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

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

Crossrefs

Cf. A256272, A000129 (Pell), A028982.

Sequence in context: A242172 A356954 A126982 * A034739 A276670 A244706

Adjacent sequences: A256278 A256279 A256280 * A256282 A256283 A256284

Keyword

nonn

Author

Paul D. Hanna, Jun 02 2015

Status

approved