A256281 - OEIS

%I #28 Jun 06 2017 02:58:54

%S 1,3,6,15,30,78,170,423,991,2410,5742,13950,33462,80954,195060,471255,

%T 1136690,2745273,6625110,15996850,38614140,93228102,225058682,

%U 543354078,1311738151,3166849426,7645371036,18457637018,44560482150,107578717860,259717522850,627014037303,1513744660692,3654504012630

%N Inverse Moebius transform of Pell numbers.

%C Here Pell(n) = A000129(n).

%F G.f.: Sum_{n>=1} Pell(n) * x^n / (1 - x^n).

%F G.f.: Sum_{n>=1} x^n / (1 - 2*x^n - x^(2*n)).

%F G.f.: Sum_{n>=1} x^n * Sum_{d|n} (2 + x^d)^(n/d-1).

%F Sum_{d|n} Moebius(n/d) * a(d) = Pell(n) for n>=1.

%F a(n) = Sum_{d|n} Pell(d).

%F a(A028982(n)) == 1 (mod 2), with a(n) == 0 (mod 2) otherwise, where A028982 lists the squares and twice squares.

%e G.f.: A(x) = x + 3*x^2 + 6*x^3 + 15*x^4 + 30*x^5 + 78*x^6 + 170*x^7 +...

%e where by definition

%e A(x) = x/(1-x) + 2*x^2/(1-x^2) + 5*x^3/(1-x^3) + 12*x^4/(1-x^4) + 29*x^5/(1-x^5) + 70*x^6/(1-x^6) + 169*x^7/(1-x^7) + 408*x^8/(1-x^8) + 985*x^9/(1-x^9) + 2378*x^10/(1-x^10) +...+ Pell(n)*x^n/(1-x^n) +...

%e The g.f. is also given by the series:

%e A(x) = x/(1-2*x-x^2) + x^2/(1-2*x^2-x^4) + x^3/(1-2*x^3-x^6) + x^4/(1-2*x^4-x^8) + x^5/(1-2*x^5-x^10) + x^6/(1-2*x^6-x^12) + x^7/(1-2*x^7-x^14) +...+ x^n/(1-x^n-x^(2*n)) +...

%e It is interesting to note that the ODD TERMS occur at A028982(n):

%e [1, 2, 4, 8, 9, 16, 18, 25, 32, 36, 49, 50, 64, 72, 81, 98, 100, ...];

%e i.e., those numbers k such that sum of divisors of k is odd.

%o (PARI) Pell(n)=polcoeff(x/(1-2*x-x^2 +x*O(x^n)), n)

%o a(n) = sumdiv(n,d, Pell(d))

%o for(n=1, 40, print1(a(n), ", "))

%o (PARI) Pell(n)=polcoeff(x/(1-2*x-x^2 +x*O(x^n)), n)

%o a(n) = polcoeff(sum(m=1, n, Pell(m)*x^m/(1-x^m +x*O(x^n))), n)

%o for(n=1, 40, print1(a(n), ", "))

%o (PARI) a(n) = polcoeff(sum(m=1, n, x^m/(1-2*x^m-x^(2*m) +x*O(x^n)) ), n)

%o for(n=1, 40, print1(a(n), ", "))

%o (PARI) a(n) = local(A=x+x^2); A=sum(m=1, n, x^m*sumdiv(m, d, (2 + x^d +x*O(x^n))^((m/d-1)))); polcoeff(A, n)

%o for(n=1, 40, print1(a(n), ", "))

%Y Cf. A256272, A000129 (Pell), A028982.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jun 02 2015