A055101 Expansion of square of continued fraction 1/ ( 1+q/ ( 1+q^2/ ( 1+q^3/ ( 1+q^4/... )))).

1, -2, 3, -2, -1, 4, -6, 6, -3, -2, 9, -16, 17, -10, -5, 24, -36, 36, -21, -10, 46, -74, 77, -42, -22, 94, -144, 142, -78, -38, 172, -266, 266, -146, -73, 312, -471, 464, -251, -122, 534, -814, 801, -432, -213, 910, -1364, 1328, -713, -344, 1485, -2234, 2178

Offset: 0

Author

N. J. A. Sloane, Jun 14 2000

Keywords

• sign
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• For the third power see G. E. Andrews, Simplicity and surprise in Ramanujan's "Lost" Notebook, Amer. Math. Monthly, 104 (No. 10, Dec. 1997), 918-925.

Crossrefs

Product_{k>0} ((1-x^{5k-1}) * (1-x^{5k-4})/((1-x^{5k-2}) * (1-x^{5k-3})))^m: A285444 (m=-4), A285443 (m=-3), A285442 (m=-2), A003823 (m=-1), A007325 (m=1), this sequence (m=2), A055102 (m=3), A055103 (m=4).

Cf. A109091, A340453, A340454, A340455, A340456.

Formula

a(0) = 1, a(n) = -(2/n)*Sum_{k=1..n} A109091(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 16 2017

Euler transform of period 5 sequence [-2, 2, 2, -2, 0, ...]. - Georg Fischer, Aug 18 2020

From Seiichi Manyama, Jul 29 2024: (Start)

G.f.: ( Sum_{k in Z} x^(3*k) / (1 - x^(5*k+1)) ) / ( Sum_{k in Z} x^k / (1 - x^(5*k+1)) ).

G.f.: ( Sum_{k in Z} x^(2*k) / (1 - x^(5*k+2)) ) / ( Sum_{k in Z} x^k / (1 - x^(5*k+2)) ). (End)

Extensions

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 20 2000