A208176 a(n) = F(n+1)^2, if n>=0 is even (F=A000045) and a(n) = (L(2n+2)+8)/5, if n is odd (L=A000204).

1, 3, 4, 11, 25, 66, 169, 443, 1156, 3027, 7921, 20738, 54289, 142131, 372100, 974171, 2550409, 6677058, 17480761, 45765227, 119814916, 313679523, 821223649, 2149991426, 5628750625, 14736260451, 38580030724, 101003831723, 264431464441, 692290561602

Offset: 0

Author

Vladimir Shevelev and Peter J. C. Moses, Feb 24 2012

Keywords

• nonn

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

Crossrefs

Cf. A000045, A000204, A007598, A207890.

Programs

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]

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