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A130624 Binomial transform of A101000. 3
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 (list; graph; refs; listen; history; text; internal format)
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) = -(2/3)*(1/2-(1/2)*i*sqrt(3))^n - (2/3)*(1/2+(1/2)*i*sqrt(3))^n + (4/3)*2^n - (1/3)*i*(1/2-(1/2)*i*sqrt(3))^n*sqrt(3) + (1/3)*i*(1/2+(1/2)*i*sqrt(3))^n*sqrt(3), with i = sqrt(-1). - Paolo P. Lava, Oct 06 2008

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: A126573 A000327 A220425 * A066869 A023172 A270681

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

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Last modified July 12 13:04 EDT 2020. Contains 335661 sequences. (Running on oeis4.)