A322675 a(n) = n * (4*n + 3)^2.

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

Author

Seiichi Manyama, Dec 23 2018

Keywords

• nonn
• easy

Examples

			(sqrt(2) - sqrt(1))^3 = 5*sqrt(2) - 7 = sqrt(50) - sqrt(49). So a(1) = 49.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

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).

Programs

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

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)