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
Keywords
Examples
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.
Programs
-
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), ", "))
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).
Comments