Short remark on the shifted Narayana's cows sequence

The logic of the shifted Narayana's cow sentences is similar to the other Fiboiacci-like ones. The meaning of 'shifted sequences' considered to be know - as well as the Fibonacci-like sequences. Thus we just summarize these results.

A000930 Narayana's cows sequence

RESULTS (constants of shifted sequences):
degree		constant (c)						a		b
1			1.4655712318771392	 		x^3 - 	1 x^2 		- 1 (aka Supergolden Ratio = c1)
2			3.147899035705882			x^3 - 	4 x^2 	+ 3 x 	- 1
3			4.613470267583021			x^3 - 	5 x^2 	+ 2 x 	- 1
4			6.761369303288903			x^3 - 	6 x^2 	- 5 x 	- 1 
5			9.909268338994785			x^3 - 	10 x^2 	+ 1 x 	- 1
6			14.522738606577805			x^3 - 	15 x^2 	+ 7 x 	- 1
7			21.28410790986671			x^3 - 	21 x^2 	- 6 x 	- 1
8			31.193376248861494			x^3 - 	31 x^2 	- 6 x 	- 1 
9			45.7161148554393			x^3 - 	46 x^2 	+ 13 x 	- 1 
10			67.000222765306				^3 - 	67 x^2 	+ 0 x	- 1
11			98.1935990141675			x^3 - 	98 x^2 	- 19 x 	- 1
12			143.9097138696068			x^3 - 	144 x^2 + 13 x 	- 1
13			210.9099366349128			x^3 - 	211 x^2 + 19 x 	- 1

Notice that  the value of the constant 
with degree 2 = c1^2; 
with degree 3 = c1^3;
with degree n = c1^n
...

Generating rule from degree 2:
x^3 + a*x^2 + b*x - 1 where
a = items of A001609
b = items of 	A112455	* n form the 4th item (because of the shifting) of the A112455 sequence where
n = +1 if the degree is even
n = -1 if the degree is odd

Zoltan Galantai
Sept. 29. 2019