A077057 -id:A077057 - 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

A078358 Non-oblong numbers: Complement of A002378.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74

Offset: 1

Wolfdieter Lang, Nov 29 2002

The (primitive) period length k(n)=A077427(n) of the (regular) continued fraction of (sqrt(4*a(n)+1)+1)/2 determines whether or not the Diophantine equation (2*x-y)^2 - (1+4*a(n))*y^2 = +4 or -4 is solvable and the approximants of this continued fraction give all solutions. See A077057.
The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446. - Jeremy Gardiner, Mar 16 2003
Infinite series 1/A078358(n) is divergent. Proof: Harmonic series 1/A000027(n) is divergent and can be distributed on two subseries 1/A002378(k+1) and 1/A078358(m). The infinite subseries 1/A002378(k+1) is convergent to 1, so Sum_{n>=1} 1/A078358(n) is divergent. - Artur Jasinski, Sep 28 2008

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Oskar Perron, Die Lehre von den Kettenbrüchen, Teubner, Leipzig, 1913.

a(n)=(A077425(n)-1)/4.

Cf. A049068 (subsequence), A144786.

a078358 n = a078358_list !! (n-1)
a078358_list = filter ((== 0) . a005369) [0..]
-- Reinhard Zumkeller, Jul 04 2014, May 08 2012

Complement[Range[930], Table[n (n + 1), {n, 0, 30}]] (* and *) Table[Ceiling[Sqrt[n]] + n - 1, {n, 900}] (* Vladimir Joseph Stephan Orlovsky, Jul 20 2011 *)

a(n)=sqrtint(n-1)+n \\ Charles R Greathouse IV, Jan 17 2013

from operator import sub
from sympy import integer_nthroot
def A078358(n): return n+sub(*integer_nthroot(n,2)) # Chai Wah Wu, Oct 01 2024

4*a(n)+1 is not a square number.
a(n) = ceiling(sqrt(n)) + n -1. - Leroy Quet, Jul 06 2007
A005369(a(n)) = 0. - Reinhard Zumkeller, Jul 05 2014

A217470 The Diophantine equation x^2 - xy - Gy^2 = -1, G a positive integer, D = 4*G + 1 not a perfect square, has no solution precisely for G = a(n).

Original entry on oeis.org

5, 8, 11, 14, 17, 19, 23, 26, 29, 32, 33, 35, 38, 40, 41, 44, 47, 50, 51, 52, 53, 54, 55, 59, 61, 62, 63, 65, 68, 71, 74, 75, 76, 77, 80, 82, 83, 85, 86, 89, 92, 94, 95, 96, 98, 101

Offset: 1

Wolfdieter Lang, Oct 04 2012

nonn

See the Perron reference for the theorem which by negation implies that this quadratic Diophantine equation has no solution if and only if A077427 is even.
See the pairs (x, y) = (A077057, A077058) which for these a(n) values are the smallest positive solutions of the Diophantine equation x^2 - x*y - a(n)*y^ = +1.
In the table on p. 108 of the Perron reference these a(n) values, called there also G, are the ones were in the third column numbers in brackets appear.
The case D = 4*G + 1 = m^2 > 1 has trivially no solutions: the equation is then X^2 - Y^2 = -4, with  X = |2*x-y|, Y = |m*y|. X and Y are either both even or both odd. In the first case one is led to the equation v^2 - w^2 = (v-w)*(v+w)= -1, with X = 2*v and Y = 2*w, and there is only the solution (v,w) = (0,1), hence 2*x = y, m*y = 2. But then m=2 and y=1 with non-integer x solution. In the other case X = 2*v+1 and Y = 2*w+1, v not w, leading to v + w + 1 = -1 with no positive integer solution. Thanks to T. D. Noe for pointing out that one has to mention that these values G = A002378(k), k>=1, with D a perfect (odd) square, are here not included.

			a(1) = 5 because 5 = A078358(4) and A077427(4) = 2, which is even.

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

Cf. A077057, A077058, A078358, A077427.

a(n) gives the increasingly ordered values for G from A078358 which appear at position k where A077427(k) is even, for k>=1. The next even number in A077427 appears for k = 6 and

A078358(6) = 8, hence a(2) = 8.

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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A078358 Non-oblong numbers: Complement of A002378.

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A217470 The Diophantine equation x^2 - xy - Gy^2 = -1, G a positive integer, D = 4*G + 1 not a perfect square, has no solution precisely for G = a(n).

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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).

Views

Author

Keywords

Comments

References

Programs

Mathematica

PARI

Extensions

A078358 Non-oblong numbers: Complement of A002378.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

A217470 The Diophantine equation x^2 - x*y - G*y^2 = -1, G a positive integer, D = 4*G + 1 not a perfect square, has no solution precisely for G = a(n).

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Author

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Comments

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Formula

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).

A217470 The Diophantine equation x^2 - xy - Gy^2 = -1, G a positive integer, D = 4*G + 1 not a perfect square, has no solution precisely for G = a(n).