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

0,2

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

From Klaus Purath, Nov 26 2025: (Start)

Alternating sums of 3 consecutive terms divided by 6 are powers of 2. The following appears to hold true for all recurrences satisfying b(n) = 3*b(n-1) - 3*b(n-2) + b(n-3): Alternating sums of 3 consecutive terms divided by the alternating sum of the first three initial terms are powers of 2.

The initial terms of the differences of this sequence give A010882. (End)

References

Paul 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).

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).

E.g.f.: 2*exp(2*x) - exp(x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/3. - Stefano Spezia, Dec 02 2025

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 *)

Prog

(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
(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

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).

Sequence in context: A360848 A002625 A027181 * A281166 A165273 A002626

Adjacent sequences: A130747 A130748 A130749 * A130751 A130752 A130753

Keyword

nonn,easy

Author

Paul Curtz, Jul 13 2007

Extensions

Edited and extended by Klaus Brockhaus, Aug 03 2007

Status

approved