A259911 -id:A259911 - cp's OEIS frontend

A259912 Discriminant of the field of the number having constant continued fraction [n,n,n,...].

5, 8, 13, 5, 29, 40, 53, 17, 85, 104, 5, 37, 173, 8, 229, 65, 293, 328, 365, 101, 445, 488, 533, 145, 629, 680, 733, 197, 5, 904, 965, 257, 1093, 1160, 1229, 13, 1373, 1448, 61, 401, 1685, 1768, 1853, 485, 2029, 2120, 2213, 577, 2405, 2504, 2605, 677, 2813

Offset: 1

Clark Kimberling, Jul 20 2015

Central numbers of the triangle at A259911.

It appears that a(n) = 5 for n in A002878 = (1,4,11,29,...), a bisection of the Lucas sequence.

			[3,3,3,...] = (1/2)(3 + sqrt(13)), so that a(3) = 13.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A259911, A259913.

t = Table[FromContinuedFraction[{n, {n}}], {n, 1, 100}];
Flatten[NumberFieldDiscriminant[t]]

A259913 Discriminant of the number field containing the number with periodic continued fraction [1,n,1,n,1,n,...].

Original entry on oeis.org

5, 12, 21, 8, 5, 60, 77, 24, 13, 140, 165, 12, 221, 28, 285, 5, 357, 44, 437, 120, 21, 572, 69, 168, 29, 780, 93, 56, 957, 1020, 1085, 8, 1221, 1292, 1365, 40, 1517, 1596, 1677, 440, 205, 1932, 2021, 33, 5, 92, 2397, 156, 53, 12, 2805, 728, 3021, 348, 3245

Offset: 1

Clark Kimberling, Jul 20 2015

a(n) is the first term in row n of the triangle at A259911.

It appears that a(n) = 5 if n is a nonzero term of A004146.

			[1,3,1,3,1,3,...] = (1/6)(3 + sqrt(21)), so that a(3) = 21.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A259911.

v = Table[FromContinuedFraction[{1, {n, 1}}], {n, 1, 60}];
Flatten[NumberFieldDiscriminant[v]]

A378873 Squarefree part of A378872(n) (the discriminant of the minimal polynomial of a number whose continued fraction expansion has periodic part given by the n-th composition (in standard order)).

Original entry on oeis.org

5, 2, 5, 13, 3, 3, 5, 5, 21, 2, 10, 21, 10, 10, 5, 29, 2, 15, 17, 15, 85, 85, 6, 2, 17, 85, 6, 17, 6, 6, 5, 10, 5, 6, 26, 13, 37, 37, 165, 6, 37, 2, 221, 37, 3, 221, 65, 5, 26, 37, 165, 37, 221, 3, 65, 26, 165, 221, 65, 165, 65, 65, 5, 53, 15, 35, 37, 3, 229

Offset: 1

Pontus von Brömssen, Dec 10 2024

nonn

Any number x whose continued fraction expansion is eventually periodic can be written uniquely as x = (c+f*sqrt(d))/b, where b, c, f, d are integers, b > 0, d > 0 is squarefree, and GCD(b,c,f) = 1. a(n) is equal to d when the periodic part of the continued fraction of x is given by the n-th composition. If two numbers have eventually periodic continued fraction expansions with the same periodic part, their respective values of d are the same.

			For n = 6, the 5th composition is (1,2). The value of the continued fraction 1+1/(2+1/(1+1/(2+...))) is (1+sqrt(3))/2, so a(6) = 3.

Wikipedia, Discriminant of an algebraic number field.

Cf. A007913, A066099 (compositions in standard order), A246904, A246922, A259911, A259912, A305311, A378872, A378874.

a(n) = A007913(A378872(n)) = A378872(n)/A378874(n)^2.
a(2^n) = A259912(n+1) if a(2^n) == 1 (mod 4), a(2^n) = A259912(n+1)/4 otherwise.
For n > k >= 0, a(2^n+2^k) = A259911(n,k+1) if a(2^n+2^k) == 1 (mod 4), a(2^n+2^k) = A259911(n,k+1)/4 otherwise.

A259532 Triangular array: row k shows the discriminant of sqrt(j) + sqrt(k) for j=1..k.

Original entry on oeis.org

1, 8, 8, 12, 8, 12, 1, 2304, 2304, 1, 5, 8, 12, 8, 5, 24, 1600, 12, 12, 1600, 24, 28, 2304, 3600, 1, 3600, 2304, 28, 8, 12544, 2304, 5, 5, 2304, 12544, 8, 1, 8, 7056, 24, 5, 24, 7056, 8, 1, 40, 8, 2304, 28, 14400, 14400, 28, 2304, 8, 40, 44, 1600, 12, 8

Offset: 1

Clark Kimberling, Jul 20 2015

			First seven rows:
1
8   8
12  8     12
1   2304  2304   1
5   8     12     8   5
24  1600  12     12  1600  24
28  2304  3600   1   3600  2304  28

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A259911, A259912, A259913.

t = Table[NumberFieldDiscriminant[Sqrt[j] + Sqrt[k + 1 - j]], {k, 1, 16}, {j, 1, k}]
TableForm[t] (* A259532 array *)
Flatten[t]   (* A259532 sequence *)

cp's OEIS Frontend

A259912 Discriminant of the field of the number having constant continued fraction [n,n,n,...].

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A259913 Discriminant of the number field containing the number with periodic continued fraction [1,n,1,n,1,n,...].

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A378873 Squarefree part of A378872(n) (the discriminant of the minimal polynomial of a number whose continued fraction expansion has periodic part given by the n-th composition (in standard order)).

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A259532 Triangular array: row k shows the discriminant of sqrt(j) + sqrt(k) for j=1..k.

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Mathematica