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
Examples
For n = 3, a(3) = 1+1+1+2 = 5.
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).
Programs
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Mathematica
LinearRecurrence[{4, -5, 3, -1}, {1, 2, 3, 5}, 50] (* Paolo Xausa, Apr 08 2024 *) -
PARI
Vec((1 - 2*x) / ((1 - x)*(1 - 3*x + 2*x^2 - x^3)) + O(x^35)) \\ Colin Barker, Nov 09 2019
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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))
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)