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 A377264 #32 Nov 04 2024 16:24:31
%S A377264 1,2,3,14,69,413,7222,90211,2577626,127385577,5092018073,655664812074,
%T A377264 78618294607139,13276948495989478,5995083279193033837,
%U A377264 1895278734817024984181,1542333923096758721461086,1867485777936169465836858947,2020248742951852823878208914098,7078136335206254534330825538868049
%N A377264 Consider the recurrence d(k) = (d(k-3)*d(k-2) + 1)/(d(k-5)*d(k-4)*d(k-3)^2*d(k-2)^2*d(k-1)), with d(0..4) = {1,1,1,2,1}. a(n) = numerator(d(2*n+1)).
%C A377264 Consider the sequence s(k) with ordinary generating function: 1/(1-d(0)*x/(1-d(1)*x/(1-d(2)*x/(...)))), the Hankel sequence transform of s(k) is A006720 starting with the third term.
%C A377264 This is a special case of a more general theorem: Consider the sequence h(k) with generating function 1/(1-c(0)*x/(1-c(1)*x/(1-c(2)*x/(...)))), if c(2*k+1) = (c(2*k-2)*c(2*k-1) + t)/(c(2*k-4)*c(2*k-3)*c(2*k-2)^2*c(2*k-1)^2*c(2*k)) for all k with c(< 0) = 1, then the Hankel sequence transform of h(k) satisfies a Somos-4 A(1, t) recurrence.
%C A377264 If we would change the start condition into d(0..4) = {1,1,-1,-2,(5/2)}, the expansion of the continued fraction generating function would give us A171416, its Hankel sequence transform is again A006720. There exist infinitely many sequences with the same Hankel sequence transform.
%F A377264 a(n) = A006720(n+1)*A006720(n+3).
%F A377264 denominator(d(2*n+1)) = A006720(n+2)^2.
%F A377264 -a(n)/A006720(n+3)^2 are the x-coordinates of (2*n+1) times [-1,0] on the curve y^2 - y = x^3 + 3*x^2 + 2*x. "Times" means here the multiplication under the elliptic group law.
%o A377264 (PARI)
%o A377264 d(n) = if(n<5, [1,1,1,2,1][n+1], (d(n-3)*d(n-2)+1)/(d(n-5)*d(n-4)*d(n-3)^2*d(n-2)^2*d(n-1)))
%o A377264 a(n) = numerator(d(2*n+1))
%o A377264 (PARI)
%o A377264 a(n) = -numerator(ellmul(ellinit([0, 3, -1, 2, 0]), [-1,0], 2*n+1)[1])
%Y A377264 Cf. A006720, A028940.
%Y A377264 The Hankel transform is directly related to A006720: A157002, A157003, A160702, A171416, A173992, A173993, A254314.
%K A377264 nonn
%O A377264 0,2
%A A377264 _Thomas Scheuerle_, Oct 22 2024