A256281 - OEIS

A256281

Inverse Moebius transform of Pell numbers.

1, 3, 6, 15, 30, 78, 170, 423, 991, 2410, 5742, 13950, 33462, 80954, 195060, 471255, 1136690, 2745273, 6625110, 15996850, 38614140, 93228102, 225058682, 543354078, 1311738151, 3166849426, 7645371036, 18457637018, 44560482150, 107578717860, 259717522850, 627014037303, 1513744660692, 3654504012630

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OFFSET

1,2

COMMENTS

Here Pell(n) = A000129(n).

LINKS

Table of n, a(n) for n=1..34.

FORMULA

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

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

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

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

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

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

EXAMPLE

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

where by definition

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) +...

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

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)) +...

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

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

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

PROG

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

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

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

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

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

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

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

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