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A379103
Expansion of (1-3*x-sqrt(9*x^2-14*x+1))/4.
2
0, 1, 5, 35, 295, 2765, 27705, 290535, 3148995, 34995065, 396602605, 4566227435, 53259218495, 627982592965, 7473163652705, 89640387354735, 1082664905352795, 13155505626756465, 160709002086562005, 1972595405313408435, 24315686632846439895, 300886761671728853565, 3736205372071338170505, 46540791299676591116535
OFFSET
0,3
COMMENTS
Problem A6 on the 2024 William Lowell Putnam Mathematical Competition was to compute the Hankel transform of this sequence, which is A110147.
LINKS
Nathaniel Johnston, Determinant involving (1 - 3x - sqrt(1 - 14x + 9x^2))/4, YouTube video, 2024.
Nathaniel Johnston, Putnam 2024 A6 solution, 2024.
FORMULA
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.
G.f.: (1-3*x-sqrt(9*x^2-14*x+1))/4.
G.f.: x/(1-5*x/(1-2*x/(1-5*x/(1-2*x/(1-5*x/(...)))))). - Thomas Scheuerle, Feb 28 2025
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
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
PROG
(MATLAB) a = 3; b = 2; c(1) = 1; last_val = 16; for j = 2:last_val
c(j) = a*c(j-1) + b*sum(c(1:j-1).*fliplr(c(1:j-1)));
end
(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
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Nathaniel Johnston, Dec 15 2024
STATUS
approved