A133726 Möbius transform of the Pell numbers (A000129).

1, 1, 4, 10, 28, 64, 168, 396, 980, 2348, 5740, 13780, 33460, 80612, 194992, 470424, 1136688, 2743160, 6625108, 15992040, 38613792, 93216616, 225058680, 543325464, 1311738092, 3166782500, 7645369060, 18457475260, 44560482148, 107578322912, 259717522848

Offset: 1

Author

Gary W. Adamson, Sep 21 2007

Keywords

• nonn

Examples

			a(4) = 10 = (0, -1, 0, 1) dot (1, 2, 5, 12) = (0, -2, 0, 12).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000129, A054525, A256281.

Programs

Maple

with(numtheory):
a:= n-> add(mobius(n/d)*(<<0|1>, <1|2>>^d. <<0,1>>)[1,1], d=divisors(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Sep 26 2011

Mathematica

a[n_] := Sum[MoebiusMu[n/d]*(MatrixPower[{{0, 1}, {1, 2}}, d]. {0, 1})[[1]], {d, Divisors[n]}];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Apr 19 2025, after Alois P. Heinz *)

Formula

a(n) = Sum_{d|n} A008683(n/d)*A000129(d) = Sum_{k=1..n} A054525(n,k) * A000129(k).

G.f.: Sum_{k>=1} mu(k) * x^k / (1 - 2*x^k - x^(2*k)). - Ilya Gutkovskiy, Feb 06 2020