A077425 -id:A077425 - cp's OEIS frontend

A077058 Minimal positive solution a(n) of Diophantine equation b(n)^2 - b(n)a(n) - G(n)a(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077057(n).

1, 1, 2, 1, 1, 8, 2, 10, 1, 1, 40, 5, 2, 3, 250, 1, 1, 106, 3, 1138, 2, 8, 25, 146, 1, 1, 2968, 15, 298, 16, 2, 5, 352, 17, 1856, 1, 1, 9384, 97, 10, 8, 253970, 2, 72664, 3, 6440, 5, 521904, 1, 1, 3034, 5, 9148450, 3, 1084152, 117, 2, 45, 746, 10, 88, 157, 126890, 1, 1

Offset: 1

Wolfdieter Lang, Nov 29 2002

nonn

This equation can also be written as (2*b(n)-a(n))^2 - D(n)*a(n)^2 = +4 or -4 with D(n) := A077425(n)=1+4*G(n).
This is from Perron's table (see reference p. 108, for n = 1..28) which gives the minimal x,y values which solve the above mentioned Diophantine equations.
For Pell equation x^2 - D*y^2 = +4, see A077428 and A078355. For Pell equation x^2 - D*y^2 = -4, see A078356 and A078357.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

g[n_] := Ceiling[ Sqrt[n] ] + n - 1; r[n_] := Reduce[an > 0 && (bn^2 - bn *an - g[n]*an^2 == 1 || bn^2 - bn *an - g[n]*an^2 == - 1), {an, bn}, Integers] /. C -> c; ab[n_] := DeleteCases[ Flatten[ Table[{an, bn} /. {ToRules[r[n]]} // Simplify, {c[1], 0, 1}] , 1] , an | bn]; a[n_] := a[n] = Min[ ab[n][[All, 1]] ]; Table[ Print[{n, a[n]}]; a[n], {n, 1, 65}] (* Jean-François Alcover, Oct 03 2012 *)

forstep(D=1,1000,4, if(issquare(D),next); u=bnfinit(x^2-D).fu[1]; k=1; while( denominator(t=polcoeff(lift(u^k),1)*2)>1, k++); print1(abs(t),", "); ) \\ Max Alekseyev, Feb 06 2010

More terms from Max Alekseyev, Feb 06 2010

A053373 Write fundamental unit for real quadratic field of discriminant n as x + y*omega; sequence gives values of y for n == 1 (mod 4).

Original entry on oeis.org

1, 1, 2, 1, 1, 8, 2, 10, 1, 40, 5, 2, 3, 250, 1, 1, 106, 3, 1138, 2, 8, 25, 146, 2968, 15, 298, 16, 2, 5, 17, 1856, 1, 1, 9384, 97, 10, 253970, 2, 72664, 3, 6440, 5, 521904, 1, 1, 3034, 5, 9148450, 1084152, 117, 2, 746, 10, 88, 157, 126890, 1, 1, 1311, 56, 287

Offset: 1

N. J. A. Sloane, Jan 06 2000

Entries are indexed by values of n from A039955.

Subsequence of A077058 excluding terms for which A077425(n) is not squarefree. - Max Alekseyev, Dec 12 2012

R. A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.

Emmanuel Vantieghem, Table of n, a(n) for n=1..1000
S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]

Cf. A053370-A053375.

2*NumberFieldFundamentalUnits[ Sqrt[#] ][[1, 2, 2]] & /@ Select[ Range[5, 309, 4], SquareFreeQ ]  (* Jean-François Alcover, Jul 09 2013 *)

forstep(n=5,1000,4, if(!issquarefree(n),next); print1( 2*polcoeff(lift(bnfinit(x^2-n).fu[1]),1), ", " )) /* Max Alekseyev */

A166409 Odd numbers corresponding to the positions of zeros in A166406.

Original entry on oeis.org

5, 13, 17, 21, 29, 33, 37, 41, 45, 53, 57, 61, 65, 69, 73, 77, 85, 89, 93, 97, 99, 101, 105, 109, 113, 117, 125, 129, 133, 137, 141, 145, 147, 149, 153, 157, 161, 165, 173, 177, 181, 185, 189, 193, 197, 201, 205, 207, 209, 213, 217, 221, 229, 233, 237, 241

Offset: 1

Antti Karttunen, Oct 21 2009, Oct 22 2009

nonn

Those odd numbers 2n+1 for which the sum of i in [1,2n+1] with J(i,2n+1)=-1 is equal to the sum of i in [1,2n+1] with J(i,2n+1)=+1. Here J(i,k) is the Jacobi symbol.

Probably a union of A077425 & A165603: It is clear that A077425 is a subsequence of this sequence. For the remaining terms to be equal to A165603, it is at least required that the intersection of A165603 and A095100 be empty.

Cf. A077425, A095100, A165603.

from sympy import jacobi_symbol as J
def a(n):
    l=0
    m=0
    for i in range(1, 2*n + 2):
        if J(i, 2*n + 1)==-1: l+=i
        elif J(i, 2*n + 1)==1: m+=i
    return l - m
print([2*n + 1 for n in range(201) if a(n)==0]) # Indranil Ghosh, Jun 12 2017

cp's OEIS Frontend

A077058 Minimal positive solution a(n) of Diophantine equation b(n)^2 - b(n)a(n) - G(n)a(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077057(n).

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A053373 Write fundamental unit for real quadratic field of discriminant n as x + y*omega; sequence gives values of y for n == 1 (mod 4).

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A166409 Odd numbers corresponding to the positions of zeros in A166406.

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cp's OEIS Frontend

A077058 Minimal positive solution a(n) of Diophantine equation b(n)^2 - b(n)*a(n) - G(n)*a(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077057(n).

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Author

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Mathematica

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Extensions

A053373 Write fundamental unit for real quadratic field of discriminant n as x + y*omega; sequence gives values of y for n == 1 (mod 4).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

A166409 Odd numbers corresponding to the positions of zeros in A166406.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

A077058 Minimal positive solution a(n) of Diophantine equation b(n)^2 - b(n)a(n) - G(n)a(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077057(n).