A345190 Number of rows with the value "true" in the Kleene truth tables of all bracketed formulae with n distinct propositions p1, ..., pn connected by the binary connective of implication.

1, 5, 30, 229, 1938, 17530, 165852, 1621133, 16242474, 165923854, 1721675460, 18095802306, 192256162740, 2061367432212, 22276538889912, 242387718986301, 2653259550491034, 29198054511893638, 322835545567447092, 3584671507685675894, 39955514234936341980, 446897274497509974508

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..925

Volkan Yildiz, Notes on algebraic structure of truth tables of bracketed formulae connected by implications, arXiv:2106.04728 [math.CO], 2021.

Formula

G.f.: (4-sqrt(1-12*x)-sqrt(5+24*x+4*sqrt(1-12*x)))/6.

a(n) = 2*A005159(n-1) - A345189(n). - G. C. Greubel, May 20 2022

Mathematica

CoefficientList[Series[(4 -Sqrt[1-12*x] -Sqrt[5 +24*x +4*Sqrt[1-12*x]])/6, {x, 0, 40}], x]//Rest (* G. C. Greubel, May 20 2022 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((4-sqrt(1-12*x)-sqrt(5+24*x+4*sqrt(1-12*x)))/6)
(SageMath)
def A345190_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (4-sqrt(1-12*x)-sqrt(5+24*x+4*sqrt(1-12*x)))/6 ).list()
a=A345190_list(40); a[1:] # G. C. Greubel, May 20 2022

Crossrefs

Cf. A005159 (unknown rows, shifted), A025226 (all rows), A345189 (false rows).

Sequence in context: A318920 A363908 A167892 * A144498 A201368 A072213

Adjacent sequences: A345187 A345188 A345189 * A345191 A345192 A345193

Keyword

nonn

Author

Michel Marcus, Jun 10 2021

Status

approved