OFFSET
0,2
COMMENTS
Consider the multivariate polynomial quotient ring K'_n = Z[x_1, x_2, x_3, ..., x_n]/I where I = <x_1^2 - P_1, x_2^2 - P_2, ..., x_n^2 - P_n> is an ideal in Z[x_1, x_2, x_3, ..., x_n]. Here, each polynomial P_i = -2x_i + x_{i+1} for 0 < i <= n, with x_{n+1} assumed to be 1. In this ring, every variable x_i for 0 < i <= n satisfies the recursive relation x_i^2 = -2x_i + x_{i+1}. The n-th term of this sequence is obtained by expanding the polynomial (2 + x_1)^n within the ring K'_n and evaluating at x_1 = x_2 = ... = x_n = 1. For a detailed explanation and proof, refer to Shunia's paper under links.
LINKS
Joseph M. Shunia, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Plot of a(n)/a(n-1) for n = 2..2000
Joseph M. Shunia, A Polynomial Ring Connecting Central Binomial Coefficients and Gould's Sequence, arXiv:2312.00302 [math.GM], 2023.
FORMULA
MATHEMATICA
Table[Sum[Binomial[n, k] * 2^DigitCount[k, 2, 1], {k, 0, n}], {n, 0, 32}] (* Vaclav Kotesovec, Apr 02 2024 *)
PROG
(PARI) {a(n) = sum(k=0, n, binomial(n, k) * 2^hammingweight(k))};
(Sage)
def a(n):
R = PolynomialRing(ZZ, n, 'x')
x = R.gens()
I_list = [x[i]^2 - (-2*x[i] + x[i+1]) if i < n-1 else x[i]^2 for i in range(n)]
I = R.ideal(I_list)
K_n = R.quotient(I, 'x')
p_n = K_n((x[0]+2)^n)
subs_dict = {x[i]: 1 for i in range(n)}
a_n = p_n.lift().subs(subs_dict)
return a_n # Joseph M. Shunia, Mar 22 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Joseph M. Shunia, Jan 02 2024
STATUS
approved