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

0,1

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).

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

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 *)

Prog

(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 */
(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

Crossrefs

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

Sequence in context: A290464 A335768 A039946 * A309331 A059529 A271318

Adjacent sequences: A130749 A130750 A130751 * A130753 A130754 A130755

Keyword

nonn,easy

Author

Paul Curtz, Jul 13 2007

Extensions

Edited and extended by Klaus Brockhaus, Aug 03 2007

Status

approved