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

2, 7, 94, 263, 3218, 8959, 109342, 304367, 3714434, 10339543, 126181438, 351240119, 4286454482, 11931824527, 145613270974, 405330793823, 4946564758658, 13769315165479, 168037588523422, 467751384832487, 5708331445037714, 15889777769139103, 193915231542758878

Offset: 0

Author

Helge Robitzsch (hrobi(AT)math.uni-goettingen.de), Aug 01 2000

Keywords

• nonn
• easy

Comments

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)}.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Seon-Hong Kim and Kenneth B. Stolarsky, Translations and extensions of the Nicomachus identity, arXiv:2306.17402 [math.NT], 2023. See also J. Int. Seq. (2024), Vol. 27, Issue 6, Art. No. 24.6.3, p. 12.
• Index entries for linear recurrences with constant coefficients, signature (1,34,-34,-1,1).

Crossrefs

Cf. A055979.

Programs

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

Maple

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

Mathematica

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 *)
LinearRecurrence[{1, 34, -34, -1, 1}, {2, 7, 94, 263, 3218}, 30] (* Vincenzo Librandi, Jan 10 2016 *)

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

Formula

a(n) = floor(A055979(n)/sqrt(2)).

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

a(n) = a(n-1) + 34*a(n-2) - 34*a(n-3) - a(n-4) + a(n-5). - Colin Barker, May 17 2015

Extensions

More terms from Alois P. Heinz, Jun 03 2009