This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A118337 #52 May 04 2025 14:35:54
%S A118337 0,12,33,69,133,252,460,832,1525,2737,4905,8944,16008,28644,52185,
%T A118337 93357,167005,304212,544180,973432,1773133,3171769,5673633,10334632,
%U A118337 18486480,33068412,60234705,107747157,192736885,351073644,627996508,1123352944,2046207205
%N A118337 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 23)^2 = y^2.
%C A118337 Also values x of Pythagorean triples (x, x+23, y).
%C A118337 Corresponding values y of solutions (x, y) are in A156567.
%C A118337 For the generic case x^2 + (x + p)^2 = y^2 with p = m^2 - 2 a (prime) number in A028871, m>=5, the x values are given by the sequence defined by a(n) = 6*a(n-3) - a(n-6) + 2p with a(1)=0, a(2) = 2m + 2, a(3) = 3*m^2 - 10m + 8, a(4)=3p, a(5) = 3*m^2 + 10m + 8, a(6) = 20*m^2 - 58m + 42. Pairs (p, m) are (23, 5), (47, 7), (79, 9), (167, 13), (223, 15), (359, 19), (439, 21), (727, 27), (839, 29), ...
%C A118337 Limit_{n -> oo} a(n)/a(n-3) = 3 + 2*sqrt(2).
%C A118337 Limit_{n -> oo} a(n)/a(n-1) = (27 + 10*sqrt(2))/23 for n mod 3 = {1, 2}.
%C A118337 Limit_{n -> oo} a(n)/a(n-1) = (3 + 2*sqrt(2))/((27 + 10*sqrt(2))/23)^2 for n mod 3 = 0.
%C A118337 For the generic case x^2 + (x + p)^2=y^2 with p = m^2 - 2 a prime number in A028871, m>=5, Y values are given by the sequence defined by b(n) = 6*b(n-3) - b(n-6) with b(1) = p, b(2) = m^2 + 2m + 2, b(3) = 5m^2 - 14m + 10, b(4) = 5p, b(5) = 5m^2 + 14m + 10, b(6) = 29m^2 - 82m + 58. - _Mohamed Bouhamida_, Sep 09 2009
%C A118337 For the generic case x^2 + (x + p)^2 = y^2 with p = m^2 - 2 a prime number, m>=5, the first three consecutive solutions are: (0;p), (2m+2; m^2+2m+2), (3*m^2-10m+8; 5*m^2-14m+10) and the other solutions are defined by: (X(n); Y(n))= (3*X(n-3)+2*Y(n-3)+p; 4*X(n-3)+3*Y(n-3)+2p). - _Mohamed Bouhamida_, Aug 19 2019
%C A118337 X(n) = 6*X(n-3) - X(n-6) + 2*p, and Y(n) = 6*Y(n-3) - Y(n-6) (can be easily proved using X(n) = 3*X(n-3) + 2*Y(n-3) + p, and Y(n) = 4*X(n-3) + 3*Y(n-3) + 2*p). - _Mohamed Bouhamida_, Aug 20 2019
%H A118337 G. C. Greubel, <a href="/A118337/b118337.txt">Table of n, a(n) for n = 1..1000</a>
%H A118337 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F A118337 a(n) = 6*a(n-3) - a(n-6) + 46 for n > 6; a(1)=0, a(2)=12, a(3)=33, a(4)=69, a(5)=133, a(6)=252.
%F A118337 G.f.: x*(12 + 21*x + 36*x^2 - 8*x^3 - 7*x^4 - 8*x^5)/((1-x)*(1 - 6*x^3 + x^6)).
%t A118337 Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+23)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,12,33,69,133,252,460},50] (* _Vladimir Joseph Stephan Orlovsky_, Feb 02 2012 *)
%o A118337 (PARI) forstep(n=0, 1124000000, [1, 3], if(issquare(2*n*(n+23)+529), print1(n, ",")))
%o A118337 (PARI) x='x+O('x^30); concat([0], Vec(x*(12+21*x+36*x^2-8*x^3-7*x^4-8*x^5)/((1-x)*(1-6*x^3+x^6)))) \\ _G. C. Greubel_, May 04 2018
%o A118337 (Magma) I:=[0,12,33,69,133,252,460]; [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..30]]; // _G. C. Greubel_, May 04 2018
%Y A118337 Cf. A156567, A028871 (primes of form n^2 - 2), A156035 (decimal expansion of 3 + 2*sqrt(2)), A156571 (decimal expansion of (27 + 10*sqrt(2))/23).
%Y A118337 Cf. A118675 (p=47), A118676 (p=79), A130608 (p=167), A130609 (p=223), A130610 (p=359), A130645 (p=439), A130646 (p=727), A130647 (p=839).
%K A118337 nonn,easy
%O A118337 1,2
%A A118337 _Mohamed Bouhamida_, May 14 2006
%E A118337 Edited by _Klaus Brockhaus_, Feb 10 2009