A325904 Generator sequence for A100982.

1, 0, -3, -8, 15, -91, -54, 2531, -17021, 43035, -66258, 1958757, -24572453, 146991979, -287482322, -3148566077, 35506973089, -198639977241, 1006345648929, -8250266425561, 76832268802555, -517564939540551, 1890772860334557, 3323588929061820, -104547561696315008, 907385094824827328, -6313246535826877248

Offset: 0

Author

Benjamin Lombardo, Sep 08 2019

Keywords

• sign

Comments

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.

Links

• Mike Winkler, The algorithmic structure of the finite stopping time behavior of the 3x + 1 function, arXiv:1709.03385 [math.GM], 2017.

Crossrefs

Cf. A020914, A076227, A100982.

Programs

Python

import math
numberOfTerms = 20
L6 = [1,0]
def c(n):
    return math.floor(n/(math.log2(3)-1))
def p(a,b):
    return math.factorial(a)/(math.factorial(a-b)*math.factorial(b))
def anotherTerm(newTermCount):
    global L6
    for a in range(newTermCount+1-len(L6)):
        y = len(L6)
        newElement = 0
        for k in range(y):
            newElement -= int(L6[k]*p(c(y)+y-k-2, c(y)-2))
        L6.append(newElement)
anotherTerm(numberOfTerms)
print("A325904")
for a in range(numberOfTerms+1):
    print(a, "|", L6[a])

SageMath

@cached_function
def a(n):
    if n < 2: return 0^n
    A = floor(n/(log(3, 2) - 1)) - 2
    return -sum(a(k)*binomial(A + n - k, A) for k in (0..n-1))
[a(n) for n in range(100)] # Peter Luschny, Sep 10 2019

Formula

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.