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 A106525 Values of x in x^2 - 49 = 2*y^2. 6
 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 (list; graph; refs; listen; history; text; internal format)
 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. = 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

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Last modified March 28 23:19 EDT 2023. Contains 361596 sequences. (Running on oeis4.)