A130750 Binomial transform of A010882.

1, 3, 8, 17, 33, 64, 127, 255, 512, 1025, 2049, 4096, 8191, 16383, 32768, 65537, 131073, 262144, 524287, 1048575, 2097152, 4194305, 8388609, 16777216, 33554431, 67108863, 134217728, 268435457, 536870913, 1073741824, 2147483647

Offset: 0

Author

Paul Curtz, Jul 13 2007

Keywords

• nonn
• easy

Comments

The first sequence of "less twisted numbers"; this sequence, A130752 and A130755 form a "suite en trio" (cf. reference, p. 130).

First differences of A130755, second differences of A130752.

Sequence equals its third differences:

1 3 8 17 33 64 127 255 512 1025

2 5 9 16 31 63 128 257 513

3 4 7 15 32 65 129 256

1 3 8 17 33 64 127

References

• P. Curtz, Exercise Book, manuscript, 1995.

Links

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

Crossrefs

Cf. A010882 (periodic (1, 2, 3)), A128834 (periodic (0, 1, 1, 0, -1, -1)), A057079 (periodic (1, 2, 1, -1, -2, -1)), A130752 (first differences), A130755 (second differences).

Programs

Magma

m:=31; S:=[ [1, 2, 3][(n-1) mod 3 +1]: n in [1..m] ]; [ &+[ Binomial(i-1, k-1)*S[k]: k in [1..i] ]: i in [1..m] ]; // Klaus Brockhaus, Aug 03 2007

Magma

I:=[1,3,8]; [n le 3 select I[n] else 3*Self(n-1) - 3*Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jan 15 2018

Mathematica

CoefficientList[Series[(1+2*x^2)/((1-2*x)*(1-x+x^2)), {x, 0, 50}], x] (* or *) LinearRecurrence[{3,-3,2}, {1,3,8}, 30] (* G. C. Greubel, Jan 15 2018 *)

PARI

{m=31; v=vector(m); v[1]=1; v[2]=3; v[3]=8; for(n=4, m, v[n]=3*v[n-1]-3*v[n-2]+2*v[n-3]); v} \\ Klaus Brockhaus, Aug 03 2007

PARI

{for(n=0, 30, print1(2^(n+1)+[ -1, -1,0, 1, 1, 0][n%6+1], ","))} \\ Klaus Brockhaus, Aug 03 2007

Formula

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

a(0) = 1; a(1) = 3; a(2) = 8; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3).

a(n) = 2^(n+1) + A128834(n+4).

a(0) = 1; for n > 0, a(n) = 2*a(n-1) + A057079(n-1).

Extensions

Edited and extended by Klaus Brockhaus, Aug 03 2007