A111075 -id:A111075 - cp's OEIS frontend

A111159 Numbers k such that A111075(k) is odd.

1, 3, 4, 6, 12, 16, 24, 25, 27, 48, 49, 54, 64, 75, 96, 100, 108, 121, 147, 150, 169, 192, 196, 216, 243, 256, 289, 294, 300, 361, 363, 384, 400, 432, 484, 486, 507, 529, 588, 600, 625, 675, 676, 726, 768, 784, 841, 864, 867, 961, 972, 1014, 1024

Offset: 1

Paul D. Hanna, Oct 12 2005

nonn

Cf. A028982, A111075.

Conjecture: A111075(n) is odd whenever: (i) n = m^2 for all m>=1 such that 3 does not divide m, or (ii) n = 3*A028982(m) for all m>=1. Also, A028982 gives "Union of nonzero squares and twice squares."

It is not difficult to show that the conjecture is true. - Max Alekseyev, Oct 12 2005

A203802 a(n) = Fibonacci(n) * Sum_{d|n} -(-1)^(n/d) / Fibonacci(d).

Original entry on oeis.org

1, 0, 3, -5, 6, -3, 14, -48, 52, -10, 90, -329, 234, -28, 1038, -2349, 1598, -1044, 4182, -12750, 17262, -198, 28658, -135285, 90031, -520, 300405, -554974, 514230, -464658, 1346270, -5188656, 5326470, -3570, 11782764, -34556612, 24157818, -9348, 95140422, -256249218, 165580142

Offset: 1

Paul D. Hanna, Jan 11 2012

sign

			G.f.: A(x) = x + 3*x^3 - 5*x^4 + 6*x^5 - 3*x^6 + 14*x^7 - 48*x^8 + 52*x^9 +...
where A(x) = x/(1+x-x^2) + x^2/(1+3*x^2+x^4) + x^3/(1+4*x^3-x^6) + x^4/(1+7*x^4+x^8) + x^5/(1+11*x^5-x^10) + x^6/(1+18*x^6+x^12) +...+ x^n/(1 + Lucas(n)*x^n + (-1)^n*x^(2*n)) +...
Illustration of terms.
a(1) = 1; a(2) = 1*(-1/1 + 1/1) = 0; a(3) = 2*(1/1 + 1/2) = 3;
a(4) = 3*(-1/1 - 1/1 + 1/3) = -5; a(5) = 5*(1/1 + 1/5) = 6;
a(6) = 8*(-1/1 + 1/1 - 1/2 + 1/8) = -3; a(7) = 13*(1/1 + 1/13) = 14;
a(8) = 21*(-1/1 - 1/1 - 1/3 + 1/21) = -48; ...

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cf. A111075, A000045, A000204 (Lucas).

{a(n)=fibonacci(n) * sumdiv(n, d, -(-1)^(n/d) / fibonacci(d))}

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1, n, x^m/(1 + Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))), n)}

G.f.: Sum_{n>=1} x^n/(1 + Lucas(n)*x^n + (-1)^n*x^(2*n)) where Lucas(n) = A000204(n).

A203797 a(n) = Pell(n) * Sum_{d|n} 1/Pell(d), where Pell(n) = A000129(n).

Original entry on oeis.org

1, 3, 6, 19, 30, 120, 170, 647, 1183, 3650, 5742, 24916, 33462, 121652, 240756, 746639, 1136690, 4707147, 6625110, 25882770, 46565244, 139849776, 225058682, 978088748, 1356970471, 4750318586, 9182205852, 29333908544, 44560482150, 188175715440, 259717522850, 994309609247

Offset: 1

Paul D. Hanna, Jan 13 2012

nonn

			G.f.: A(x) = x + 3*x^2 + 6*x^3 + 19*x^4 + 30*x^5 + 120*x^6 + 170*x^7 + ... where A(x) = x/(1-2*x-x^2) + x^2/(1-6*x^2+x^4) + x^3/(1-14*x^3-x^6) + x^4/(1-34*x^4+x^8) + x^5/(1-82*x^5-x^10) + x^6/(1-198*x^6+x^12) + ... + x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) + ...

Cf. A204059, A111075, A203802, A000129, A002203.

a[n_] := Fibonacci[n, 2] * DivisorSum[n, 1/Fibonacci[#, 2] &]; Array[a, 32] (* Amiram Eldar, Aug 18 2023 *)

{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=Pell(n) * sumdiv(n, d, 1/Pell(d))}

/* G.f. using companion Pell numbers: */
{A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=polcoeff(sum(m=1, n, x^m/(1 - A002203(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))), n)}

G.f.: Sum_{n>=1} x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203 is the companion Pell numbers.

A204059 a(n) = Pell(n) * Sum_{d|n} (-1)^(n/d) / Pell(d), where Pell(n) = A000129(n).

Original entry on oeis.org

1, -1, 6, -17, 30, -48, 170, -645, 1183, -1270, 5742, -22604, 33462, -40868, 240756, -746637, 1136690, -1884529, 6625110, -23217030, 46565244, -46627416, 225058682, -975425316, 1356970471, -1583502622, 9182205852, -26257649200, 44560482150, -77433044928

Offset: 1

Paul D. Hanna, Jan 13 2012

sign

			G.f.: A(x) = x - x^2 + 6*x^3 - 17*x^4 + 30*x^5 - 48*x^6 + 170*x^7 +...
where A(x) = x/(1+2*x-x^2) + x^2/(1+6*x^2+x^4) + x^3/(1+14*x^3-x^6) + x^4/(1+34*x^4+x^8) + x^5/(1+82*x^5-x^10) + x^6/(1+198*x^6+x^12) +...+ x^n/(1 + A002203(n)*x^n + (-1)^n*x^(2*n)) +...

Cf. A203797, A111075, A203802, A000129, A002203.

{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=Pell(n) * sumdiv(n, d, -(-1)^(n/d)/Pell(d))}

/* G.f. using companion Pell numbers: */
{A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=polcoeff(sum(m=1, n, x^m/(1 + A002203(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))), n)}

G.f.: Sum_{n>=1} x^n/(1 + A002203(n)*x^n + (-1)^n*x^(2*n)) where A002203 is the companion Pell numbers.

a(2*n-1) = A203797(2*n-1).

cp's OEIS Frontend

A111159 Numbers k such that A111075(k) is odd.

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A203802 a(n) = Fibonacci(n) * Sum_{d|n} -(-1)^(n/d) / Fibonacci(d).

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A203797 a(n) = Pell(n) * Sum_{d|n} 1/Pell(d), where Pell(n) = A000129(n).

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Formula

A204059 a(n) = Pell(n) * Sum_{d|n} (-1)^(n/d) / Pell(d), where Pell(n) = A000129(n).

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