A147658 (1, 2, -4, 6, -8, ...) interleaved with (3, -3, 3, -3, 3, ...).

1, 3, 2, -3, -4, 3, 6, -3, -8, 3, 10, -3, -12, 3, 14, -3, -16, 3, 18, -3, -20, 3, 22, -3, -24, 3, 26, -3, -28, 3, 30, -3, -32, 3, 34, -3, -36, 3, 38, -3, -40, 3, 42, -3, -44, 3, 46, -3, -48, 3, 50, -3, -52, 3, 54, -3, -56, 3, 58, -3, -60, 3, 62, -3, -64

Offset

1,2

Comments

POLYMOTZKINT A147657 = [1,2,3,...].

POLYMOTZKINTINV operation on [1,3,5,7,...], such that POLYMOTZKINT a(n) = [1,3,5,7,...].

Cf. A005717 for an example of the POLYMOTZKINT operation.

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

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

Formula

a(1) = 1; a(2*k) = 3*(-1)^(k-1); a(2*k+1) = 2*(-1)^(k-1)*k for k >= 1. - Georg Fischer, Nov 02 2021

From G. C. Greubel, Dec 13 2025: (Start)

a(n+2) = (6*a(n+1) - (n^2 + n + 9)*a(n))/(n^2 - n + 9), with a(1) = 1, a(2) = 3, and a(3) = 2.

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

Maple

with(ListTools): Flatten([1, seq([(-1)^(k-1)*3, (-1)^(k-1)*2*k], k=1..72)]); # Georg Fischer, Nov 02 2021

Mathematica

LinearRecurrence[{0, -2, 0, -1}, {1, 3, 2, -3, -4}, 70] (* G. C. Greubel, Dec 13 2025 *)

Prog

(Magma)
A147658:= function(n)
    if n eq 1 then return 1;
    elif (n mod 2) eq 0 then return 3*(-1)^(Floor(n/2)-1);
    else return (-1)^Floor((n-3)/2)*(n-1);
end if; end function;
[A147658(n): n in [1..70]]; // G. C. Greubel, Dec 13 2025
(SageMath)
def A147658(n):
    if n==1: return 1
    elif (n%2==0): return 3*(-1)^((n//2)-1)
    else: return (-1)^((n-3)//2)*(n-1)
print([A147658(n) for n in range(1, 71)]) # G. C. Greubel, Dec 13 2025

Crossrefs

Cf. A005717, A147657.

Sequence in context: A233386 A093407 A349351 * A221529 A105161 A383972

Adjacent sequences: A147655 A147656 A147657 * A147659 A147660 A147661

Keyword

easy,sign

Author

Gary W. Adamson, Nov 09 2008

Extensions

a(25) ff. corrected by Georg Fischer, Nov 02 2021

Status

approved