A230000 -id:A230000 - cp's OEIS frontend

A231733 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)x + ... + c(n)x^(n) which is the denominator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 2)/(x + 1).

1, 1, 2, 3, 1, 5, 11, 8, 2, 12, 34, 36, 17, 3, 29, 101, 141, 99, 35, 5, 70, 289, 499, 462, 242, 68, 8, 169, 807, 1659, 1905, 1320, 552, 129, 13, 408, 2212, 5272, 7218, 6210, 3438, 1196, 239, 21, 985, 5977, 16198, 25738, 26427, 18183, 8383, 2497, 436, 34

Offset: 1

Clark Kimberling, Nov 13 2013

Sum of numbers in row n: 2*A015521(n). Left edge: A000129. Right edge: A000045 (Fibonacci numbers).

			First 3 rows:
1 ... 1
2 ... 3 ... 1
5 ... 11 .. 8 .. 2
First 3 polynomials:  1 + x, 2 + 3*x + x^2, 5 + 11*x + 8*x^2 + 2*x^3.

Cf. A230000, A231732.

t[n_] := t[n] = Table[(x + 2)/(x + 1), {k, 0, n}];
b = Table[Factor[Convergents[t[n]]], {n, 0, 10}];
p[x_, n_] := p[x, n] = Last[Expand[Denominator[b]]][[n]];
u = Table[p[x, n], {n, 1, 10}]
v = CoefficientList[u, x]; Flatten[v]

A231774 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)x + ... + c(n)x^(n) which is the numerator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 1)/(x + 2).

Original entry on oeis.org

2, 1, 5, 6, 2, 9, 19, 13, 3, 29, 72, 69, 30, 5, 65, 213, 278, 182, 60, 8, 181, 682, 1084, 928, 451, 118, 13, 441, 1975, 3795, 4065, 2625, 1023, 223, 21, 1165, 5868, 13015, 16590, 13290, 6852, 2221, 414, 34, 2929, 16697, 42404, 63020, 60435, 38799, 16682

Offset: 1

Clark Kimberling, Nov 13 2013

Sum of numbers in row n: A002534(n). Left edge: A006131. Right edge: A000045 (Fibonacci numbers).

			First 3 rows:
2 ... 1
5 ... 6 .... 2
9 ... 19 ... 13 ... 3
First 3 polynomials:  2 + x, 5 + 6*x + 2*x^2, 9 + 19*x + 13*x^2 + 3*x^3.

Cf. A230000, A231775, A000045.

t[n_] := t[n] = Table[(x + 1)/(x + 2), {k, 0, n}];
b = Table[Factor[Convergents[t[n]]], {n, 0, 10}];
p[x_, n_] := p[x, n] = Last[Expand[Numerator[b]]][[n]];
u = Table[p[x, n], {n, 1, 10}]
v = CoefficientList[u, x]; Flatten[v]

A231775 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)x + ... + c(n)x^(n) which is the denominator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 1)/(x + 2).

Original entry on oeis.org

2, 1, 2, 3, 1, 10, 17, 10, 2, 18, 47, 45, 19, 3, 58, 173, 210, 129, 40, 5, 130, 491, 769, 642, 302, 76, 8, 362, 1545, 2850, 2940, 1830, 687, 144, 13, 882, 4391, 9565, 11925, 9315, 4671, 1469, 265, 21, 2330, 12901, 31898, 46195, 43170, 26994, 11294, 3049, 482

Offset: 1

Clark Kimberling, Nov 13 2013

Sum of numbers in row n: 3*A002534(n). Left edge: 2*A006131. Right edge: A000045 (Fibonacci numbers).

			First 3 rows:
2 .... 1
2 .... 3 .... 1
10 ... 17 ... 10 ... 2
First 3 polynomials:  2 + x, 2 + 3*x + x^2, 10 + 17*x + 10*x^2 + 2*x^3.

Cf. A230000, A231774, A000045.

t[n_] := t[n] = Table[(x + 1)/(x + 2), {k, 0, n}];
b = Table[Factor[Convergents[t[n]]], {n, 0, 10}];
p[x_, n_] := p[x, n] = Last[Expand[Denominator[b]]][[n]];
u = Table[p[x, n], {n, 1, 10}]
v = CoefficientList[u, x]; Flatten[v]

cp's OEIS Frontend

A231733 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)x + ... + c(n)x^(n) which is the denominator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 2)/(x + 1).

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A231774 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)x + ... + c(n)x^(n) which is the numerator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 1)/(x + 2).

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A231775 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)x + ... + c(n)x^(n) which is the denominator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 1)/(x + 2).

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cp's OEIS Frontend

A231733 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)*x + ... + c(n)*x^(n) which is the denominator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 2)/(x + 1).

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A231774 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)*x + ... + c(n)*x^(n) which is the numerator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 1)/(x + 2).

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Programs

Mathematica

A231775 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)*x + ... + c(n)*x^(n) which is the denominator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 1)/(x + 2).

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Author

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Programs

Mathematica

A231733 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)x + ... + c(n)x^(n) which is the denominator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 2)/(x + 1).

A231774 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)x + ... + c(n)x^(n) which is the numerator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 1)/(x + 2).

A231775 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)x + ... + c(n)x^(n) which is the denominator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 1)/(x + 2).