A329178 Sum of the products of pairs of Padovan numbers which are two apart, starting from A000931(5).

1, 3, 5, 11, 19, 34, 62, 107, 191, 335, 587, 1035, 1812, 3184, 5589, 9803, 17213, 30199, 52999, 93014, 163214, 286439, 502655, 882095, 1547991, 2716503, 4767160, 8365776, 14680889, 25763219, 45211237, 79340227, 139232411, 244335770, 428779502, 752455475

Offset: 0

Author

David Nacin, Nov 08 2019

Keywords

• nonn

Examples

			For n=3, a(3) = 1*1 + 1*2 + 1*2 + 2*3 = 11.
		

Crossrefs

Cf. A000931, A133037, A329227.

Programs

Python

p = lambda x:[1, 1, 1][x] if x<3 else p(x-2)+p(x-3)
a = lambda x:sum(p(i)*p(i+2) for i in range(x+1))

Formula

a(n) = Sum_{i=5..n+5} A000931(i)*A000931(i+2).

a(n) = A329227(n+7) - 1.

Conjectures from Colin Barker, Nov 09 2019: (Start)

G.f.: (1 + x - x^2 + x^3 - x^4) / ((1 - x)*(1 - 2*x + x^2 - x^3)*(1 + x - x^3)).

a(n) = 2*a(n-1) - 2*a(n-4) + 2*a(n-5) - 2*a(n-6) + a(n-7) for n>6.

(End)