cp's OEIS Frontend

A133726 Möbius transform of the Pell numbers (A000129).

%I A133726 #23 Apr 19 2025 09:24:46
%S A133726 1,1,4,10,28,64,168,396,980,2348,5740,13780,33460,80612,194992,470424,
%T A133726 1136688,2743160,6625108,15992040,38613792,93216616,225058680,
%U A133726 543325464,1311738092,3166782500,7645369060,18457475260,44560482148,107578322912,259717522848
%N A133726 Möbius transform of the Pell numbers (A000129).
%H A133726 Alois P. Heinz, <a href="/A133726/b133726.txt">Table of n, a(n) for n = 1..1000</a>
%F A133726 a(n) = Sum_{d|n} A008683(n/d)*A000129(d) = Sum_{k=1..n} A054525(n,k) * A000129(k).
%F A133726 G.f.: Sum_{k>=1} mu(k) * x^k / (1 - 2*x^k - x^(2*k)). - _Ilya Gutkovskiy_, Feb 06 2020
%e A133726 a(4) = 10 = (0, -1, 0, 1) dot (1, 2, 5, 12) = (0, -2, 0, 12).
%p A133726 with(numtheory):
%p A133726 a:= n-> add(mobius(n/d)*(<<0|1>, <1|2>>^d. <<0,1>>)[1,1], d=divisors(n)):
%p A133726 seq(a(n), n=1..40);  # _Alois P. Heinz_, Sep 26 2011
%t A133726 a[n_] := Sum[MoebiusMu[n/d]*(MatrixPower[{{0, 1}, {1, 2}}, d]. {0, 1})[[1]], {d, Divisors[n]}];
%t A133726 Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Apr 19 2025, after _Alois P. Heinz_ *)
%Y A133726 Cf. A000129, A054525, A256281.
%K A133726 nonn
%O A133726 1,3
%A A133726 _Gary W. Adamson_, Sep 21 2007