A332058 a(1) = 1; a(n+1) = a(n) +- (sum of digits of a(1) up to a(n)), with "+" when a(n) is odd, or "-" if even.

1, 2, -1, 3, 10, 2, -8, -26, -52, -85, -39, 19, 87, 170, 79, 186, 64, -68, -214, -367, -198, -385, -182, -396, -628, -876, -1145, -865, -566, -882, -1216, -1560, -1916, -2289, -1895, -1478, -1915, -1462, -1928, -2414, -2911, -2401

Offset: 1

Author

Eric Angelini and M. F. Hasler, Feb 24 2020

Keywords

• sign
• base

Comments

The graph appears to have a shape similar to that of Mertens function A002321, with increasingly large "mountains" and "valleys":

Successive record values of opposite sign are a(2) = 2, a(3) = -1, a(5) = 10, a(10) = -85, a(16) = 186, a(222) = -75573, a(391) = 26186, a(658) = -341791, a(987) = 134304, a(1831) = -1820815, a(2476) = 393048, a(2692) = -2089141, a(3321) = 1816290, a(6114) = -8650189, ...

Examples

			a(1) = 1 is odd, so we add the partial sum (so far equal to a(1)) to get the next term, a(2) = 2.
Now a(2) = 2 is even, so we subtract the sum of the digits of a(1) and a(2), 1 + 2 = 3 to get a(3) = -1.
Since a(3) = -1 is odd, we add the sum of the digits of a(1), a(2) and a(3), 1 + 2 + 1 = 4 to get a(4) = 3.
And so on.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..10000
• Eric Angelini, Re: Add or subtract my cumulative sum of digits, SeqFan list, Feb 24 2020.

Crossrefs

See A332056 for the variant considering sum of a(n) instead of digits.

Programs

Mathematica

Nest[Append[#, #[[-1]] + (2 Boole[OddQ@ #[[-1]] ] - 1)*Total[Flatten@ IntegerDigits[#]] ] &, {1}, 41] (* Michael De Vlieger, Feb 25 2020 *)

PARI

A332058_vec(N,a=1,s=-a)={vector(N,n, a-=(-1)^a*s+=sumdigits(a))}

Python

from itertools import count, islice
def agen(): # generator of terms
    an, s = 1, 1
    while True:
        yield an
        an = an + s if an&1 else an - s
        s += sum(map(int, str(abs(an))))
print(list(islice(agen(), 42))) # Michael S. Branicky, Oct 14 2024