A347365 a(n) = n * (2-(-1)^n), or zero together with first differences of even triangular numbers halved (A074378).

0, 3, 2, 9, 4, 15, 6, 21, 8, 27, 10, 33, 12, 39, 14, 45, 16, 51, 18, 57, 20, 63, 22, 69, 24, 75, 26, 81, 28, 87, 30, 93, 32, 99, 34, 105, 36, 111, 38, 117, 40, 123, 42, 129, 44, 135, 46, 141, 48, 147, 50, 153, 52, 159, 54, 165, 56, 171, 58, 177, 60, 183, 62, 189, 64

Offset: 0

Author

Federico Provvedi, Aug 29 2021

Keywords

• nonn
• easy

Comments

This sequence and A165998 form a complementary pair as solutions of alternating sequences a(n) + b(n) = 4*n (A008586), and a(n)*b(n) = 3*n^2 (A033428).

This is the particular case of the two integer sequences x(n)=2n and y(n)=n, where more generally, x(n) + y(n) = 2*a(n) and x(n)*y(n) = (a(n) + b(n)) * (a(n) - b(n)) give the two conjugate binomials a(n) = x(n) + (-1)^n*y(n) and b(n) = x(n) - (-1)^n*y(n) as solutions over the integer domain.

a(n) is also A005843 and A016945 interleaved.

For every integer k: a(n*k) = n*k is multiplicative for nonnegative even integers n and a(n*k) = n*a(k) for nonnegative odd integers n.

For every nonnegative odd integer k, the k-th difference of a(k*n)/k = (2n+1)*(-1)^n + 2 = A166519(n), and 1 for all nonnegative even integers.

a(6n+1)/3 = 6n+1, and a(6n+5)/3 = 6n+5, related to Collatz Conjecture.

Half-periods of a(k) mod n is A026741(n).

Links

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

Crossrefs

Cf. A074378, A165998, A008586, A033428, A237420, A193356, A016921, A016969, A166519, A014682, A005843, A016945, A010684, A008585, A000027, A026741.

Programs

Mathematica

Table[n(2-(-1)^n),{n,0,99}] (* or *)
LinearRecurrence[{0,2,0,-1}, {0,3,2,9}, 100] (* or *)
If[EvenQ@#,#,3#]&/@Range[0,99]  (* or *)
Drop[Flatten@Transpose[{2#,6#+3}&@Range[0,Quotient[#,2]]],-Boole@EvenQ@#]&@(10^2)

PARI

a(n) = n*(2-(-1)^n); \\ Michel Marcus, Sep 13 2021

Sage

(x*(3+2*x+3*x^2)/(1-x^2)^2).series(x,65).coefficients(x,sparse=False) # Stefano Spezia, Aug 30 2021

Formula

G.f.: x*(3 + 2*x + 3*x^2)/(1 - x^2)^2.

E.g.f.: x*(3*cosh(x) + sinh(x)).

Dirichlet g.f.: 2^(-s) * (3*2^s - 4) * zeta(s-1) = (3 - 4/2^s) * zeta(s-1) = (3 - 1/2^(s-2)) * zeta(s-1).

a(n) = n*(2-(-1)^n) = 3*n / (2+(-1)^n).

a(n) = 3*n if n odd, a(n) = n if n even, implies a(a(2n)) = 2n, a(a(2n+1)) = 9*a(2n+1).

a(n) = 3*b(n), if n odd and a(n) = b(n)/3, if n even, with b(n) = A165998(n).

a(n) = a(a(2k*n)/(2k)) = a((2k+1)*n) / (2k+1), since a(2*k*n) / (2*k) = n.

a(n) = 4*n - A165998(n).

a(n+1) = a(n) + A086970(n+1)*(-1)^n.

a(n) = 2*A014682(n) - A000035(n).

a(n) = n*A010684(n). - Michel Marcus, Sep 13 2021

For positive integers k and n, a(n) = A(n,1) = n * (A(n,k)/n)^(1/k), where the k-th nesting composition A(n,k) = a(a(...a(a(n))...)) = n * ( a(n) / n )^k, and d.g.f. of A(n,k) = (2^(1-s) + (1-2^(1-s))*3^k) * zeta(s-1). - Federico Provvedi, Sep 18 2021

a(n+1) = A165998(n)*(1 + 1/n). - Federico Provvedi, Sep 19 2021