1,1

Arises in solving Pell equations of the form X^2 - A*Y^2 = 1.

If A=[A156853] 2025*n^2-649*n+52 (52,1428,6854,..,], or A=[A156854] 2025*n^2-3401*n+1428 (1428,52,2726,...,), or A=[A156855] 2025*n^2-n , n>0, (2024,8098,18222), or A=[A156856] 2025*n^2+n , n>0, (2026,8102,18228); Y=[A156865] 729000*n-612180, n>0, (116820,845820,1574820,), or Y=[A156866] 729000*n-116820, n>0, (612180,1341180, 2070180,...,), or Y=[A156867] 729000*n-180, n>0, (728820,1457820,2186820,...), or Y=[A156868] 729000*n+180, n>0, (729180,1458180,2187180,...,); X=[A157078] 32805000*n^2-55096200*n+23133601, (23133601,842401,44161201,..,), or X=[A157079] 32805000*n^2 - 10513800*n+842401, (842401,23133601,111034801,..,), or X=[A157080] 32805000*n^2- 16200*n+1, n>0, (327888801, 131187601,...,), or X=[A157081] 32805000*n^2+16200*n+1, n>0, (32821201, 131252401,...,)

then we have, for all terms, Pell's equation X^2-A*Y^2=1. Example: 842401^2-52*116820^2=1; 23133601^2-1428*612180^2=1; 32788801^2-2024*728820^2=1; 32821201^2-2026*729180^2=1; 44161201^2-2726*845820^2=1 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 22 2009]

Vincenzo Librandi, X^2-AY^2=1 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 22 2009]

For n=1, a(1)=612180; n=2, a(2)=1341180; n=3, a(3)=2070180

Cf. A156865

Cf. A156867, A156868, A156853, A156854, A156855, A156856, A157078, A157079, A157080, A157081 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 22 2009]

Sequence in context: A022257 A106779 A106780 this_sequence A122131 A161533 A053877

Adjacent sequences: A156863 A156864 A156865 this_sequence A156867 A156868 A156869

nonn

Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 17 2009