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a(n) = ((n^2+1)^3) - s, where s is the nearest square to (n^2+1)^3.
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%I #19 Nov 04 2024 03:13:58

%S -1,4,-24,13,-113,28,-316,49,-681,76,-1256,109,-2089,148,-3228,193,

%T -4721,244,-6616,301,-8961,364,-11804,433,-15193,508,-19176,589,

%U -23801,676,-29116,769,-35169,868,-42008,973,-49681,1084,-58236,1201,-67721,1324,-78184,1453,-89673,1588,-102236,1729,-115921,1876

%N a(n) = ((n^2+1)^3) - s, where s is the nearest square to (n^2+1)^3.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-6,0,4,0,-1).

%F a(n) = (n^2+1)^3 - (round(sqrt((n^2+1)^3)))^2.

%F Recurrence formula: a(n)= - a(n-2) + 4*a(n-4) - 6*a(n-6) + 4*a(n-8).

%F a(n) = -A077119(n^2+1). - _R. J. Mathar_, Jan 18 2021

%F a(2*n) = A056107(n). - _R. J. Mathar_, Jan 18 2021

%e Table of n, n^2, n^2+1, (n^2+1)^3, closest square, difference:

%e 1 1 2 8 9 -1

%e 2 4 5 125 121 4

%e 3 9 10 1000 1024 -24

%e 4 16 17 4913 4900 1

%e ...

%t aa = {}; Do[AppendTo[aa, (n^2 + 1)^3 - Round[Sqrt[(n^2 + 1)^3]]^2], {n, 1, 50}]; aa

%Y Cf. A077119, A056107.

%K sign,easy

%O 1,2

%A _Artur Jasinski_, Dec 05 2013