A242574 -id:A242574 - cp's OEIS frontend

A242308 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.

1, 1, 2, 2, 1, 3, 3, 2, 5, 3, 5, 3, 1, 3, 4, 8, 5, 4, 8, 5, 2, 5, 3, 7, 13, 7, 8, 4, 7, 13, 7, 8, 4, 1, 3, 4, 8, 5, 5, 11, 11, 21, 12, 7, 13, 7, 5, 11, 11, 21, 12, 7, 13, 7, 2, 5, 3, 7, 13, 7, 8, 4, 9, 18, 10, 19, 34, 18, 19, 9, 12, 21, 11, 11, 5, 9, 18, 10

Offset: 1

Clark Kimberling, Jun 07 2014

Decree that row 1 is (1) and row 2 is (1/2). For n >=3, row n consists of numbers in increasing order generated as follows: 1/(x + 1) for each x in row n-1 together with x + 1 for each x in row n-2. It is easy to prove that row n consists of F(n) numbers, where F = A000045 (the Fibonacci numbers), and that every positive rational number occurs exactly once.

			First 6 rows of the array of rationals:
1/1
1/2
2/3 ... 2/1
1/3 ... 3/5 ... 3/2
2/5 ... 5/8 ... 3/4 ... 5/3 ... 3/1
1/4 ... 3/8 ... 4/7 ... 8/13 .. 5/7 .. 4/3 .. 8/5 .. 5/2
The numerators, by rows:  1,1,2,2,1,3,3,2,5,3,5,3,1,3,4,8,5,4,8,5,...

Clark Kimberling, Table of n, a(n) for n = 1..6000

Cf. A242574, A242360, A000045.

z = 18; g[1] = {1}; f1[x_] := 1/x; f2[x_] := 1/(x + 1); h[1] = g[1];
b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
u = Table[g[n], {n, 1, z}]; v = Flatten[u];
Denominator[v]; (* A243574 *)
Numerator[v];   (* A242308 *)

A242573 a(n) = [x^n] G(n-1,x) where G(n,x) is the n-th iteration of G(1,x) = x/(1-x+x^2), so that G(n,x) = G(n-1, G(1,x)) with G(0,x)=x.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 702, 4445, 27812, 187911, 4154105, 226545132, 11811552612, 567839904255, 26530164469576, 1244353584654296, 59633342751369016, 2947881116272213508, 151083714566902161495, 8048911065786420441543, 446230730213409483222040

Offset: 1

Paul D. Hanna, May 17 2014

nonn

			Given x/(1-x+x^2) = x + x^2 - x^4 - x^5 + x^7 + x^8 - x^10 - x^11 + x^13 +...
form a table of coefficients in the iterations of x/(1-x+x^2) like so:
[1,  0,  0,   0,    0,     0,      0,       0,        0,        0, ...];
[1,  1,  0,  -1,   -1,     0,      1,       1,        0,       -1, ...];
[1,  2,  2,  -1,   -8,   -15,    -10,      22,       79,      112, ...];
[1,  3,  6,   6,  -11,   -73,   -201,    -309,       37,     1913, ...];
[1,  4, 12,  26,   24,  -116,   -808,   -3000,    -7566,    -9882, ...];
[1,  5, 20,  65,  155,   120,  -1379,  -10761,   -51202,  -183269, ...];
[1,  6, 30, 129,  464,  1225,    702,  -18978,  -169139,  -994138, ...];
[1,  7, 42, 224, 1057,  4235,  12411,    4445,  -301321, -3076795, ...];
[1,  8, 56, 356, 2064, 10752,  48000,  156416,    27812, -5458012, ...];
[1,  9, 72, 531, 3639, 23064, 132633,  658197,  2388060,   187911, ...];
[1, 10, 90, 755, 5960, 44265, 306742, 1942198, 10676571, 43159172, ...]; ...
then this sequence forms the main diagonal in the above table.

Paul D. Hanna, Table of n, a(n) for n = 1..200

Cf. A242574, A242575.

{a(n)=local(A=x, G=x/(1-x+x^2)); for(i=1, n-1, A=subst(G, x, A+x*O(x^(n)))); polcoeff(A, n)}
for(n=1,30,print1(a(n),", "))

A242575 a(n) = [x^n] G(n+1,x) where G(n,x) is the n-th iteration of G(1,x) = x/(1-x+x^2), so that G(n+1,x) = G(n, G(1,x)) with G(0,x)=x.

Original entry on oeis.org

1, 3, 12, 65, 464, 4235, 48000, 658197, 10676571, 200994145, 4321369524, 104676808756, 2824120458186, 84038944594677, 2735391623889696, 96690702098948611, 3688764754855986702, 151064345601782722492, 6609564176073032406148, 307682727619722793499662

Offset: 1

Paul D. Hanna, May 17 2014

nonn

			Given x/(1-x+x^2) = x + x^2 - x^4 - x^5 + x^7 + x^8 - x^10 - x^11 + x^13 +...
form a table of coefficients in the iterations of x/(1-x+x^2) like so:
[1,  1,  0,  -1,   -1,     0,      1,       1,        0,       -1, ...];
[1,  2,  2,  -1,   -8,   -15,    -10,      22,       79,      112, ...];
[1,  3,  6,   6,  -11,   -73,   -201,    -309,       37,     1913, ...];
[1,  4, 12,  26,   24,  -116,   -808,   -3000,    -7566,    -9882, ...];
[1,  5, 20,  65,  155,   120,  -1379,  -10761,   -51202,  -183269, ...];
[1,  6, 30, 129,  464,  1225,    702,  -18978,  -169139,  -994138, ...];
[1,  7, 42, 224, 1057,  4235,  12411,    4445,  -301321, -3076795, ...];
[1,  8, 56, 356, 2064, 10752,  48000,  156416,    27812, -5458012, ...];
[1,  9, 72, 531, 3639, 23064, 132633,  658197,  2388060,   187911, ...];
[1, 10, 90, 755, 5960, 44265, 306742, 1942198, 10676571, 43159172, ...]; ...
then this sequence forms a diagonal in the above table.

Paul D. Hanna, Table of n, a(n) for n = 1..200

Cf. A242573, A242574.

{a(n)=local(A=x, G=x/(1-x+x^2)); for(i=1, n+1, A=subst(G, x, A+x*O(x^(n)))); polcoeff(A, n)}
for(n=1,30,print1(a(n),", "))

cp's OEIS Frontend

A242308 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.

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A242573 a(n) = [x^n] G(n-1,x) where G(n,x) is the n-th iteration of G(1,x) = x/(1-x+x^2), so that G(n,x) = G(n-1, G(1,x)) with G(0,x)=x.

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A242575 a(n) = [x^n] G(n+1,x) where G(n,x) is the n-th iteration of G(1,x) = x/(1-x+x^2), so that G(n+1,x) = G(n, G(1,x)) with G(0,x)=x.

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Author

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Programs

PARI