A332057 -id:A332057 - cp's OEIS frontend

A332056 a(1) = 1, then a(n+1) = a(n) - (-1)^a(n) Sum_{k=1..n} a(k): if a(n) is odd, add the partial sum, else subtract.

1, 2, -1, 1, 4, -3, 1, 6, -5, 1, 8, -7, 1, 10, -9, 1, 12, -11, 1, 14, -13, 1, 16, -15, 1, 18, -17, 1, 20, -19, 1, 22, -21, 1, 24, -23, 1, 26, -25, 1, 28, -27, 1, 30, -29, 1, 32, -31, 1, 34, -33, 1, 36, -35, 1, 38, -37, 1, 40, -39

Offset: 1

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

sign

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

Conjecture from Barker confirmed by recurrence formula. - Ray Chandler, Jul 31 2025

			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 partial sum 1 + 2 = 3 to get a(3) = -1.
And so on.

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 (-1,-1,1,1,1).

See A332057 for the partial sums.

PARI
```
s=-a=1; vector(100,n, a-=(-1)^a*s+=a)
```

apply( {A332056(n)=[1-n\3*2,1,n\/3*2][n%3+1]}, [1..99])

a(3k-2) = 1, a(3k-1) = 2k, a(3k) = 1 - 2k, for all k >= 1.
From Colin Barker, Feb 25 2020: (Start)
G.f.: x*(1 + x)*(1 + 2*x + x^3) / ((1 - x)*(1 + x + x^2)^2).
a(n) = -a(n-1) - a(n-2) + a(n-3) + a(n-4) + a(n-5) for n>5.
(End)

A385938 a(n) = 2n/3 if n == 0 (mod 3), (2n+1)/3 if n == 1 (mod 3), (7*n+1)/3 if n == 2 (mod 3).

Original entry on oeis.org

0, 1, 5, 2, 3, 12, 4, 5, 19, 6, 7, 26, 8, 9, 33, 10, 11, 40, 12, 13, 47, 14, 15, 54, 16, 17, 61, 18, 19, 68, 20, 21, 75, 22, 23, 82, 24, 25, 89, 26, 27, 96, 28, 29, 103, 30, 31, 110, 32, 33, 117, 34, 35, 124, 36, 37, 131, 38, 39, 138, 40, 41, 145, 42, 43, 152, 44, 45, 159

Offset: 0

Miquel Cerda, Jul 13 2025

Ternary modular function with three cases based on residue modulo 3.
The function defines a dynamical system with multiple periodic attractors.
Fixed point at 1: a(1) = 1.
From Miquel Cerda, Aug 06 2025: (Start)
Every nonnegative integer k appears at least once as a value in the sequence.
Inverse formulas (for possible preimages):
If k is even: one preimage is n = 3*k/2.
If k is odd: one preimage is n = (3*k - 1)/2.
If k == 5 (mod 7): there is an additional preimage: n = 3*(k - 5)/7 + 2. (End)

			a(0) = 2*0/3 = 0. a(1) = (2*1+1)/3 = 1. a(2) = (7*2+1)/3 = 5. a(3) = 2*3/3 = 2.

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A385893 (cycle of length 130 in this dynamical system).

Cf. A332057 (near definition).

a[x_] := Which[Mod[x, 3] == 0, 2*x/3, Mod[x, 3] == 1, (2*x + 1)/3, Mod[x, 3] == 2, (7*x + 1)/3]; Table[a[n], {n, 0, 50}]

a(n) = if(n%3==0, 2*n/3, if(n%3==1, (2*n+1)/3, (7*n+1)/3))

def A385938(n):
    q, r = divmod(n,3)
    return (q<<1)+r if r<2 else 7*q+5 # Chai Wah Wu, Jul 17 2025

G.f.: x*(1+5*x+2*x^2+x^3+2*x^4) / ( (x-1)^2*(1+x+x^2)^2 ). - R. J. Mathar, Jul 30 2025

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

A332056 a(1) = 1, then a(n+1) = a(n) - (-1)^a(n) Sum_{k=1..n} a(k): if a(n) is odd, add the partial sum, else subtract.

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A385938 a(n) = 2n/3 if n == 0 (mod 3), (2n+1)/3 if n == 1 (mod 3), (7*n+1)/3 if n == 2 (mod 3).

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Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

cp's OEIS Frontend

A332056 a(1) = 1, then a(n+1) = a(n) - (-1)^a(n) Sum_{k=1..n} a(k): if a(n) is odd, add the partial sum, else subtract.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A385938 a(n) = 2*n/3 if n == 0 (mod 3), (2*n+1)/3 if n == 1 (mod 3), (7*n+1)/3 if n == 2 (mod 3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A385938 a(n) = 2n/3 if n == 0 (mod 3), (2n+1)/3 if n == 1 (mod 3), (7*n+1)/3 if n == 2 (mod 3).