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%I #33 Oct 17 2019 13:57:55
%S 1,0,-3,-8,15,-91,-54,2531,-17021,43035,-66258,1958757,-24572453,
%T 146991979,-287482322,-3148566077,35506973089,-198639977241,
%U 1006345648929,-8250266425561,76832268802555,-517564939540551,1890772860334557,3323588929061820,-104547561696315008,907385094824827328,-6313246535826877248
%N Generator sequence for A100982.
%C The name of this sequence is derived from its main purpose as a formula for A100982 (see link). Both formulas below stem from Mike Winkler's 2017 paper on the 3x+1 problem (see below), in which a recursive definition of A100982 and A076227 is created in 2-D space. These formulas redefine the sequences in terms of this 1-D recursive sequence.
%H Mike Winkler, <a href="https://arxiv.org/abs/1709.03385">The algorithmic structure of the finite stopping time behavior of the 3x + 1 function</a>, arXiv:1709.03385 [math.GM], 2017.
%F a(0)=1, a(1)=0, a(n) = -Sum_{k=0..n-1} a(k)*binomial(A325913(n)+n-k-2, A325913(n)-2) for n>1.
%o (Python)
%o import math
%o numberOfTerms = 20
%o L6 = [1,0]
%o def c(n):
%o return math.floor(n/(math.log2(3)-1))
%o def p(a,b):
%o return math.factorial(a)/(math.factorial(a-b)*math.factorial(b))
%o def anotherTerm(newTermCount):
%o global L6
%o for a in range(newTermCount+1-len(L6)):
%o y = len(L6)
%o newElement = 0
%o for k in range(y):
%o newElement -= int(L6[k]*p(c(y)+y-k-2, c(y)-2))
%o L6.append(newElement)
%o anotherTerm(numberOfTerms)
%o print("A325904")
%o for a in range(numberOfTerms+1):
%o print(a, "|", L6[a])
%o (SageMath)
%o @cached_function
%o def a(n):
%o if n < 2: return 0^n
%o A = floor(n/(log(3, 2) - 1)) - 2
%o return -sum(a(k)*binomial(A + n - k, A) for k in (0..n-1))
%o [a(n) for n in range(100)] # _Peter Luschny_, Sep 10 2019
%Y Cf. A020914, A076227, A100982.
%K sign
%O 0,3
%A _Benjamin Lombardo_, Sep 08 2019
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