A117997 -id:A117997 - cp's OEIS frontend

A061347 Period 3: repeat [1, 1, -2].

1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2

Offset: 1

Jason Earls, Jun 07 2001

WARNING: It is unclear whether this sequence should start at offset 1 (as written) or offset 0 (in analogy to many similar sequences, which seems to be assumed in many of the given formulas).
Inverse binomial transform of A057079. - Paul Barry, May 15 2003
The unsigned version, with g.f. (1 + x + 2*x^2)/(1 - x^3), has a(n) = 4/3 -cos(2*Pi*n/3)/3 - sqrt(3)*sin(2*Pi*n/3)/3 = gcd(Fib(n+4), Fib(n+1)). - Paul Barry, Apr 02 2004
a(n) = L(n-2,-1), where L is defined as in A108299; see also A010892 for L(n,+1). - Reinhard Zumkeller, Jun 01 2005
From the Taylor expansion of log(1 + x + x^2) at x = 1, Sum_{k > 0} a(k)/k = log(3) = A002391. This is case n = 3 of the general expression Sum_{k > 0} (1-n*!(k mod n))/k = log(n). - Jaume Oliver Lafont, Oct 16 2009
If used with offset zero, a non-simple continued fraction representation of 2+sqrt(2). - R. J. Mathar, Mar 08 2012
Periodic sequences of this type can be also calculated by a(n) = c + floor(q/(p^m-1)*p^n) mod p, where c is a constant, q is the number representing the periodic digit pattern and m is the period length. c, p and q can be calculated as follows: Let D be the array representing the number pattern to be repeated, m = size of D, max = maximum value of elements in D, min = minimum value of elements in D. Than c := min, p := max - min + 1 and q := p^m*Sum_{i=1..m} (D(i)-min)/p^i. Example: D = (1, 1, -2), c = -2, p = 4 and q = 60 for this sequence. - Hieronymus Fischer, Jan 04 2013
This is the Dirichlet inverse of A117997. - Petros Hadjicostas, Jul 25 2020

			G.f.: x + x^2 - 2*x^3 + x^4 + x^5 - 2*x^6 + x^7 + x^8 - 2*x^9 + ... - _Michael Somos_, Nov 27 2019

W. Florek, A class of generalized Tribonacci sequences applied to counting problems, Appl. Math. Comput., 338 (2018), 809-821.
Ralph E. Griswold, Shaft Sequences, 2001 (see also here).
Tanya Khovanova, Recursive Sequences.
W. O. J. Moser, Cyclic binary strings without long runs of like (alternating) bits, Fibonacci Quart. 31(1) (1993), 2-6.
Index entries for linear recurrences with constant coefficients, signature (-1,-1).

Apart from signs, same as A057079 (also bin. Transf), A100063. Cf. A000045, A010892 for the rules a(n) = a(n - 1) + a(n - 2), a(n) = a(n - 1) - a(n - 2). a(n) = - a(n - 1) + a(n - 2) gives a signed version of Fibonacci numbers.
Alternating row sums of A130777: repeat(1,-2,1).
Cf. A000032, A002391, A007040, A100886, A108299, A117997, A320769.

Flat(List([1..50],n->[1,1,-2])); # Muniru A Asiru, Aug 02 2018

&cat [[1, 1, -2]^^30]; // Wesley Ivan Hurt, Jul 01 2016

seq(op([1, 1, -2]), n=1..50); # Wesley Ivan Hurt, Jul 01 2016

a[n_] := {1, 1, -2}[[Mod[n - 1, 3] + 1]]; Table[a[n], {n, 108}] (* Jean-François Alcover, Jul 19 2013 *)
PadRight[{}, 90, {1, 1, -2}] (* After Harvey P. Dale, or *)
CoefficientList[ Series[(2x + 1)/(x^2 + x + 1), {x, 0, 89}], x]  (* or *)
LinearRecurrence[{-1, -1}, {1, 1}, 90] (* Robert G. Wilson v, Jul 30 2018 *)

a(n)=1-3*!(n%3) \\ Jaume Oliver Lafont, Oct 16 2009

def A061347():
    x, y = -1, -1
    while True:
        yield -x
        x, y = y, -x -y
a = A061347(); [next(a) for i in range(40)] # Peter Luschny, Jul 11 2013

With offset zero, a(n) = A057079(2n). a(n) = -a(n-1) - a(n-2) with a(0) = a(1) = 1.
From Mario Catalani (mario.catalani(AT)unito.it), Jan 07 2003: (Start)
G.f.: x*(1 + 2*x)/(1 + x + x^2).
a(n) = (-1)^floor(2n/3) + ((-1)^floor((2n-1)/3) + (-1)^floor((2n+1)/3))/2. (End)
a(n) = -2*cos(2*Pi*n/3). - Jaume Oliver Lafont, May 06 2008
Dirichlet g.f.: zeta(s)*(1-1/3^(s-1)). - R. J. Mathar, Feb 09 2011
a(n) = n * Sum_{k=1..n} binomial(k,n-k)/k*(-1)^(k+1). - Dmitry Kruchinin, Jun 03 2011
a(n) = -2 + floor(110/333*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 04 2013
a(n) = -2 + floor(20/21*4^(n+1)) mod 4. - Hieronymus Fischer, Jan 04 2013
a(n) = a(n-3) for n > 3. - Wesley Ivan Hurt, Jul 01 2016
E.g.f.: 2 - 2*cos(sqrt(3)*x/2)*exp(-x/2). - Ilya Gutkovskiy, Jul 01 2016
a(n) = (-1)^n*hypergeom([-n/2-1, -n/2-3/2], [-n-2], 4). - Peter Luschny, Dec 17 2016
a(n) = A000032(n) - A007040(n), for n > 1. - Wojciech Florek, Feb 20 2018

Better definition from M. F. Hasler, Jan 13 2013

A067856 Sum_{n >= 1} a(n)/n^s = 1/(Sum_{n >= 1} (-1)^(n + 1)/n^s).

Original entry on oeis.org

1, 1, -1, 2, -1, -1, -1, 4, 0, -1, -1, -2, -1, -1, 1, 8, -1, 0, -1, -2, 1, -1, -1, -4, 0, -1, 0, -2, -1, 1, -1, 16, 1, -1, 1, 0, -1, -1, 1, -4, -1, 1, -1, -2, 0, -1, -1, -8, 0, 0, 1, -2, -1, 0, 1, -4, 1, -1, -1, 2, -1, -1, 0, 32, 1, 1, -1, -2, 1, 1, -1, 0, -1, -1, 0, -2, 1, 1, -1, -8, 0, -1, -1, 2, 1, -1, 1, -4, -1, 0, 1, -2, 1, -1, 1, -16, -1, 0, 0, 0

Offset: 1

Leroy Quet, Feb 15 2002

Dirichlet inverse of A062157. - R. J. Mathar, Jul 15 2010
The first 31 terms equal the values of the Ramanujan sum c_n(8) -- see for example A085906 -- but a(32) <> c_{32}(8). - R. J. Mathar, Apr 02 2011
From  Peter Bala, Mar 12 2019: (Start)
Let Mu(n) = (-1)^(n+1)*a(n), an analog of the Möbius function mu(n). Then for arithmetic functions f(n) and g(n) we have the following analog of the Möbius inversion formula: f(n) = Sum_{d divides n} (-1)^((1+d)*(1+n/d))*g(d) iff g(n) = Sum_{d divides n} (-1)^((1+d)*(1+n/d))*Mu(n/d)*f(d).
Each of the following two equations implies the other: F(x) = Sum_{n >= 1} (-1)^(n+1)*G(n*x); G(x) = Sum_{n >= 1} a(n)*F(n*x). See G. Pólya and G. Szegő, Part V111, Chap. 1, Nos. 66-68.2. (End)
Let D(n) denote the set of partitions of n into distinct parts. Then Sum_{parts k in D(n)} a(k) = |D(n-1)| = A000009(n-1). For example, D(6) = {6, 1 + 5, 2 + 4, 1 + 2 + 3} and the sum a(6) + (a(1) + a(5)) + (a(2) + a(4)) + (a(1) + a(2) + a(3)) = 3 = |D(5)|. - Peter Bala, Mar 14 2019
From Petros Hadjicostas, Jul 25 2020: (Start)
For p prime >= 2, Petrogradsky (2003) defined the multiplicative functions 1_p and mu_p in the following way:
1_p(n) = 1 when gcd(n,p) = 1 and 1_p(n) = 1 - p when gcd(n,p) = p;
mu_p(n) = mu(n) when gcd(n,p) = 1 and mu_p(n) = mu(m)*(p^s - p^(s-1)) when n = m*p^s with gcd(m,p) = 1 and s >= 1.
We have 1_2(n) = A062157(n), 1_3(n) = A061347(n) (with offset 1), a(n) = mu_2(n), and A117997(n) = mu_3(n) for n >= 1.
Some of the results by other contributors can be generalized:
(i) Rogel's (1897) formula becomes Sum_{d | n} 1_p(d) * mu_p(n/d) = 0 for n > 1. Thus, 1_p is the Dirichlet inverse of mu_p.
(ii) R. J. Mathar's Dirichlet g.f. for mu_p becomes 1/(zeta(s) * (1 - p^(1-s))). The Dirichlet g.f. for 1_p is zeta(s) * (1 - p^(1-s)).
(iii) Benoit Cloitre's formula becomes 1 = Sum_{k=1..n} mu_p(k)*g_p(n/k), where g_p(x) = floor(x) - p*floor(x/p) = floor(x) mod p.
(iv) Paul D. Hanna's formula becomes Sum_{n >= 1} (mu_p(n)/n)*log((1 - x^(n*p))/(1 - x^n)) = x.
(v) The definition of A117997 generalizes to Sum_{d | n} mu_p(d) = n, if n = p^s for s >= 0, and = 0, otherwise.
(vi) By differentiating both sides of (iv) w.r.t. x and multiplying both sides by x, we get Sum_{n >= 1} mu_p(n)*(x^n + 2*x^(2*n) + ... + (p-1)*x^(n*(p-1)))/(1 + x^n + x^(2*n) + ... + x^(n*(p-1))) = x, which generalizes one of Peter Bala's formulas. It can be thought as a "generalized Lambert series".
(vii) Obviously, f(n) = Sum_{d | n} 1_p(n/d)*g(d) if and only if g(n) = Sum_{d | n} mu_p(n/d)*f(d). (End)

G. Pólya and G. Szegő, Problems and Theorems in Analysis Volume II. Springer_Verlag 1976.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Peter Bala, A signed Dirichlet product of arithmetical functions
V. M. Petrogradsky, Witt's formula for restricted Lie algebras, Advances in Applied Mathematics, 30 (2003), 219-227.
V. M. Petrogradsky, Witt's formula for restricted Lie algebras, Advances in Applied Mathematics, 30 (2003), 219-227.
Franz Rogel, Transformationen arithmetischer Reihen, S.-B. Kgl. Bohmischen Ges. Wiss. Article LI (1897), Prague (31 pages); see pp. 10-11 and especially Eqs. (21) - (24). [There are some obvious typos there; especially Eq. (24) should become Sum_{t | v} (-1)^(v/t) * c(t) = 0 for v > 1, which is the equation a(n) = Sum{k | n, 1 < k} (-1)^k a(n/k), for n >= 2, in the Formula section below. - _Petros Hadjicostas_, Jul 21 2019]

Cf. A000009, A038712, A038838, A048298 (inverse Mobius transform), A061347, A062157, A085906, A117997, A321088 (Euler transform), A321558.

a[n_] := If[n == 1, 1, Product[{p, e} = pe; If[2 == p, e--, If[e > 1, p = 0, p = -1]]; p^e, {pe, FactorInteger[n]}]];
a /@ Range[1, 100] (* Jean-François Alcover, Oct 01 2019 *)

{a(n)=local(k); if(n<1, 0, k=valuation(n, 2); moebius(n/2^k)*2^max(0, k-1))} /* Michael Somos, Aug 22 2006 */

A067856(n) =  { my(f=factor(n)); for(i=1,#f~,if(2==f[i,1],f[i,2]--,if(f[i,2]>1,f[i,1]=0,f[i,1]=-1))); factorback(f); }; \\ Antti Karttunen, Sep 27 2018, after Vladeta Jovovic_'s multiplicative formula.

a(1) = 1 and a(n) = Sum{k | n, 1 < k} (-1)^k a(n/k) for n >= 2; the sum is over the divisors, k, of n, where k > 1. If n is odd, a(n) = mu(n), where mu(.) is the Moebius function. If n is even, a(n) = mu(m)* 2^(k-1), where n = m*2^k, m is odd integer, and k is a positive integer.
Sum_{n > 0} a(n)*x^n/(1 + x^n) = x. Moebius transform of A048298. Multiplicative with a(2^e) = 2^(e - 1), a(p) = -1 for p > 2, a(p^e) = 0 for p > 2 and e > 1. - Vladeta Jovovic, Jan 02 2003
Sum_{n > 0} a(n)*log(1 + x^n)/n = x. - Paul D. Hanna, May 06 2003
a(n) = 0 if and only if n is divisible by the square of an odd prime (A038838). - Michael Somos, Aug 22 2006
1 = Sum_{k=1..n} a(k)*g(n/k), where g(x) = floor(x) - 2*floor(x/2). - Benoit Cloitre, Nov 11 2010
Dirichlet g.f.: 1/( zeta(s) * (1 - 2^(1-s)) ). - R. J. Mathar, Apr 02 2011
From Peter Bala, Mar 13 2019: (Start)
Sum_{n >= 1} a(n)*x^n/(1 + x^n) = x
Sum_{n >= 1} a(n)*x^n/(1 - x^n) = x + 2*x^2 + 4*x^4 + 8*x^8 + 16*x^16 + ...
Sum_{n >= 1} a(n)*x^n/(1 + (-x)^n) = x + 2*(x^2 + x^4 + x^8 + x^16 + ...)
Sum_{n >= 1} a(n)*x^n/(1 - (-x)^n) = x + 2*(x^4 + 3*x^8 + 7*x^16 + 15*x^32 + ...). (End)
G.f. A(x) satisfies: A(x) = x + A(x^2) - A(x^3) + A(x^4) - A(x^5) + ... - Ilya Gutkovskiy, May 11 2019
Sum_{k=1..n} abs(a(k)) ~ 2*n*(log(n) - 1 + gamma + 11*log(2)/6 - 12*zeta'(2)/Pi^2) / (log(2)*Pi^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 30 2024

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A061347 Period 3: repeat [1, 1, -2].

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A067856 Sum_{n >= 1} a(n)/n^s = 1/(Sum_{n >= 1} (-1)^(n + 1)/n^s).

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