A141073 List of central integer pairs in Pascal-like triangles with index of asymmetry y = 3 and index of obliqueness z = 0 or z = 1.

1, 1, 4, 2, 8, 4, 17, 8, 35, 17, 72, 35, 149, 72, 308, 149, 636, 308, 1314, 636, 2715, 1314, 5609, 2715, 11588, 5609, 23941, 11588, 49462, 23941, 102188, 49462, 211120, 102188, 436173, 211120, 901131, 436173, 1861732, 901131, 3846329, 1861732, 7946496, 3846329

Offset: 1

Author

Juri-Stepan Gerasimov, Jul 16 2008

Keywords

• nonn
• easy

Comments

For the Pascal-like triangle G(n, k) with index of asymmetry y = 3 and index of obliqueness z = 0, which is read by rows, we have G(n, 0) = G(n+1, n+1) = 1, G(n+2, n+1) = 2, G(n+3, n+1) = 4, G(n+4, n+1) = 8, and G(n+5, k) = G(n+1, k-1) + G(n+1, k) + G(n+2, k) + G(n+3, k) + G(n+4, k) for n >= 0 and k = 1..(n+1). (This is array A140996.)

For the Pascal-like triangle G(n, k) with index of asymmetry y = 3 and index of obliqueness z = 1, which is read by rows, we have G(n, n) = G(n+1, 0) = 1, G(n+2, 1) = 2, G(n+3, 2) = 4, G(n+4, 3) = 8, and G(n+5, k) = G(n+1, k-3) + G(n+1, k-4) + G(n+2, k-3) + G(n+3, k-2) + G(n+4, k-1) for n > = 0 and k = 4..(n+4). (This is array A140995.)

Arrays A140995 and A140996 are mirror images of each other. For discussion about their properties and their connection to Stepan's triangles, see their documentation. See also the documentation of the sequences in the CROSSREFS. - Petros Hadjicostas, Jun 13 2019

Examples

			Pascal-like triangle with y = 3 and z = 0 (i.e., A140996) begins as follows:
  1, so no central pair.
  1 1, so a(1) = 1 and a(2) = 1.
  1 2 1, so no central pair.
  1 4 2 1, so a(3) = 4 and a(4) = 2.
  1 8 4 2 1, so no central pair.
  1 16 8 4 2 1, so a(5) = 8 and a(6) = 4.
  1 31 17 8 4 2 1, so no central pair.
  1 60 35 17 8 4 2 1, so a(7) = 17 and a(8) = 8.
  1 116 72 35 17 8 4 2 1, so no central pair.
  1 224 148 72 35 17 8 4 2 1, so a(9) = 35 and a(10) = 17.
  1 432 303 149 72 35 17 8 4 2 1, so no central pair.
  1 833 618 308 149 72 35 17 8 4 2 1, so a(11) = 72 and a(12) = 35.
... [edited by _Petros Hadjicostas_, Jun 13 2019]
		

Links

• Juri-Stepan Gerasimov, Stepan's triangles and Pascal's triangle are connected by the recurrence relation ...
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Cf. A129847, A140993, A140994, A140995, A140996, A140997, A140998, A141065, A141066, A141067, A141068, A141069, A141070, A141072.

Programs

Mathematica

Rest[CoefficientList[Series[x*(x^8 + 3*x^6 + x^5 + 3*x^4 + x^3 + 3*x^2 + x + 1)/(1 - x^2 - x^4 - 2*x^6 -x^8),{x,0,44}],x]] (* James C. McMahon, Jul 16 2025 *)

Formula

From Petros Hadjicostas, Jun 13 2019: (Start)

a(2*n - 1) = A140996(2*n - 1, n - 1) = A140995(2*n - 1, n) and a(2*n) = A140996(2*n - 1, n) = A140995(2*n - 1, n - 1) for n >= 1.

a(2*n) = a(2*n - 3) for n >= 3.

a(n) = 2*a(n-2) + A129847(floor(n/2) - (4 + (-1)^n)) for n >= 9.

G.f.: x*(x^8 + 3*x^6 + x^5 + 3*x^4 + x^3 + 3*x^2 + x + 1)/(1 - x^2 - x^4 - 2*x^6 -x^8). (End)

Extensions

Partially edited by N. J. A. Sloane, Jul 18 2008

More terms from Petros Hadjicostas, Jun 13 2019