A129954 Second differences of A129952.

1, 3, 6, 14, 32, 72, 160, 352, 768, 1664, 3584, 7680, 16384, 34816, 73728, 155648, 327680, 688128, 1441792, 3014656, 6291456, 13107200, 27262976, 56623104, 117440512, 243269632, 503316480, 1040187392, 2147483648, 4429185024

Offset: 0

Author

Paul Curtz, Jun 10 2007

Keywords

• nonn
• easy

Comments

First differences of A129953: a(n) = A129953(n+1) - A129953(n).

Essentially the same as A078836: a(n) = A078836(n+4) for n > 1.

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, A078836.

Programs

Magma

m:=16; 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] ]; [ U[n+1]-U[n]: n in[1..2*m-2] ]; // Klaus Brockhaus, Jun 17 2007

PARI

{m=29; print1(1, ",", 3, ","); for(n=2, m, print1((n+4)*2^(n-2), ","))} \\ Klaus Brockhaus, Jun 17 2007

Python

def A129954(n): return n+4<1 else 2*n+1 # Chai Wah Wu, Oct 03 2024

Formula

a(0) = 1, a(1) = 3; for n > 1, a(n) = (n+4)*2^(n-2).

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

Binomial transform of [1, 2, 1, 4, 1, 6, 1, 8, ...]. - Gary W. Adamson, Sep 29 2007

E.g.f.: (x + exp(2*x)*(2 + x))/2. - Stefano Spezia, Oct 04 2024

Extensions

Edited and extended by Klaus Brockhaus, Jun 17 2007