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

Author

Paul Curtz, Jun 18 2007

Keywords

• nonn

Links

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

Crossrefs

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

Programs

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

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 *)

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 */

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

Extensions

Edited and extended by Klaus Brockhaus, Jun 21 2007