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A056161 Solutions (value of x) of Diophantine equation 2*x^2 + 3*x + 2 = r^2.

%I A056161 #30 Jul 05 2024 10:24:17
%S A056161 2,7,94,263,3218,8959,109342,304367,3714434,10339543,126181438,
%T A056161 351240119,4286454482,11931824527,145613270974,405330793823,
%U A056161 4946564758658,13769315165479,168037588523422,467751384832487,5708331445037714,15889777769139103,193915231542758878
%N A056161 Solutions (value of x) of Diophantine equation 2*x^2 + 3*x + 2 = r^2.
%C A056161 The same equation also has negative solutions x=-c(n), where c would be the sequence {1,2,17,46,553,1538,18761,52222,...} with the corresponding values of r being {1,2,23,64,781,2174,26531,73852,...}. Moreover, replacing x with x+K, one obtains the Diophantine equation 2*x^2+(4*K+3)*x+(2*K^2+3*K+2)=r^2. Since K can be any integer (for example K=-1, giving 2*x^2-x+1=r^2), this amounts to an infinite family of Diophantine equations with closely related solutions. For example, if the present equation has a solution pair {a(n), A055979(n)}, the one with x replaced by x+K will have a solution {a(n)-K, A055979(n)}.
%H A056161 Colin Barker, <a href="/A056161/b056161.txt">Table of n, a(n) for n = 0..1000</a>
%H A056161 Seon-Hong Kim and Kenneth B. Stolarsky, <a href="https://arxiv.org/abs/2306.17402">Translations and extensions of the Nicomachus identity</a>, arXiv:2306.17402 [math.NT], 2023. See also <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Kim/kim8.html">J. Int. Seq.</a> (2024), Vol. 27, Issue 6, Art. No. 24.6.3, p. 12.
%H A056161 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,34,-34,-1,1).
%F A056161 a(n) = floor(A055979(n)/sqrt(2)).
%F A056161 G.f.: (x^4 + x^3 - 19*x^2 - 5*x - 2) / (x^5 - x^4 - 34*x^3 + 34*x^2 + x - 1). - _Alois P. Heinz_, Jun 03 2009
%F A056161 a(n) = a(n-1) + 34*a(n-2) - 34*a(n-3) - a(n-4) + a(n-5). - _Colin Barker_, May 17 2015
%p A056161 a:= n-> (Matrix([94,7,2,-1,-2]). Matrix([[1,1,0,0,0], [34,0,1,0,0], [ -34,0,0,1,0], [ -1,0,0,0,1], [1,0,0,0,0]])^n)[1,3]: seq(a(n), n=0..25); # _Alois P. Heinz_, Jun 03 2009
%t A056161 CoefficientList[Series[(x^4 + x^3 - 19 x^2 - 5 x - 2)/(x^5 - x^4 - 34 x^3 + 34 x^2 + x - 1), {x, 0, 22}], x] (* _Michael De Vlieger_, Jan 09 2016 *)
%t A056161 LinearRecurrence[{1, 34, -34, -1, 1}, {2, 7, 94, 263, 3218}, 30] (* _Vincenzo Librandi_, Jan 10 2016 *)
%o A056161 (PARI) Vec((x^4+x^3-19*x^2-5*x-2)/((x-1)*(x^2-6*x+1)*(x^2+6*x+1)) + O(x^100)) \\ _Colin Barker_, May 17 2015
%o A056161 (Magma) I:=[2,7,94,263,3218]; [n le 5 select I[n] else Self(n-1)+34*Self(n-2)-34*Self(n-3)-Self(n-4)+Self(n-5): n in [1..30]]; // _Vincenzo Librandi_, Jan 10 2016
%Y A056161 Cf. A055979.
%K A056161 nonn,easy
%O A056161 0,1
%A A056161 Helge Robitzsch (hrobi(AT)math.uni-goettingen.de), Aug 01 2000
%E A056161 More terms from _Alois P. Heinz_, Jun 03 2009