A106525 Values of x in x^2 - 49 = 2*y^2.

9, 11, 21, 43, 57, 119, 249, 331, 693, 1451, 1929, 4039, 8457, 11243, 23541, 49291, 65529, 137207, 287289, 381931, 799701, 1674443, 2226057, 4660999, 9759369, 12974411, 27166293, 56881771, 75620409, 158336759, 331531257, 440748043

Offset

1,1

Comments

From Wolfdieter Lang, Sep 27 2016: (Start)

These are the x members of all positive solutions (x(n), y(n)), proper and improper, of the Pell equation x^2 - 2*y^2 = 7^2.

The y(n) members are given in 2*A276600(n+2).

This sequence is composed of the two y members of the two proper classes of solutions of the Pell equation x^2 - 2*y^2 = 7^2 and of 7 times the proper solutions X of the Pell equation X^2 - 2*Y^2 = +1. See A275793, A275795 and 7*A001541. See A275793 for further information, and the Nagell reference. (End)

The sums of the consecutive integers in the following sequences will be squares: for n, i >= 1, if mod(i,3)=0 then 7*n+1, 7*n+2, ..., a(i)*n + (A001541(i/3)-1)/2; otherwise, if mod(i,3)=1 or 2 then 7*n+4, 7*n+5, ..., a(i)*n + (a(i)-1)/2.

Links

Colin Barker, Table of n, a(n) for n = 1..1001 (offset adapted by Georg Fischer, Jan 31 2019).

Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Formula

a(3*k) = A001541(k)*7 for k >= 2.

a(3*k+1) = (A001541(k+2) + A001541(k-1) + A001541(k) - A001541(k+1))/2;

a(3*k+2) = (A001541(k+2) + A001541(k-1) - A001541(k) + A001541(k+1))/2.

a(3*n) = A275793(n), a(3*n+1) = A275795(n), a(3*n+2) = 7*A001541(n+1), n >= 0. - Wolfdieter Lang, Sep 27 2016

From Colin Barker, Mar 29 2012: (Start)

a(n) = 6*a(n-3) - a(n-6).

G.f.: x*(9 + 11*x + 21*x^2 - 11*x^3 - 9*x^4 - 7*x^5)/(1 - 6*x^3 + x^6). (End)

Example

In the following, aa(n) denotes A001541(n):
a(9)=693; as mod(9,3)=0, a(9)=aa(3)*7=99*7=693, also 693^2-49=2*490^2
a(10)=1451; as mod(10,3)=1, a(10)=(aa(5)+aa(2)+aa(3)-aa(4))/2 =(3363+17+99-577)/2=1451, also 1451^2-49=2*1026^2.
The solutions (proper and every third pair improper) of x^2 - 2*y^2 = +49 begin [9, 4], [11, 6], [21, 14], [43, 30], [57, 40], [119, 84], [249, 176], [331, 234], [693, 490], [1451, 1026], [1929, 1364], [4039, 2856], [8457, 5980], [11243, 7950], [23541, 16646], ... - Wolfdieter Lang, Sep 27 2016

Mathematica

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {9, 11, 21, 43, 57, 119}, 50] (* Vincenzo Librandi, Oct 26 2018 *)

Prog

(PARI) Vec((9+11*x+21*x^2-11*x^3-9*x^4-7*x^5)/(1-6*x^3+x^6) + O(x^50)) \\ Colin Barker, Sep 28 2016
(Magma) I:=[9, 11, 21, 43, 57, 119]; [n le 6 select I[n] else 6*Self(n-3)-Self(n-6): n in [1..40]]; // Vincenzo Librandi, Oct 26 2018
(Sage)
def A106525_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( x*(9+11*x+21*x^2-11*x^3-9*x^4-7*x^5)/(1-6*x^3+x^6) ).list()
a=A106525_list(41); a[1:] # G. C. Greubel, Sep 15 2021

Crossrefs

Cf. 14*A001109, 7*A001541, A106525, A106526.

Cf. A275793, A275794, A275795, A275796, 2*A276600.

Sequence in context: A299250 A074345 A022323 * A103510 A233402 A276406

Adjacent sequences: A106522 A106523 A106524 * A106526 A106527 A106528

Keyword

nonn,easy

Author

Andras Erszegi (erszegi.andras(AT)chello.hu), May 07 2005

Extensions

Entry revised by N. J. A. Sloane, Oct 26 2018 at the suggestion of Georg Fischer.

Status

approved