A322675 - OEIS

%I #30 Dec 28 2018 09:07:56

%S 0,49,242,675,1444,2645,4374,6727,9800,13689,18490,24299,31212,39325,

%T 48734,59535,71824,85697,101250,118579,137780,158949,182182,207575,

%U 235224,265225,297674,332667,370300,410669,453870,499999,549152,601425,656914,715715,777924,843637

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

%H Colin Barker, <a href="/A322675/b322675.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(n+1) + sqrt(n))^3.

%F sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(n+1) - sqrt(n))^3.

%F 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

%F From _Colin Barker_, Dec 23 2018: (Start)

%F G.f.: x*(49 + 46*x + x^2) / (1 - x)^4.

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3.

%F (End)

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

%o (PARI) {a(n) = n*(4*n+3)^2}

%o (PARI) concat(0, Vec(x*(49 + 46*x + x^2) / (1 - x)^4 + O(x^40))) \\ _Colin Barker_, Dec 23 2018

%Y Column 3 of A322699.

%Y 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).

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Dec 23 2018