A100321 The trinomial transform (A027907) gives powers of 2, while the trinomial transform of this sequence shift one place left gives powers of 3.

1, 1, 0, 2, -3, 8, -16, 35, -72, 150, -307, 628, -1276, 2587, -5228, 10546, -21235, 42704, -85784, 172179, -345344, 692286, -1387155, 2778492, -5563748, 11138443, -22294596, 44617850, -89282067, 178639160, -357399712, 714995843, -1430309496, 2861133222, -5723098483, 11447543236

Offset: 0

Author

Paul D. Hanna, Nov 15 2004

Keywords

• sign

Examples

			2^3 = 1*(1) + 3*(1) + 6*(0) + 7*(2) + 6*(-3) + 3*(8) + 1*(-16).
3^3 = 1*(1) + 3*(0) + 6*(2) + 7*(-3) + 6*(8) + 3*(-16) + 1*(35).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-2,2,3,-2).

Crossrefs

Cf. A000045, A027907.

Cf. A000079, A000244.

Programs

Magma

[((-1)^n*(3*Fibonacci(n-1) -2^n) +1)/3: n in [0..40]]; // G. C. Greubel, Feb 01 2023

Mathematica

LinearRecurrence[{-2,2,3,-2}, {1,1,0,2}, 41] (* G. C. Greubel, Feb 01 2023 *)

PARI

a(n)=polcoeff((1+3*x-3*x^3)/(1+2*x-2*x^2-3*x^3+2*x^4+x*O(x^n)),n)

SageMath

def A100321(n): return ((-1)^n*(3*fibonacci(n-1) -2^n) +1)/3
[A100321(n) for n in range(41)] # G. C. Greubel, Feb 01 2023

Formula

G.f.: (1 + 3*x - 3*x^3) / (1 + 2*x - 2*x^2 - 3*x^3 + 2*x^4).

2^n = Sum_{k=0..2*n} A027907(n, k)*a(k).

3^n = Sum_{k=0..2*n} A027907(n, k)*a(k+1).

a(n) = (1/3)*((-1)^n*(3*Fibonacci(n-1) - 2^n) + 1). - Ralf Stephan, May 15 2007