A130624 Binomial transform of A101000.

0, 1, 5, 12, 23, 43, 84, 169, 341, 684, 1367, 2731, 5460, 10921, 21845, 43692, 87383, 174763, 349524, 699049, 1398101, 2796204, 5592407, 11184811, 22369620, 44739241, 89478485, 178956972, 357913943, 715827883, 1431655764, 2863311529, 5726623061, 11453246124

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3, -3, 2).

Formula

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

a(0)=0; a(n+1) = 2*a(n) + A119910(n).

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

a(n) = 2^n + a(n-1) - a(n-2). - Jon Maiga, Nov 14 2018

Mathematica

LinearRecurrence[{3, -3, 2}, {0, 1, 5}, 40] (* Harvey P. Dale, Mar 05 2013 *)
RecurrenceTable[{a[0]==0, a[1]==1, a[n]==(2^n) + a[n-1] - a[n-2]}, a, {n, 50}] (* Vincenzo Librandi, Nov 15 2018 *)

Prog

(PARI) {m=32; v=concat([0, 1, 5], vector(m-3)); for(n=4, m, v[n]=3*v[n-1]-3*v[n-2]+2*v[n-3]); v} /* Klaus Brockhaus, Jun 21 2007 */
(Magma) m:=32; S:=[[0, 1, 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, Jun 21 2007 */
(Magma) I:=[0, 1, 5]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+2*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Nov 15 2018

Crossrefs

Cf. A101000, A119910, A130625 (first differences), A130626 (second differences).

Sequence in context: A341209 A000327 A220425 * A344846 A066869 A023172

Adjacent sequences: A130621 A130622 A130623 * A130625 A130626 A130627

Keyword

nonn

Author

Paul Curtz, Jun 18 2007

Extensions

Edited and extended by Klaus Brockhaus, Jun 21 2007

Status

approved