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A256281
Inverse Moebius transform of Pell numbers.
3
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).
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 02 2015
STATUS
approved