|
|
A322677
|
|
a(n) = 16 * n * (n+1) * (2*n+1)^2.
|
|
3
|
|
|
0, 288, 2400, 9408, 25920, 58080, 113568, 201600, 332928, 519840, 776160, 1117248, 1560000, 2122848, 2825760, 3690240, 4739328, 5997600, 7491168, 9247680, 11296320, 13667808, 16394400, 19509888, 23049600, 27050400, 31550688, 36590400, 42211008, 48455520
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
LINKS
|
|
|
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
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
|
sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(n+1) + sqrt(n))^k: A033996(n) (k=2), A322675 (k=3), this sequence (k=4).
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|