|
%I #39 May 10 2017 11:33:32
%S 1,4,4,9,16,25,36,36,49,64,81,100,100,121,144,169,196,196,225,256,289,
%T 324,324,361,400,441,484,484,529,576,625,676,676,729,784,841,900,900,
%U 961,1024,1089,1156,1156,1225,1296,1369
%N Squares with doubled (4*n+2)^2.
%C Also nondecreasing ordered values of A226008 (except 0).
%C Consider A225948/A226008 ordered according to a(n): 0/1, -15/4, -3/4, 2/9, 3/16, 6/25, -7/36, 5/36, 12/49, 15/64, 20/81, ... = b(n)/a(n), and consider the sequence with period 5: 1, 64, 16, 1, 4, ... = t(n); then a(n) = 4*b(n) + t(n).
%C The recurrences in Formula lines are also valid for b(n).
%C Note that the fractions b(n)/a(n) of rank 0, 3,4,5, 8,9,10, ... = A047205:
%C 0, 2/9, 3/16, 6/25, 12/49, 15/64, 20/81, ... are all in A226023(n).
%H Vincenzo Librandi, <a href="/A226096/b226096.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,2,-2,0,0,0,-1,1).
%F a(n+5) - a(n) = 8*A090223(n+4).
%F a(n) = 1 followed by (A090223(n) + 2)^2.
%F a(n) = 3*a(n-5) -3*a(n-10) +a(n-15).
%F G.f.: (x^9 + 3*x^8 + 5*x^6 + 7*x^5 + 7*x^4 + 5*x^3 + 3*x + 1)/((1 - x)*(1 - x^5)^2). [_Ralf Stephan_, May 30 2013]
%F a(n) = a(n-1) +2*a(n-5) -2*a(n-6) -a(n-10) +a(n-11). [_Bruno Berselli_, May 30 2013]
%F a(n) = (24*(16*floor(n/5)^2 + 8*floor(n/5) + 1) - (11 + 24*floor(n/5))*(n - 5*floor(n/5))^4 + 2*(49 + 104*floor(n/5))*(n - 5*floor(n/5))^3 - 23*(11 + 24*floor(n/5))*(n - 5*floor(n/5))^2 + 2*(119 + 280*floor(n/5))*(n - 5*floor(n/5)))/24. - _Luce ETIENNE_, May 08 2017
%t MapIndexed[ If [Mod[First[#2], 4] == 2, Sequence @@ {#1, #1}, #1] &, Range[40]]^2 (* _Jean-François Alcover_, May 28 2013 *)
%Y Cf. A016826, A226008.
%K nonn,easy
%O 0,2
%A _Paul Curtz_, May 26 2013
|
