 48, 40, 32, 24, 16, 8, 0, 81, 72, 63, 54, 45, 36, 27, 18, 9, 0, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, 0, 121, 110, 99, 88, 77, 66, 55, 44, 33, 22, 11, 0

Offset: 0

Ross La Haye, Mar 02 2004

			Table begins
   0;
   1,  0;
   4,  2,  0;
   9,  6,  3,  0;
  16, 12,  8,  4,  0;
  25, 20, 15, 10,  5,  0;
  36, 30, 24, 18, 12,  6,  0;
  ...
a(5,3) = 40 because 5 * (5 + 3) = 5 * 8 = 40.

Muniru A Asiru, Table of n, a(n) for n = 0..5150 (rows n = 0..100,flattened)
P. De Geest, Palindromic Quasipronics of the form n(n+x)

Columns: a(n, 0) = A000290(n), a(n, 1) = A002378(n), a(n, 2) = A005563(n), a(n, 3) = A028552(n), a(n, 4) = A028347(n+2), a(n, 5) = A028557(n), a(n, 6) = A028560(n), a(n, 7) = A028563(n), a(n, 8) = A028566(n). Diagonals: a(n, n-4) = A054000(n-1), a(n, n-3) = A014107(n), a(n, n-2) = A046092(n-1), a(n, n-1) = A000384(n), a(n, n) = A001105(n), a(n, n+1) = A014105(n), a(n, n+2) = A046092(n), a(n, n+3) = A014106(n), a(n, n+4) = A054000(n+1), a(n, n+5) = A033537(n). Also note that the sums of the antidiagonals = A002411.

Cf. A056536, A056537, A082156.

Flat(List([0..11],j->List([0..j],i->j*(j-i)))); # Muniru A Asiru, Sep 11 2018

Maple
```
seq(seq((j-i)*j,i=0..j),j=0..14);
```

Table[# (# + k) &[m - k], {m, 0, 11}, {k, 0, m}] // Flatten (* Michael De Vlieger, Oct 15 2018 *)

G.f.: x*(1+x-2*x^2*y)/((1-x*y)^2*(1-x)^3). - Vladeta Jovovic, Mar 05 2004

More terms from Emeric Deutsch, Mar 15 2004

A132773 a(n) = n*(n + 31).

Original entry on oeis.org

0, 32, 66, 102, 140, 180, 222, 266, 312, 360, 410, 462, 516, 572, 630, 690, 752, 816, 882, 950, 1020, 1092, 1166, 1242, 1320, 1400, 1482, 1566, 1652, 1740, 1830, 1922, 2016, 2112, 2210, 2310, 2412, 2516, 2622, 2730, 2840, 2952, 3066, 3182, 3300, 3420, 3542, 3666

Offset: 0

Omar E. Pol, Aug 28 2007

Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569.
Cf. A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132758, A132759, A132760.
Cf. A132761, A132762, A132763, A132764, A132765, A132766, A132767, A132768, A132769, A132770.
Cf. A132771, A132772.

Table[n*(n + 31), {n, 0, 100}] (* Wesley Ivan Hurt, Apr 16 2024 *)

a(n)=n*(n+31) \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 2*x*(-16+15*x)/(-1+x)^3. - R. J. Mathar, Nov 14 2007
a(n) = 2*A132758(n). - R. J. Mathar, Jul 22 2009
a(n) = 2*n + a(n-1) + 30, with n > 0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(31)/31 = A001008(31)/A102928(31) = 290774257297357/2238255069850800, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/31 - 7313175618421/319750724264400. (End)
From Elmo R. Oliveira, Dec 13 2024: (Start)
E.g.f.: exp(x)*x*(32 + x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A380149 Characteristic polynomial of the tesseract graph: a(n) = n^6(n^2-16)(n^2-4)^4.

Original entry on oeis.org

0, -1215, 0, -3189375, 0, 27348890625, 978447237120, 15920336210625, 163074539520000, 1214314872035265, 7134511104000000, 34856907746165505, 146828238520320000, 547377978676010625, 1841813423998894080, 5678883183381890625, 16238028554439229440, 43474602051830210625, 109846357522513920000

Offset: 0

Darío Clavijo, Jan 13 2025

Given that the eigenvalues of the adjacency matrix of the tesseract graph are: {4,2,0,-2,-4} and their multiplicities are defined by binomial(4,k) for k = 0..4 which results in {1,4,6,4,1}, and the characteristic polynomial is given by P(x) = Prod_{k=1..j} (x-lambda_k)^m_k with j=5, substitute the eigenvalues and their multiplicities as:
 k | eigenvalue(lambda_k) | multiplicity(m_k) | term
---+----------------------+-------------------+---------
 1 |          4           | 1                 | (x-4)^1
 2 |          2           | 4                 | (x-2)^4
 3 |          0           | 6                 | (x)^6
 4 |         -2           | 4                 | (x+2)^4
 5 |         -4           | 1                 | (x+4)^1
This results in the characteristic polynomial:
P(x) = (x-4) * (x-2)^4 * (x)^6  * (x+2)^4  * (x+4).
Also all terms are a(n) = 0 mod 5.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Characteristic Polynomial.
Eric Weisstein's World of Mathematics, Tesseract Graph.
Eric Weisstein's World of Mathematics, Graph Spectrum.
Wikipedia, Hypercube Graph.

Cf. A001014, A028347, A028566.

A380149[n_] := n^6*(n^2 - 16)*(n^2 - 4)^4; Array[A380149, 20, 0] (* Paolo Xausa, Jan 21 2025 *)

a = lambda n: (n**6)*(n**2-16)*(n**2-4)**4
print([a(n) for n in range(0,19)])

a(n) = -4096*n^6 + 4352*n^8 - 1792*n^10 + 352*n^12 - 32*n^14 + n^16.

A382310 Array read by ascending antidiagonals: A(n,m) is the squared distance between the roots of the 2nd degree equations z^2 +- n*z + m = 0 on the complex plane.

Original entry on oeis.org

0, 1, 4, 4, 3, 8, 9, 0, 7, 12, 16, 5, 4, 11, 16, 25, 12, 1, 8, 15, 20, 36, 21, 8, 3, 12, 19, 24, 49, 32, 17, 4, 7, 16, 23, 28, 64, 45, 28, 13, 0, 11, 20, 27, 32, 81, 60, 41, 24, 9, 4, 15, 24, 31, 36, 100, 77, 56, 37, 20, 5, 8, 19, 28, 35, 40, 121, 96, 73, 52, 33, 16, 1, 12, 23, 32, 39, 44

Offset: 0

Stefano Spezia, Mar 21 2025

			The array begins as:
   0,  4,  8, 12, 16, 20, 24, 28, 32, 36, 40, 44, ...
   1,  3,  7, 11, 15, 19, 23, 27, 31, 35, 39, 43, ...
   4,  0,  4,  8, 12, 16, 20, 24, 28, 32, 36, 40, ...
   9,  5,  1,  3,  7, 11, 15, 19, 23, 27, 31, 35, ...
  16, 12,  8,  4,  0,  4,  8, 12, 16, 20, 24, 28, ...
  25, 21, 17, 13,  9,  5,  1,  3,  7, 11, 15, 19, ...
  ...
A(2,0) = 4 since z^2 - 2*z = 0 and z^2 + 2*z = 0 have respectively roots 0, 2, and -2, 0 with squared distance equal to 4;
A(1,2) = 7 since z^2 - z + 2 = 0 and z^2 + z + 2 = 0 have respectively roots (1 +- i*sqrt(7))/2 and (-1 +- i*sqrt(7))/2 with squared distance equal to 7, where i denotes the imaginary unit.

Cf. A000290 (m=0), A008586 (n=0), A028347, A028566, A028884, A131098, A134594, A145917, A382311 (antidiagonal sums).

A[n_,m_]:=Abs[n^2-4m]; Table[A[n-m,m],{n,0,11},{m,0,n}]//Flatten

A(n,m) = abs(n^2 - 4*m).
A(n,n) = A028347(n-2) for n > 3.
A(n,1) = A028347(n) for n > 1.
A(n,2) = A028884(n-3) for n > 2.
A(n,4) = A028566(n-4) for n > 3.
A(n,5) = A134594(n-5) for n > 4.
A(1,n) = A131098(n+1).

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A091435 Array T(n,k) = n*(n+k), read by antidiagonals.

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A132773 a(n) = n*(n + 31).

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A380149 Characteristic polynomial of the tesseract graph: a(n) = n^6*(n^2-16)*(n^2-4)^4.

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A382310 Array read by ascending antidiagonals: A(n,m) is the squared distance between the roots of the 2nd degree equations z^2 +- n*z + m = 0 on the complex plane.

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A380149 Characteristic polynomial of the tesseract graph: a(n) = n^6(n^2-16)(n^2-4)^4.