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 A379103 #31 Apr 28 2025 12:18:25
%S A379103 0,1,5,35,295,2765,27705,290535,3148995,34995065,396602605,4566227435,
%T A379103 53259218495,627982592965,7473163652705,89640387354735,
%U A379103 1082664905352795,13155505626756465,160709002086562005,1972595405313408435,24315686632846439895,300886761671728853565,3736205372071338170505,46540791299676591116535
%N A379103 Expansion of (1-3*x-sqrt(9*x^2-14*x+1))/4.
%C A379103 Problem A6 on the 2024 William Lowell Putnam Mathematical Competition was to compute the Hankel transform of this sequence, which is A110147.
%C A379103 Given constants X and Y, let A(x) = (1 - x*(X - Y) - sqrt(1 - 2*x*(X + Y) + x^2*(X - Y)^2))/(2*Y) = x*(1) + x^2*(X) + x^3*X*(X + Y) + x^4*X*(X^2 + 3*X*Y + Y^2) + ... where the coefficients of A(x) is the Narayana triangle A090181. A(x) satisfies 0 = x - A(x)*(1 - x*(X-Y)) + A(x)^2*Y. The Hankel transform of the coefficients 1, X, X*(X + Y), ... is the sequence 1, (X*Y), (X*Y)^2, ... while the Hankel transform of X, X*(X + Y), X*(X^2 + 3*X*Y + Y^2), ... is the sequence X, X^3*Y, X^6*Y^3, X^10*Y^6, .... In the case of this sequence, X = 5 and Y = 2. - _Michael Somos_, Apr 26 2025
%H A379103 Nathaniel Johnston, <a href="https://www.youtube.com/watch?v=siTb64vXP6A">Determinant involving (1 - 3x - sqrt(1 - 14x + 9x^2))/4</a>, YouTube video, 2024.
%H A379103 Nathaniel Johnston, <a href="https://njohnston.ca/videos/siTb64vXP6A.pdf">Putnam 2024 A6 solution</a>, 2024.
%F A379103 a(0) = 0, a(1) = 1, a(n) = 3*a(n-1) + 2*Sum_{k=0..n} a(k)*a(n-k) for n >= 2.
%F A379103 G.f.: (1-3*x-sqrt(9*x^2-14*x+1))/4.
%F A379103 G.f.: x/(1-5*x/(1-2*x/(1-5*x/(1-2*x/(1-5*x/(...)))))). - _Thomas Scheuerle_, Feb 28 2025
%F A379103 a(n) = (1/4)*(-1)^(n+1) * Sum_{k=0..n} binomial(1/2,k) * binomial(1/2,n-k) * (7+2*sqrt(10))^k * (7-2*sqrt(10))^(n-k) for n >= 2. - _Ehren Metcalfe_, Feb 26 2025
%F A379103 a(n) ~ 5^(1/4) * (7 + 2*sqrt(10))^(n - 1/2) / (2^(7/4) * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 27 2025
%F A379103 The g.f. A(x) satisfies 0 = x - (1 - 3*x)*A(x) + 2*A(x)^2 and A(x) = x + 3*x*A(x) + 2*A(x)^2. - _Michael Somos_, Apr 26 2025
%e A379103 G.f. = x + 5*x^2 + 35*x^3 + 295*x^4 + 2765*x^5 + 27705*x^6 + ... - _Michael Somos_, Apr 26 2025
%t A379103 a[ n_] := SeriesCoefficient[ (1 - 3*x - Sqrt[1 - 14*x + 9*x^2])/4, {x, 0, n}]; (* _Michael Somos_, Apr 26 2025 *)
%t A379103 a[ n_] := With[{X = 5, Y = 2}, SeriesCoefficient[ Nest[x/(1 - (X-Y)*x - Y*#)&, O[x], n], {x, 0, n}]]; (* _Michael Somos_, Apr 28 2025 *)
%t A379103 a[ n_] := With[{X = 5, Y = 2}, SeriesCoefficient[ Nest[x/(1 - X*x/(1 - Y*#))&, O[x], Ceiling[n/2]], {x, 0, n}]]; (* _Michael Somos_, Apr 28 2025 *)
%o A379103 (MATLAB) a = 3;b = 2;c(1) = 1;last_val = 16;for j = 2:last_val
%o A379103 c(j) = a*c(j-1) + b*sum(c(1:j-1).*fliplr(c(1:j-1)));
%o A379103 end
%o A379103 (PARI) my(x='x+O('x^33)); concat([0],Vec((1-3*x-sqrt(9*x^2-14*x+1))/4)) \\ _Joerg Arndt_, Dec 15 2024
%o A379103 (PARI) a(n) = my(A = O(x)); for(k=1, n, A = x + 3*x*A + 2*A^2); polcoeff(A, n); /* _Michael Somos_, Apr 26 2025 */
%o A379103 (PARI) a(n) = my(A = O(x), X = 5, Y = 2); for(k = 1, n, A = x/(1 - (X-Y)*x - Y*A)); polcoeff(A, n); /* _Michael Somos_, Apr 28 2025 */
%o A379103 (PARI) a(n) = my(A = O(x), X = 5, Y = 2); for(k = 1, (n+1)\2, A = x/(1 - X*x/(1 - Y*A))); polcoeff(A, n); /* _Michael Somos_, Apr 28 2025 */
%Y A379103 Cf. A025231, A047891, A082298, A110147.
%K A379103 nonn,easy
%O A379103 0,3
%A A379103 _Nathaniel Johnston_, Dec 15 2024