OFFSET
0,2
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(n+1) + sqrt(n))^4.
sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(n+1) - sqrt(n))^4.
Sum_{n>=1} 1/a(n) = (5 - Pi^2/2)/16 = 0.004074862465957543161422156253870277... - Vaclav Kotesovec, Dec 23 2018
From Colin Barker, Dec 23 2018: (Start)
G.f.: 96*x*(3 + x)*(1 + 3*x) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4.
(End)
EXAMPLE
(sqrt(2) - sqrt(1))^4 = (sqrt(9) - sqrt(8))^2 = sqrt(289) - sqrt(288). So a(1) = 288.
PROG
(PARI) {a(n) = 16*n*(n+1)*(2*n+1)^2}
(PARI) concat(0, Vec(96*x*(3 + x)*(1 + 3*x) / (1 - x)^5 + O(x^40))) \\ Colin Barker, Dec 23 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Dec 23 2018
STATUS
approved