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

%I A106525 #35 Sep 08 2022 08:45:18
%S A106525 9,11,21,43,57,119,249,331,693,1451,1929,4039,8457,11243,23541,49291,
%T A106525 65529,137207,287289,381931,799701,1674443,2226057,4660999,9759369,
%U A106525 12974411,27166293,56881771,75620409,158336759,331531257,440748043
%N A106525 Values of x in x^2 - 49 = 2*y^2.
%C A106525 From _Wolfdieter Lang_, Sep 27 2016: (Start)
%C A106525 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.
%C A106525 The y(n) members are given in 2*A276600(n+2).
%C A106525 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)
%C A106525 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.
%H A106525 Colin Barker, <a href="/A106525/b106525.txt">Table of n, a(n) for n = 1..1001</a> (offset adapted by _Georg Fischer_, Jan 31 2019).
%H A106525 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-1).
%F A106525 a(3*k)   = A001541(k)*7 for k >= 2.
%F A106525 a(3*k+1) = (A001541(k+2) + A001541(k-1) + A001541(k) - A001541(k+1))/2;
%F A106525 a(3*k+2) = (A001541(k+2) + A001541(k-1) - A001541(k) + A001541(k+1))/2.
%F A106525 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
%F A106525 From _Colin Barker_, Mar 29 2012: (Start)
%F A106525 a(n) = 6*a(n-3) - a(n-6).
%F A106525 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)
%e A106525 In the following, aa(n) denotes A001541(n):
%e A106525 a(9)=693; as mod(9,3)=0, a(9)=aa(3)*7=99*7=693, also 693^2-49=2*490^2
%e A106525 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.
%e A106525 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
%t A106525 LinearRecurrence[{0,0,6,0,0,-1}, {9,11,21,43,57,119}, 50] (* _Vincenzo Librandi_, Oct 26 2018 *)
%o A106525 (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
%o A106525 (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
%o A106525 (Sage)
%o A106525 def A106525_list(prec):
%o A106525     P.<x> = PowerSeriesRing(ZZ, prec)
%o A106525     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()
%o A106525 a=A106525_list(41); a[1:] # _G. C. Greubel_, Sep 15 2021
%Y A106525 Cf. 14*A001109, 7*A001541, A106525, A106526.
%Y A106525 Cf. A275793, A275794, A275795, A275796, 2*A276600.
%K A106525 nonn,easy
%O A106525 1,1
%A A106525 Andras Erszegi (erszegi.andras(AT)chello.hu), May 07 2005
%E A106525 Entry revised by _N. J. A. Sloane_, Oct 26 2018 at the suggestion of _Georg Fischer_.