

A309462


Limiting row sequence for Pascallike triangle A140995 (Stepan's triangle with index of asymmetry s = 3).


2



1, 2, 4, 8, 17, 35, 72, 149, 308, 636, 1314, 2715, 5609, 11588, 23941, 49462, 102188, 211120, 436173, 901131, 1861732, 3846329, 7946496, 16417420, 33918306, 70075047, 144774689, 299103768, 617946857, 1276675050, 2637604132, 5449276664, 11258177753, 23259337731
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OFFSET

0,2


COMMENTS

In the attached photograph, we see that the index of asymmetry is denoted by s and the index of obliqueness by e. The general recurrence is G(n+s+2, k) = G(n+1, ke*s+e1) + Sum_{1 <= m <= s+1} G(n+m, ke*s+m*e2*e) for n >= 0 with k = 1..(n+1) when e = 0 and k = (s+1)..(n+s+1) when e = 1. The initial conditions are G(n+x+1, ne*n+e*xe+1) = 2^x for x=0..s and n >= 0. There is one more initial condition, namely, G(n, e*n) = 1 for n >= 0.
For s = 0, we get Pascal's triangle A007318. For s = 1, we get A140998 (e = 0) and A140993 (e = 1). For s = 2, we get A140997 (e = 0) and A140994 (e = 1). For s = 3, we get A140996 (e = 0) and A140995 (e = 1). For s = 4, we have arrays A141020 (with e = 0) and A141021 (with e = 1). In some of these arrays, the indices n and k are shifted.
For the triangular array G(n, k) = A140995(n, k), 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, k3) + G(n+1, k4) + G(n+2, k3) + G(n+3, k2) + G(n+4, k1) for n >= 0 and k = 4..(n+4).
With G(n, k) = A140995(n, k), the current sequence (a(k): k >= 0) is defined by a(k) = lim_{n > infinity} G(n, k) for k >= 0. Then a(k) = a(k4) + 2*a(k3) + a(k2) + a(k1) for k >= 4 with a(x) = 2^x for x = 0..3.


LINKS

Table of n, a(n) for n=0..33.
JuriStepan Gerasimov, Stepan's triangles and Pascal's triangle are connected by the recurrence relation ...


FORMULA

a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 8, and a(k) = a(k1) + a(k2) + 2*a(k3) + a(k4) for k >= 4.
G.f.: (x^2 + x + 1)/(x^4  2*x^3  x^2  x + 1).


CROSSREFS

Cf. A007318, A140993, A140994, A140995, A140996, A140997, A140998, A141020, A141021, A141031, A141065, A141066, A141067, A141068, A141069, A141070, A141072, A141073, A308808.
Sequence in context: A293331 A309908 A001357 * A058520 A127680 A136750
Adjacent sequences: A309459 A309460 A309461 * A309463 A309464 A309465


KEYWORD

nonn,easy


AUTHOR

Petros Hadjicostas, Aug 03 2019


STATUS

approved



