Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #24 Feb 16 2024 06:34:58
%S 0,9,60,93,140,429,620,893,2576,3689,5280,15089,21576,30849,88020,
%T 125829,179876,513093,733460,1048469,2990600,4274993,6111000,17430569,
%U 24916560,35617593,101592876,145224429,207594620,592126749,846430076
%N Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 31)^2 = y^2.
%C Also values x of Pythagorean triples (x, x+31, y).
%C Corresponding values y of solutions (x, y) are in A157646.
%C For the generic case x^2 + (x + p)^2 = y^2 with p = 2*m^2 - 1 a (prime) number in A066436 see A118673 or A129836.
%C lim_{n -> infinity} a(n)/a(n-3) = 3 + 2*sqrt(2).
%C lim_{n -> infinity} a(n)/a(n-1) = (33 + 8*sqrt(2))/31 for n mod 3 = {1, 2}.
%C lim_{n -> infinity} a(n)/a(n-1) = (1539 + 850*sqrt(2))/31^2 for n mod 3 = 0.
%H G. C. Greubel, <a href="/A118674/b118674.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F a(n) = 6*a(n-3) - a(n-6) + 62 for n > 6; a(1)=0, a(2)=9, a(3)=60, a(4)=93, a(5)=140, a(6)=429.
%F G.f.: x*(9 + 51*x + 33*x^2 - 7*x^3 - 17*x^4 - 7*x^5)/((1-x)*(1 - 6*x^3 + x^6)).
%F a(3*k + 1) = 31*A001652(k) for k >= 0.
%t ClearAll[a]; Evaluate[Array[a, 6]] = {0, 9, 60, 93, 140, 429}; a[n_] := a[n] = 6*a[n-3] - a[n-6] + 62; Table[a[n], {n, 1, 31}] (* _Jean-François Alcover_, Dec 27 2011, after given formula *)
%t LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,9,60,93,140,429,620}, 50] (* _G. C. Greubel_, Mar 31 2018 *)
%o (PARI) {forstep(n=0, 850000000, [1, 3], if(issquare(2*n^2+62*n+961), print1(n, ",")))};
%o (Magma) I:=[0,9,60,93,140,429,620]; [n le 7 select I[n] else Self(n-1) - 6*Self(n-3) - 6*Self(n-4) - Self(n-6) + Self(n-7): n in [1..50]]; // _G. C. Greubel_, Mar 31 2018
%Y cf. A157646, A066436 (primes of the form 2*n^2-1), A118673, A129836, A001652, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3 + 2*sqrt(2)), A157647 (decimal expansion of (33 + 8*sqrt(2))/31), A157648 (decimal expansion of (1539 + 850*sqrt(2))/31^2).
%K nonn,easy
%O 1,2
%A _Mohamed Bouhamida_, May 19 2006
%E Edited by _Klaus Brockhaus_, Mar 11 2009