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A336640 a(n) is the minimal value of Sum x_i when Sum binomial(x_i, 2) = n. 1
0, 2, 4, 3, 5, 7, 4, 6, 8, 7, 5, 7, 8, 8, 10, 6, 8, 10, 9, 11, 10, 7, 9, 11, 10, 11, 13, 11, 8, 10, 12, 11, 13, 15, 12, 14, 9, 11, 13, 12, 14, 16, 13, 14, 16, 10, 12, 14, 13, 15, 17, 14, 16, 18, 17, 11, 13, 15, 14, 16, 16, 15, 17, 19, 17, 16, 12, 14, 16, 15, 17 (list; graph; refs; listen; history; text; internal format)
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

Cf. A000217, A061336.

Sequence in context: A338835 A210536 A102767 * A258238 A166266 A011295

Adjacent sequences:  A336637 A336638 A336639 * A336641 A336642 A336643

KEYWORD

nonn,easy

AUTHOR

Mara Hashuga, Megan Herbine, Alathea Jensen, Jul 27 2020

EXTENSIONS

More terms from Jinyuan Wang, Jul 29 2020

STATUS

approved

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Last modified January 17 19:43 EST 2022. Contains 350403 sequences. (Running on oeis4.)