A332056 -id:A332056 - cp's OEIS frontend

A332057 Partial sums (and absolute value of first differences) of A332056: if odd (resp. even) add (resp. subtract) the partial sum to get the next term.

1, 3, 2, 3, 7, 4, 5, 11, 6, 7, 15, 8, 9, 19, 10, 11, 23, 12, 13, 27, 14, 15, 31, 16, 17, 35, 18, 19, 39, 20, 21, 43, 22, 23, 47, 24, 25, 51, 26, 27, 55, 28, 29, 59, 30, 31, 63, 32, 33, 67, 34, 35, 71, 36, 37, 75, 38, 39, 79, 40

Offset: 1

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

The terms show a 3-quasiperiodic pattern (2m-1, 4m-1, 2m), m = 1, 2, 3, ...

Or: group positive integers by pairs, then insert the sum of the pair between the two terms.

Antti Karttunen, Table of n, a(n) for n = 1..19683
Eric Angelini, Re: Add or subtract my cumulative sum of terms, SeqFan list, Feb 24 2020.
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A332056.

PARI
```
apply( {A332057(n)=n<
				
```

Vec(x*(1 + x)*(1 + 2*x + x^3) / ((1 - x)^2*(1 + x + x^2)^2) + O(x^60)) \\ Colin Barker, Feb 26 2020

a(3k-2) = 2k - 1, a(3k-1) = 4k - 1, a(3k) = 2k, for all k >= 1.
From Colin Barker, Feb 25 2020: (Start)
G.f.: x*(1 + x)*(1 + 2*x + x^3) / ((1 - x)^2*(1 + x + x^2)^2).
a(n) = 2*a(n-3) - a(n-6) for n>6.
(End)

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.

Original entry on oeis.org

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

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

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, ...

			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.

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.

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

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

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

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

A332059 Absolute value of first differences, or sum of digits of the first n terms of A332058.

Original entry on oeis.org

1, 3, 4, 7, 8, 10, 18, 26, 33, 46, 58, 68, 83, 91, 107, 122, 132, 146, 153, 169, 187, 203, 214, 232, 248, 269, 280, 299, 316, 334, 344, 356, 373, 394, 417, 437, 453, 466, 486, 497, 510, 517, 538, 548, 566, 583, 598, 609, 623

Offset: 1

Eric Angelini and M. F. Hasler, Feb 25 2020

Eric Angelini, Add or subtract my cumulative sum of digits, SeqFan list, Feb 24 2020.

See A332057 for the variant corresponding to A332056 instead of A332058.

A332059_vec(N)=abs((N=A332058_vec(N+1))[^1]-N[^-1])

cp's OEIS Frontend

A332057 Partial sums (and absolute value of first differences) of A332056: if odd (resp. even) add (resp. subtract) the partial sum to get the next term.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

A332059 Absolute value of first differences, or sum of digits of the first n terms of A332058.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI