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 A145542 #53 Apr 09 2025 04:22:31
%S A145542 1,3,7,24,55,189,433,1488,3409,11715,26839,92232,211303,726141,
%T A145542 1663585,5716896,13097377,45009027,103115431,354355320,811826071,
%U A145542 2789833533,6391493137,21964312944,50320119025,172924670019,396169459063,1361433047208,3119035553479
%N A145542 Numerators in continued fraction expansion of sqrt(3/5).
%C A145542 a(n)/A145543(n) tends to sqrt(3/5).
%C A145542 A strong divisibility sequence, that is gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Related to the Lehmer sequence U_n(sqrt(R),Q) with parameters R = 6 and Q = -1. See A041023. - _Peter Bala_, Jun 06 2014
%H A145542 Peter Bala, <a href="/A243469/a243469_1.pdf">Notes on 2-periodic continued fractions and Lehmer sequences</a>
%H A145542 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LehmerNumber.html">Lehmer Number</a>
%F A145542 Numerators in continued fraction expansion of sqrt(3/5); i.e., of [1, 3, 2, 3, 2, 3, 2, 3, 2, ...].
%F A145542 [a(2*n - 1), a(2*n)] = X^n * [1,0], where X is the 2 X 2 matrix [1,2; 3,7].
%F A145542 Empirical G.f.: x*(1+3*x-x^2)/(1-8*x^2+x^4). - _Colin Barker_, Jan 04 2012
%F A145542 From _Peter Bala_, Jun 06 2014: (Start)
%F A145542 a(2*n + 1) = Product_{k=1..n} (6 + 4*cos^2(k*Pi/(2*n+1))).
%F A145542 a(2*n) = 3*Product_{k=1..n-1} (6 + 4*cos^2(k*Pi/(2*n))).
%F A145542 a(2*n + 1) = A070997(n); a(2*n) = 3*A001090(n). (End)
%e A145542 [a(7), a(8)] = [433, 1488] X^4 * [1, 0] = [433, 1488].
%e A145542 a(5) = 55 = 2*a(4) + a(3) = 2*24 + 7.
%e A145542 G.f. = x + 3*x^2 + 7*x^3 + 24*x^4 + 55*x^5 + 189*x^6 + 433*x^7 + 1488*x^8 + ...
%t A145542 Numerator[Convergents[Sqrt[3/5], 30]] (* gives terms with 0 prepended *) (* _Wesley Ivan Hurt_, Jun 15 2014 *)
%o A145542 (PARI) {a(n) = if( n<1, 0, polcoeff( x * (1 + 3*x - x^2) / (1 - 8*x^2 + x^4) + x * O(x^n), n))}; /* _Michael Somos_, Nov 14 2015 */
%Y A145542 Cf. A001090, A041203, A070997, A145543.
%K A145542 nonn,easy
%O A145542 1,2
%A A145542 _Gary W. Adamson_, Oct 12 2008
%E A145542 More terms from _Wesley Ivan Hurt_, Jun 15 2014