Thomas Baruchel - cp's OEIS frontend

A102708 Composite numbers a(n) having a GCD with Pell_x(a(n))-1 equal to 1 (where Pell_x(n) is the x solution of the Pell-Fermat equation: x^2 - n y^2 = 1). There is no square in the sequence because the Pell-Fermat equation has no solution for them.

27, 51, 65, 85, 123, 125, 145, 171, 185, 187, 243, 265, 267, 291, 325, 339, 363, 365, 387, 411, 425, 445, 451, 459, 481, 485, 493, 531, 533, 565, 603, 627, 629, 685, 697, 699, 731, 771, 779, 785, 803, 843, 845, 865, 867, 901, 925, 949, 963, 965, 985, 1003

Offset: 0

Thomas Baruchel, Feb 05 2005

nonn

A101443 Continued fraction expansion of (I_0(1/2)/I_1(1/2)-1)/2 = 1.56185896... (where I_n is the modified Bessel function of the first kind).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 5, 1, 1, 7, 1, 1, 9, 1, 1, 11, 1, 1, 13, 1, 1, 15, 1, 1, 17, 1, 1, 19, 1, 1, 21, 1, 1, 23, 1, 1, 25, 1, 1, 27, 1, 1, 29, 1, 1, 31, 1, 1, 33, 1, 1, 35, 1, 1, 37, 1, 1, 39, 1, 1, 41, 1, 1, 43, 1, 1, 45, 1, 1, 47, 1, 1, 49, 1, 1, 51, 1, 1, 53, 1, 1, 55, 1, 1, 57, 1, 1, 59, 1, 1, 61, 1

Offset: 0

Thomas Baruchel, Jan 18 2005

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

LinearRecurrence[{0, 0, 2, 0, 0, -1}, {1, 1, 1, 3, 1, 1}, 92] (* Georg Fischer, Feb 25 2022 *)

contfrac((besseli(0,1/2)/besseli(1,1/2)-1)/2)

PARI
```
a(n) = 2/3*n*!(n%3)+1
```

G.f.: 1 + x*U(0) where U(k)= 1 + x/(1 - x*(2*k+2)/(1+x*(2*k+2) - 1/((2*k+2) + 1 - (2*k+2)*x/(x + 1/U(k+1))))) ; (continued fraction, 5-step). - Sergei N. Gladkovskii, Oct 07 2012

A092142 Compute the continued fraction expansion of Pi; multiply each term by i, the square root of -1, compute this new continued fraction and get a number with a real part equal to 0. Then compute the regular continued fraction of the imaginary part of that new number.

Original entry on oeis.org

2, 1, 5, 1, 12, 1, 301, 2, 78, 1, 14, 1, 1, 1, 10, 1, 1, 2, 4, 1, 4, 1, 94, 1, 3, 1, 1, 1, 8, 1, 5, 1, 2, 2, 4, 1, 10, 1, 8, 1, 10, 1, 13, 1, 158, 1, 42, 1, 18, 1, 21, 1, 8, 2, 2, 1, 3, 1, 2, 3, 23, 1, 8, 2, 39, 1, 3, 1, 1, 1, 7, 2, 2, 1, 7, 1, 5, 3, 53, 1, 14, 1, 6, 1, 15, 1, 14, 2, 5, 1, 28, 1, 1, 2, 4

Offset: 0

Thomas Baruchel, Mar 31 2004

k=contfracpnqn(contfrac(Pi,500)*I);contfrac(imag(k[1,1]/k[2,1]),200)

A094234 a(n) = period of terms in quasi-periodic continued fraction expansion of 2^n*tanh(1).

Original entry on oeis.org

1, 5, 10, 32, 76, 184, 408, 944, 2088, 4680, 10168, 22192, 47952

Offset: 0

Thomas Baruchel, Jun 03 2004

			E.g. tanh(1)=[0,1,3,5,7,9,...] pattern being /2n+1/ (length=1), so a(1) = 1.
2*tanh(1) = [1; 1, 1, 10, 3, 1, 1, 4, 22, 6, 1, 1, 7, 34, 9...]. It is the concatenation of parts of the form [1, 1, 3*m-2, 12*m-2, 3*m] for m = 1,2,3..., so a(2) = 5.

More terms from Thomas Baruchel, Aug 26 2004

A087642 Sequence of squarefree n such that Q(sqrt(n)) has no element with a fully periodical continued fraction of period 1.

Original entry on oeis.org

3, 6, 7, 11, 14, 15, 19, 21, 22, 23, 30, 31, 33, 34, 35, 38, 39, 42, 43, 46, 47, 51, 55, 57, 59, 62, 66, 67, 69, 70, 71, 77, 78, 79, 83, 86, 87, 91, 93, 94, 95, 102, 103, 105, 107, 110, 111, 114, 115, 118, 119, 123, 127, 129, 131, 133, 134, 138, 139, 141, 142, 143, 146

Offset: 3

Thomas Baruchel, Sep 16 2003

Diophantine equation x^2 - n.y^2 + 4 = 0 has no solution (x,y) for a given squarefree n. Squarefree n not in the sequence A013946. Same sequence with square factors allowed is A087643.

			3 is in the sequence because no [k,k,k,k,...] is in Q(sqrt(3))
5 is not in the sequence since Q(sqrt(5)) contains [1,1,1,1,...]

Cf. A087643, A013946.

A087414 Numbers n such that 2nk(n) is rational but not an integer, where k(n) is sum of successive remainders when computing the Euclidean algorithm for (1, 1/sqrt(n)) as defined in A086378 (MuPAD program is given there); numbers belonging to A086378 but not to A088900.

Original entry on oeis.org

153, 1717, 2244, 2340, 3525, 3650, 6460, 7119, 7475, 10074, 14490, 19147, 20008, 20862, 21424, 21747, 24453, 25400, 26039, 27346, 28028, 28371, 31484, 35483, 37008, 44275, 44678, 45974, 50389, 52155, 62187, 63724, 64752

Offset: 1

Thomas Baruchel, Oct 21 2003

nonn

Cf. A086378 and A088900.

/* z(n)!=0 iff n is in the sequence */
z(n)= { local(a,b,c,d,e,f,g,h,i,j,k);
b=a=sqrtint(n);d=f=i=1;e=g=h=0;j=c=n-a^2;if(!c,return(0));
until((a==b)&&(c==j),k=d+a*e;f*=c;d=a*d+e*n;e=k;g+=i;i*=c;
k=g+a*h;g=a*g+h*n;h=k;k=(a+b)\c;g-=i*k;a=c*k-a;c=(n-a^2)/c);
d=d/f-1;e/=f;g/=i;h/=i;i=d^2-n*e^2;k=h*d-g*e;g=g*d-h*e*n;
b=n-a^2;a=b*g-c*a*i;c=b*k+i*c;b*=i;!a*(2%(b/gcd(b,n*c))); }

A087295 Successive remainders when computing the Euclidean algorithm for (n,m) where m is any positive integer having no common factor with n, gives a list ending with a sublist of Fibonacci sequence. Find m such that this sublist has the greatest length and define a(n) as this length.

Original entry on oeis.org

0, 0, 1, 2, 1, 3, 1, 2, 4, 2, 1, 3, 2, 5, 3, 2, 2, 3, 4, 3, 3, 6, 2, 4, 2, 3, 3, 3, 4, 5, 3, 4, 3, 4, 7, 3, 3, 5, 4, 3, 2, 4, 2, 4, 4, 5, 3, 6, 4, 4, 5, 4, 3, 5, 3, 8, 3, 4, 4, 4, 6, 5, 3, 4, 4, 3, 5, 4, 4, 5, 4, 5, 3, 6, 4, 4, 7, 5, 4, 5, 4, 6, 5, 4, 3, 5, 6, 4, 4, 9, 3, 4, 5, 5, 4, 5, 4, 7, 5, 6, 4, 5, 3, 5, 4

Offset: 0

Thomas Baruchel, Oct 19 2003

			a(5) = 3 because computing Euclidean algorithm for (5,8) gives 3, 2, 1 as successive remainders, all three belonging to Fibonacci sequence.

A087947 Sum of successive remainders in computing Euclidean algorithm for (1, 1/sqrt(-n)) has real and imaginary parts equal.

Original entry on oeis.org

1, 4, 5, 6, 8, 9, 16, 17, 18, 20, 24, 25, 36, 37, 38, 39, 40, 42, 48, 49, 64, 65, 66, 68, 72, 78, 80, 81, 100, 101, 102, 104, 105, 110, 117, 120, 121, 144, 145, 146, 147, 148, 150, 152, 155, 156, 164, 168, 169, 196, 197, 198, 200, 203, 210, 220, 222, 224, 225, 256

Offset: 1

Thomas Baruchel, Sep 07 2003

nonn

Since the computation of the algorithm needs an extension of the integer part over a subset of C, the rule: floor(I*x) = i*floor(x) is used (which is what MuPAD does). The following program computes the exact value of the sum.

For all a(n) in the sequence, the relation: (2k)^2 <= a(n) <= (2k+1)^2 is true.

			kappa(1/sqrt(-203)) = (1/2 + (1/2)i) - (1/29 + (1/29)i)*sqrt(203).

Cf. A086378, A087948.

kappa_1_over_comp_sqrt := proc(n) local a,b,i,p; begin if (a := sqrt(-n)-isqrt(-n)) = 0 then return(0) end_if: a := simplify(1/a,sqrt); i := a := simplify(1/(a - floor(a)),sqrt); p := 1; b := 0; repeat p := p*a; b := b*a+a-floor(a); until (a := simplify(1/(a-floor(a)),sqrt)) = i end_repeat: return(simplify((1-isqrt(n)/sqrt(n))*(1+b/(p-1)+1/a-floor(1/a)),sqrt)); end_proc:

A087948 Sum of successive remainders in computing euclidean algorithm for (1, -1/sqrt(-n)) has real and imaginary parts equal.

Original entry on oeis.org

1, 4, 5, 9, 16, 17, 18, 25, 36, 37, 39, 49, 64, 65, 66, 68, 81, 100, 101, 105, 121, 126, 144, 145, 146, 147, 150, 169, 196, 197, 203, 225, 256, 257, 258, 260, 264, 289, 324, 325, 327, 333, 361, 400, 401, 402, 405, 410, 441, 484, 485, 495, 529, 576, 577, 578, 579

Offset: 1

Thomas Baruchel, Sep 07 2003

nonn

Since the computation of the algorithm needs an extension of the integer part over a subset of C, the rule: floor(i*x) = i*floor(x) is used (which is what MuPAD does). The following program computes the exact value of the sum.

			kappa(-1/sqrt(-105)) = -(1/210 + (1/210)i)*sqrt(105).

Cf. A086378, A087947.

kappa_neg_1_over_comp_sqrt := proc(n) local a,b,i,p; begin if (a := -sqrt(-n)+ceil(sqrt(-n))) = 0 then return(0) end_if: i := a := simplify(1/a,sqrt); p := 1; b := 0; repeat p := p*a; b := b*a+a-floor(a); until (a := simplify(1/(a-floor(a)),sqrt)) = i end_repeat: return(simplify(-(b/(p-1) + 1/a)/sqrt(-n),sqrt)); end_proc:

A088900 Numbers n such that 2nk(n) is an integer, where k(n) is sum of successive remainders when computing the Euclidean algorithm for (1, 1/sqrt(n)) as defined in A086378 (MuPAD program is given there).

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 55, 56, 64, 70, 72, 81, 90, 100, 110, 121, 132, 144, 155, 156, 169, 180, 182, 196, 210, 225, 240, 256, 272, 289, 305, 306, 324, 342, 361, 377, 380, 400, 420, 441, 462, 484, 504, 505, 506, 529, 546, 552, 576, 600, 625

Offset: 1

Thomas Baruchel, Oct 21 2003

nonn

Subset of A086378.

Cf. A086378.

cp's OEIS Frontend

User: Thomas Baruchel

A102708 Composite numbers a(n) having a GCD with Pell_x(a(n))-1 equal to 1 (where Pell_x(n) is the x solution of the Pell-Fermat equation: x^2 - n y^2 = 1). There is no square in the sequence because the Pell-Fermat equation has no solution for them.

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A101443 Continued fraction expansion of (I_0(1/2)/I_1(1/2)-1)/2 = 1.56185896... (where I_n is the modified Bessel function of the first kind).

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A092142 Compute the continued fraction expansion of Pi; multiply each term by i, the square root of -1, compute this new continued fraction and get a number with a real part equal to 0. Then compute the regular continued fraction of the imaginary part of that new number.

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PARI

A094234 a(n) = period of terms in quasi-periodic continued fraction expansion of 2^n*tanh(1).

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Extensions

A087642 Sequence of squarefree n such that Q(sqrt(n)) has no element with a fully periodical continued fraction of period 1.

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A087414 Numbers n such that 2*n*k(n) is rational but not an integer, where k(n) is sum of successive remainders when computing the Euclidean algorithm for (1, 1/sqrt(n)) as defined in A086378 (MuPAD program is given there); numbers belonging to A086378 but not to A088900.

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PARI

A087295 Successive remainders when computing the Euclidean algorithm for (n,m) where m is any positive integer having no common factor with n, gives a list ending with a sublist of Fibonacci sequence. Find m such that this sublist has the greatest length and define a(n) as this length.

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Examples

A087947 Sum of successive remainders in computing Euclidean algorithm for (1, 1/sqrt(-n)) has real and imaginary parts equal.

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MuPAD

A087948 Sum of successive remainders in computing euclidean algorithm for (1, -1/sqrt(-n)) has real and imaginary parts equal.

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MuPAD

A088900 Numbers n such that 2*n*k(n) is an integer, where k(n) is sum of successive remainders when computing the Euclidean algorithm for (1, 1/sqrt(n)) as defined in A086378 (MuPAD program is given there).

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A087414 Numbers n such that 2nk(n) is rational but not an integer, where k(n) is sum of successive remainders when computing the Euclidean algorithm for (1, 1/sqrt(n)) as defined in A086378 (MuPAD program is given there); numbers belonging to A086378 but not to A088900.

A088900 Numbers n such that 2nk(n) is an integer, where k(n) is sum of successive remainders when computing the Euclidean algorithm for (1, 1/sqrt(n)) as defined in A086378 (MuPAD program is given there).