A103457 a(n) = 3^n + 1 - 0^n.

1, 4, 10, 28, 82, 244, 730, 2188, 6562, 19684, 59050, 177148, 531442, 1594324, 4782970, 14348908, 43046722, 129140164, 387420490, 1162261468, 3486784402, 10460353204, 31381059610, 94143178828, 282429536482, 847288609444

Offset: 0

Author

Paul Barry, Feb 07 2005

Keywords

• easy
• nonn

Links

• Patrick De Geest, World!Of Numbers
• Index entries for linear recurrences with constant coefficients, signature (4,-3).

Crossrefs

Cf. A011782, A027649, A034472, A094388, A110593.

Programs

Magma

[1] cat [3^n + 1: n in [1..30]]; // G. C. Greubel, Jun 22 2021

Mathematica

Join[{1},LinearRecurrence[{4,-3},{4,10},30]] (* Harvey P. Dale, Mar 29 2015 *)

PARI

my(x='x+O('x^50)); Vec((1-3*x^2)/((1-x)*(1-3*x))) \\ Altug Alkan, Dec 04 2015

Sage

[1]+[3^n + 1 for n in (1..30)] # G. C. Greubel, Jun 22 2021

Formula

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

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

From R. J. Mathar, Aug 04 2008: (Start)

a(n) = A034472(n), n>0.

a(n) = A094388(n-1), n>1.

a(n+1) - a(n) = A110593(n+1). (End)

a(n) = 3*a(n-1) - 2, with a(1)=4. - Vincenzo Librandi, Dec 29 2010

From J. Conrad, Nov 25 2015: (Start)

For n>0, a(n) = 2 * (A011782(0) + A011782(n) + Sum_{x=1..n-1} Sum_{k=0..x-1}(binomial(x-1,k)*(A011782(k+1) + A011782(n-x+k)))).

Alternatively, for n>0, a(n) = A027649(n) - 2 * Sum_{x=1..n-1}Sum_{k=0..x-1}(binomial(x-1,k)*(A011782(k+1) + A011782(n-x+k))). (End)

E.g.f.: -1 + exp(x) + exp(3*x). - G. C. Greubel, Jun 22 2021