OFFSET
0,2
COMMENTS
The row sums of Pascal-like triangle with the left side {1,1,1,...} and the right side (a(0), a(1), a(2),...) are F(n+2)^2 (cf. A007598).
Triangle begins:
n/k.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....3
.2..|..1.....4.....4
.3..|..1.....5.....8....11
.4..|..1.....6....13....19....25
.5..|..1.....7....19....32....44....66
.6..|..1.....8....26....51....76...110...164
.7..|..1.....9....34....77...127...186...279....443
.8..|
This sequence is associated with the identity arctan(1/(a+b)) = arctan(1/a) - arctan(b/(a^2+a*b+1)) (which is due to Euler, see the reference to Beckman), let a = F(n) and b = F(n+1). - Gary Detlefs, Apr 18 2012
REFERENCES
Petr Beckman, The History of Pi, Golem Press, 1977, p. 154
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..725
FORMULA
G.f.: -(2*x^3-5*x^2+1)/(x^4-3*x^3+3*x-1). - Alois P. Heinz, Feb 24 2012
a(n) = F(n)^2 + F(n)*F(n+1) + 1. - Gary Detlefs, Apr 18 2012
MAPLE
a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <1|-3|0|3>>^n.
<<1, 3, 4, 11>>)[1, 1]:
seq (a(n), n=0..30); # Alois P. Heinz, Feb 24 2012
MATHEMATICA
rows={{1}, {1, 3}}; Table[(x=Flatten[{1, 2MovingAverage[rows[[n]], 2]}]; z=If[EvenQ[n], Fibonacci[n+1]^2, (8+LucasL[(2n+2)])/5]; rows=Append[rows, Append[x, z]]), {n, 2, 15}]; A208176 = Map[Last[#] &, rows]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Shevelev and Peter J. C. Moses, Feb 24 2012
STATUS
approved