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 A294972 #24 Nov 22 2017 10:36:18 %S A294972 1,4,41,127,295,1012,10415,32257,74929,257044,2645369,8193151, %T A294972 19031671,65288164,671913311,2081028097,4833969505,16582936612, %U A294972 170663335625,528572943487,1227809222599,4212000611284,43347815335439,134255446617601,311858708570641,1069831572329524,11010174431865881 %N A294972 Numerators of continued fraction convergents to sqrt(7)/2. %C A294972 The denominators are given in A294973. %C A294972 The continued fraction expansion of sqrt(7)/2 is 1, repeat(3, 10, 3, 2). %H A294972 Colin Barker, <a href="/A294972/b294972.txt">Table of n, a(n) for n = 0..1000</a> %H A294972 <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 A294972 From _Colin Barker_, Nov 19 2017: (Start) %F A294972 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 A294972 a(n) = 254*a(n-4) - a(n-8) for n > 7. %F A294972 (End) %F A294972 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 A294972 Numerator[Convergents[Sqrt[7]/2, 30]] (* _Vaclav Kotesovec_, Nov 19 2017 *) %o A294972 (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 A294972 Cf. A242703, A294973. %K A294972 nonn,cofr,frac,easy %O A294972 0,2 %A A294972 _Wolfdieter Lang_, Nov 18 2017