A103820 Whitney transform of 3^n.

1, 4, 16, 61, 232, 880, 3337, 12652, 47968, 181861, 689488, 2614048, 9910609, 37573972, 142453744, 540083149, 2047610680, 7763081488, 29432076505, 111585473980, 423052651456, 1603914376309, 6080901083296, 23054446378816

Offset: 0

Author

Paul Barry, Feb 16 2005

Keywords

• easy
• nonn

Comments

Partial sums of A030195. The Whitney transform maps the sequence with g.f. g(x) to that with g.f. (1/(1-x))g(x(1+x)).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,0,-3).

Crossrefs

Cf. A004070, A030195.

Equals (A108306(n+1) - 1)/5.

Programs

Magma

I:=[1,4,16]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Aug 18 2017

Maple

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=3*a[n-1]+3*a[n-2]+1 od: seq(a[n], n=1..33); # Zerinvary Lajos, Dec 14 2008

Mathematica

Join[{a=0,b=1},Table[c=3*b+3*a+1;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)
LinearRecurrence[{4, 0, -3}, {1, 4, 16}, 40] (* Vincenzo Librandi, Aug 18 2017 *)

Formula

G.f.: 1/((1-x)(1-3x-3x^2));

a(n) = 4a(n-1) - 3a(n-3);

a(n) = Sum_{k=0..n} (Sum_{i=0..n} C(k, i-k))*3^k.

a(n) = 3(a(n-1) + a(n-2)) + 1, n > 1. [Gary Detlefs, Jun 21 2010]