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%I #24 Nov 22 2017 10:36:18
%S 1,4,41,127,295,1012,10415,32257,74929,257044,2645369,8193151,
%T 19031671,65288164,671913311,2081028097,4833969505,16582936612,
%U 170663335625,528572943487,1227809222599,4212000611284,43347815335439,134255446617601,311858708570641,1069831572329524,11010174431865881
%N Numerators of continued fraction convergents to sqrt(7)/2.
%C The denominators are given in A294973.
%C The continued fraction expansion of sqrt(7)/2 is 1, repeat(3, 10, 3, 2).
%H Colin Barker, <a href="/A294972/b294972.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,254,0,0,0,-1).
%F From _Colin Barker_, Nov 19 2017: (Start)
%F G.f.: (1 + 4*x + 41*x^2 + 127*x^3 + 41*x^4 - 4*x^5 + x^6 - x^7) / ((1 - 16*x^2 + x^4)*(1 + 16*x^2 + x^4)).
%F a(n) = 254*a(n-4) - a(n-8) for n > 7.
%F (End)
%F The proof of the g.f. runs like the one given for the denominators in A294973. The recurrence for a(n) is the same but the input is now a(0) = b(0) = 1 and a(-1) = 1, (a(-2) = 0). - _Wolfdieter Lang_, Nov 19 2017
%t Numerator[Convergents[Sqrt[7]/2, 30]] (* _Vaclav Kotesovec_, Nov 19 2017 *)
%o (PARI) Vec((1 + 4*x + 41*x^2 + 127*x^3 + 41*x^4 - 4*x^5 + x^6 - x^7) / ((1 - 16*x^2 + x^4)*(1 + 16*x^2 + x^4)) + O(x^40)) \\ _Colin Barker_, Nov 21 2017
%Y Cf. A242703, A294973.
%K nonn,cofr,frac,easy
%O 0,2
%A _Wolfdieter Lang_, Nov 18 2017
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