A329244 Sum of every third term of the Padovan sequence A000931.

1, 2, 3, 5, 10, 22, 50, 115, 266, 617, 1433, 3330, 7740, 17992, 41825, 97230, 226031, 525457, 1221538, 2839730, 6601570, 15346787, 35676950, 82938845, 192809421, 448227522, 1042002568, 2422362080, 5631308625, 13091204282, 30433357675, 70748973085, 164471408186

Offset

0,2

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-5,3,-1).

Formula

a(n) = Sum_{i=0..n} A000931(3*i).

a(n) = A000931(3n+2)+1.

From Colin Barker, Nov 09 2019: (Start)

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

a(n) = 4*a(n-1) - 5*a(n-2) + 3*a(n-3) - a(n-4) for n>3. (End)

Example

For n = 3, a(3) = 1+1+1+2 = 5.

Mathematica

LinearRecurrence[{4, -5, 3, -1}, {1, 2, 3, 5}, 50] (* Paolo Xausa, Apr 08 2024 *)

Prog

(Python)
p = lambda x:[1, 0, 0][x] if x<3 else p(x-2)+p(x-3)
a = lambda x:sum(p(3*i) for i in range(x+1))
(PARI) Vec((1 - 2*x) / ((1 - x)*(1 - 3*x + 2*x^2 - x^3)) + O(x^35)) \\ Colin Barker, Nov 09 2019

Crossrefs

Partial sums of A034943.

Cf. A000931.

Sequence in context: A293842 A014844 A307264 * A173271 A326574 A280019

Adjacent sequences: A329241 A329242 A329243 * A329245 A329246 A329247

Keyword

nonn,easy

Author

David Nacin, Nov 09 2019

Status

approved