OFFSET
1,3
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
LINKS
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)
EXAMPLE
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Juri-Stepan Gerasimov, Jul 16 2008
EXTENSIONS
Partially edited by N. J. A. Sloane, Jul 18 2008
More terms from Petros Hadjicostas, Jun 13 2019
STATUS
approved