A130752 Binomial transform of periodic sequence (2, 3, 1).

2, 5, 9, 16, 31, 63, 128, 257, 513, 1024, 2047, 4095, 8192, 16385, 32769, 65536, 131071, 262143, 524288, 1048577, 2097153, 4194304, 8388607, 16777215, 33554432, 67108865, 134217729, 268435456, 536870911, 1073741823, 2147483648, 4294967297, 8589934593

Offset: 0

Author

Paul Curtz, Jul 13 2007

Keywords

• nonn
• easy

Comments

The second sequence of "less twisted numbers"; this sequence, A130750 and A130755 form a "suite en trio" (cf. reference, p. 130).

First differences of A130750, second differences of A130755.

Sequence equals its third differences:

2.....5.....9....16....31....63...128...257...513..1024...

...3.....4.....7....15....32....65...129...256...511...

......1.....3.....8....17....33....64...127...255...

..........2.....5.....9....16....31....63...128...

References

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

Links

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

Crossrefs

Cf. A010882, A130755 (first differences), A130750 (second differences).

Programs

Magma

m:=31; S:=[ [2, 3, 1][(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 */

Mathematica

a[n_] := 2^(n+1) + 2*Sin[n*Pi/3]/Sqrt[3]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 13 2012 *)
LinearRecurrence[{3,-3,2},{2,5,9},40] (* Harvey P. Dale, Jun 21 2017 *)

PARI

{m=31; v=vector(m); v[1]=2; v[2]=5; v[3]=9; 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)+[0, 1, 1, 0, -1, -1][n%6+1], ","))} \\ Klaus Brockhaus, Aug 03 2007

PARI

Vec((2-x) / ((1-2*x)*(1-x+x^2)) + O(x^40)) \\ Colin Barker, Jan 20 2017

Formula

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

a(0) = 2; a(1) = 5; a(2) = 9; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3).

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

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

E.g.f.: 2*(sqrt(3)*exp(2*x) + sin(sqrt(3)*x/2)*exp(x/2))/sqrt(3). - Ilya Gutkovskiy, Jun 20 2016

a(n) = 2^(n+1) + (2*sin((Pi*n)/3))/sqrt(3). - Colin Barker, Jan 20 2017

Extensions

Edited and extended by Klaus Brockhaus, Aug 03 2007