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 A077447 #22 Jul 14 2025 15:19:55
%S A077447 4,8,16,44,92,256,536,1492,3124,8696,18208,50684,106124,295408,618536,
%T A077447 1721764,3605092,10035176,21012016,58489292,122467004,340900576,
%U A077447 713790008,1986914164,4160273044,11580584408,24247848256,67496592284
%N A077447 Numbers k such that (k^2 - 14)/2 is a square.
%C A077447 The equation "(n^2 - 14)/2 is a square" is a version of the generalized Pell Equation x^2 - D*y^2 = C where x^2 - 2*y^2 = 14.
%D A077447 A. H. Beiler, "The Pellian." Ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.
%D A077447 L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.
%D A077447 Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, pp. 139-147.
%H A077447 J. J. O'Connor and E. F. Robertson, <a href="https://mathshistory.st-andrews.ac.uk/HistTopics/Pell/">Pell's Equation</a>
%H A077447 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PellEquation.html">Pell Equation</a>
%H A077447 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,0,-1).
%F A077447 Limit_{k -> oo} a(2*k+1)/a(2*k) = 2.09383632135605431360 = (9 + 4*sqrt(2))/7 = R1.
%F A077447 Limit_{k -> oo} a(2*k)/a(2*k-1) = 2.78361162489122432754 = (11 + 6*sqrt(2))/7 = R2.
%F A077447 Limit_{k -> oo} a(n)/a(n-2) = 3 + 2*sqrt(2) = RG (Grand Ratio); RG = R1*R2.
%F A077447 For n = 2*k-1, a(n) = [ 2*[(3+2*sqrt(2))^n + (3-2*sqrt(2))^n] - [(3+2*sqrt(2))^(n-1) + (3-2*sqrt(2))^(n-1)] + [(3+2*sqrt(2))^(n-2) + (3-2*sqrt(2))^(n-2)] ] / 4.
%F A077447 For n = 2*k, a(n) = [ 5*[(3+2*sqrt(2))^n + (3-2*sqrt(2))^n] + [(3+2*sqrt(2))^(n-1) + (3-2*sqrt(2))^(n-1)] ] / 4.
%F A077447 a(n) = 6*a(n-2) - a(n-4) = 4*A006452(n).
%F A077447 G.f.: -4*x*(x-1)*(x^2+3*x+1) / ( (x^2+2*x-1)*(x^2-2*x-1) ). - _R. J. Mathar_, Jul 03 2011
%t A077447 LinearRecurrence[{0,6,0,-1},{4,8,16,44},40] (* _Harvey P. Dale_, Jul 22 2013 *)
%Y A077447 Cf. A156649 (R1).
%K A077447 nonn
%O A077447 1,1
%A A077447 _Gregory V. Richardson_, Nov 09 2002