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A322675
a(n) = n * (4*n + 3)^2.
3
0, 49, 242, 675, 1444, 2645, 4374, 6727, 9800, 13689, 18490, 24299, 31212, 39325, 48734, 59535, 71824, 85697, 101250, 118579, 137780, 158949, 182182, 207575, 235224, 265225, 297674, 332667, 370300, 410669, 453870, 499999, 549152, 601425, 656914, 715715, 777924, 843637
OFFSET
0,2
FORMULA
sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(n+1) + sqrt(n))^3.
sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(n+1) - sqrt(n))^3.
Sum_{n>=1} 1/a(n) = 8/27 + 2*c/3 + Pi/18 - Pi^2/12 - log(2)/3 = 0.027956857336446942649782759291008857522041405948099294509008..., where c is the Catalan constant A006752. - Vaclav Kotesovec, Dec 23 2018
From Colin Barker, Dec 23 2018: (Start)
G.f.: x*(49 + 46*x + x^2) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3.
(End)
EXAMPLE
(sqrt(2) - sqrt(1))^3 = 5*sqrt(2) - 7 = sqrt(50) - sqrt(49). So a(1) = 49.
PROG
(PARI) {a(n) = n*(4*n+3)^2}
(PARI) concat(0, Vec(x*(49 + 46*x + x^2) / (1 - x)^4 + O(x^40))) \\ Colin Barker, Dec 23 2018
CROSSREFS
Column 3 of A322699.
sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(n+1) + sqrt(n))^k: A033996(n) (k=2), this sequence (k=3), A322677 (k=4), A322745 (k=5).
Sequence in context: A211761 A373707 A373680 * A260198 A243904 A017246
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Dec 23 2018
STATUS
approved