OFFSET
0,2
COMMENTS
a(n) = Min_{m in the integers such that m*c+n*b is in S} where n is greater than or equal to 0, n is less than c, where S is an infinite numerical semigroup generated by {y_0, y_1, ...}, and c and b are set natural number values, y_n = n*c + binomial(n, 2)*b. a(n) can be used to find the Apéry set of S. Ap(s,c) = {a(n)*c+n*b for n = 0, 1, ..., c-1}.
Ap(S,c) = {a(n)*c+n*b | n = 0, 1, 2, ...}.
a(n) is a general value, however for some n, b, and c values, there is an m value less than the general a(n). This value is denoted a_c,b(n). For (c,b,n) = (29,1,26), (45,1,33), (47,1,44), (50,1,41), (55,1,50), (67,1,53), (73,1,63), or (79,1,74), a_c,b(n) = a(n)-1.
LINKS
David A. Corneth, Table of n, a(n) for n = 0..10000
Mara Hashuga, Megan Herbine, Alathea Jensen, Numerical Semigroups Generated by Quadratic Sequences, arXiv:2009.01981 [math.GR], 2020.
EXAMPLE
If n = 2, then n = binomial(2,2) + binomial(2,2) is the only way to write n = 2 as a sum of binomial coefficients. So x_1 = 2 and x_2 = 2, making a(n) = x_1 + x_2 = 4.
For n=273, x's list 23, 5, 5 has binomial(23,2) + binomial(5,2) + binomial(5,2) = 273 = n. The sum of these x's is 23+5+5 = 33. No x's with a smaller sum (of x's) gives 273, so a(273) = 33.
PROG
(Python)
f = open("mu(n, mu).txt", "a")
N = 10000
mu = [0]
x = []
f.write("0 0\n")
for n in range(1, N):
for i in range(2, N):
iChoose2 = (i*(i-1))/2
if iChoose2 <= n:
x.append(mu[int(n-iChoose2)]+i)
mu.append(min(x))
f.write(str(n)+" "+str(min(x))+"\n")
x.clear()
f.close()
(PARI) lista(nn) = {my(mu=vector(nn), t, x); for(n=2, nn, x=[]; for(i=2, n, if((t=binomial(i, 2))<n, x=concat(x, mu[n-t]+i))); mu[n]=vecmin(x)); mu; } \\ Jinyuan Wang, Jul 29 2020
(Haskell)
a336640_list = map a336640 [0..]
a336640 0 = 0
a336640 n = minimum $ map (\(i, t) -> i + (a336640_list !! (n - t))) triangular where
triangular = takeWhile (\(_, m) -> m <= n) $ map t [2..] where
t i = (i, i*(i-1) `div` 2)
-- Peter Kagey, Sep 20 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Jinyuan Wang, Jul 29 2020
STATUS
approved