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 A129836 #22 Oct 21 2022 21:58:47
%S A129836 0,15,228,291,368,1575,1940,2387,9416,11543,14148,55115,67512,82695,
%T A129836 321468,393723,482216,1873887,2295020,2810795,10922048,13376591,
%U A129836 16382748,63658595,77964720,95485887,371029716,454411923,556532768
%N A129836 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 97)^2 = y^2.
%C A129836 Also values x of Pythagorean triples (x, x + 97, y).
%C A129836 Corresponding values y of solutions (x, y) are in A157469.
%C A129836 For the generic case x^2 + (x + p)^2 = y^2 with p = 2*m^2 - 1 a (prime) number in A066436, 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 + 1, a(3) = 6m^2 - 10m + 4, a(4) = 3p, a(5) = 6m^2 + 10m + 4, a(6) = 40m^2 - 58m + 21 (cf. A118673).
%C A129836 Pairs (p, m) are (7, 2), (17, 3), (31, 4), (71, 6), (97, 7), (127, 8), (199, 10), (241, 11), (337, 13), (449, 15), (577, 17), (647, 18), (881, 21), (967, 22), ...
%C A129836 lim_{n -> infinity} a(n)/a(n-3) = 3 + 2*sqrt(2).
%C A129836 lim_{n -> infinity} a(n)/a(n-1) = (99 + 14*sqrt(2))/97 for n mod 3 = {1, 2}.
%C A129836 lim_{n -> infinity} a(n)/a(n-1) = (19491 + 12070*sqrt(2))/97^2 for n mod 3 = 0.
%C A129836 For the generic case x^2 + (x + p)^2 = y^2 with p = 2*m^2 - 1 a prime number in A066436, m>=2, Y values are given by the sequence defined by b(n) = 6*b(n-3) - b(n-6) with b(1) = p, b(2) = 2m^2 + 2m + 1, b(3) = 10m^2 - 14m + 5, b(4) = 5p, b(5) = 10m^2 + 14m + 5, b(6) = 58m^2 - 82m + 29. - _Mohamed Bouhamida_, Sep 09 2009
%H A129836 G. C. Greubel, <a href="/A129836/b129836.txt">Table of n, a(n) for n = 1..1000</a>
%H A129836 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F A129836 a(n) = 6*a(n-3) - a(n-6) + 194 for n > 6; a(1)=0, a(2)=15, a(3)=228, a(4)=291, a(5)=368, a(6)=1575.
%F A129836 G.f.: x*(15 + 213*x + 63*x^2 - 13*x^3 - 71*x^4 - 13*x^5)/((1-x)*(1 - 6*x^3 + x^6)).
%F A129836 a(3*k + 1) = 97*A001652(k) for k >= 0.
%t A129836 ClearAll[a]; Evaluate[Array[a, 6]] = {0, 15, 228, 291, 368, 1575}; a[n_] := a[n] = 6*a[n-3] - a[n-6] + 194; Table[a[n], {n, 1, 29}] (* _Jean-François Alcover_, Dec 27 2011, after given formula *)
%t A129836 LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,15,228,291,368,1575,1940}, 50] (* _G. C. Greubel_, May 07 2018 *)
%o A129836 (PARI) forstep(n=0, 600000000, [3, 1], if(issquare(2*n^2+194*n+9409), print1(n, ",")))
%o A129836 (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(15+213*x+63*x^2-13*x^3-71*x^4-13*x^5)/((1-x)*(1-6*x^3 + x^6)))); // _G. C. Greubel_, May 07 2018
%Y A129836 Cf. A157469, A066436 (primes of the form 2*n^2 - 1), A001652, A118673, A118674, A156035 (decimal expansion of 3 + 2*sqrt(2)), A157470 (decimal expansion of (99 + 14*sqrt(2))/97), A157471 (decimal expansion of (19491 + 12070*sqrt(2))/97^2).
%K A129836 nonn,easy
%O A129836 1,2
%A A129836 _Mohamed Bouhamida_, May 21 2007
%E A129836 Edited and two terms added by _Klaus Brockhaus_, Mar 12 2009