A129955 Third differences of A129952.

2, 3, 8, 18, 40, 88, 192, 416, 896, 1920, 4096, 8704, 18432, 38912, 81920, 172032, 360448, 753664, 1572864, 3276800, 6815744, 14155776, 29360128, 60817408, 125829120, 260046848, 536870912, 1107296256, 2281701376, 4697620480

Offset: 0

Author

Paul Curtz, Jun 10 2007

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4, -4).

Crossrefs

Cf. A129952, A129953, A129954, A034007.

Programs

Magma

m:=17; S:=&cat[ [ 1, 2*i ]: i in [0..m] ]; T:=[ &+[ Binomial(j-1, k-1)*S[k]: k in [1..j] ]: j in [1..2*m] ]; U:=[ T[n+1]-T[n]: n in[1..2*m-1] ]; V:=[ U[n+1]-U[n]: n in[1..2*m-2] ]; [ V[n+1]-V[n]: n in[1..2*m-3] ]; // Klaus Brockhaus, Jun 17 2007

Mathematica

Differences[LinearRecurrence[{4,-4},{1,1,2,6},40],3] (* Harvey P. Dale, Sep 04 2020 *)

PARI

{m=29; print1(2, ",", 3, ","); for(n=2, m, print1((n+6)*2^(n-2), ","))} \\ Klaus Brockhaus, Jun 17 2007

Formula

First differences of A129954: a(n) = A129954(n+1) - A129954(n).

a(n) = A034007(n+2)-2^(n-2) for n > 1.

a(0) = 2, a(1) = 3; for n > 1, a(n) = (n+6)*2^(n-2).

G.f.: (2-5*x+4*x^2-2*x^3)/(1-2*x)^2.

From Amiram Eldar, Jan 13 2021: (Start)

Sum_{n>=0} 1/a(n) = 256*log(2) - 12347/70.

Sum_{n>=0} (-1)^n/a(n) = 21851/210 - 256*log(3/2). (End)

Extensions

Edited and extended by Klaus Brockhaus, Jun 17 2007