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%I #44 Sep 26 2022 20:04:07
%S 1,1,2,2,3,3,4,5,5,6,7,7,8,8,9,10,10,11,12,12,13,14,14,15,15,16,17,17,
%T 18,19,19,20,21,21,22,22,23,24,24,25,26,26,27,28,28,29,29,30,31,31,32,
%U 33,33,34,35,35,36,36,37,38,38,39,40,40
%N a(n) = floor(A000111(n)/A000111(n-1)).
%C 1, 2, 3, 4, ... first appear at n = 1, 3, 5, 7, 8, 10, 11, 13, ... . a(500) = 318.
%C Numbers appearing only once: interleave 4+7*n, 6+7*n, 9+7*n = 4, 6, 9, 11, 13, 16, ... .
%C This is a nondecreasing sequence.
%C The ratio a(n)/n asymptotically tends to 7/11 = 0.6363... - _Jean-François Alcover_, Jul 21 2015
%H Colin Barker, <a href="/A253671/b253671.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,0,0,1,-1).
%F a(n+2) = a(n+1) + (0, 1, 0, followed by a sequence of period 11: repeat 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1).
%F a(n+12) = a(n+1) + (6, 7, 6, followed by 7's = A010727).
%F a(n) = a(n-1) + a(n-11) - a(n-12) for n>15. - _Colin Barker_, Jan 22 2015
%F G.f.: x*(x^14-x^13+x^12-x^11+x^10+x^9+x^7+x^6+x^4+x^2+1) / ((x-1)^2*(x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)). - _Colin Barker_, Jan 22 2015
%e Floor of 1/1, 1/1, 2/1, 5/2, 16/5, 61/16, ... .
%e 1=1*1+0, 1=1*1+0, 2=2*1+0, 5=2*2+1, 16=3*5+1, 61=3*16+13, 272=4*61+28, ... .
%t max = 500; ee = Table[2^n*EulerE[n, 1] + EulerE[n] - 1, {n, 0, max}]; A000111 = Table[Differences[ee, n] // First // Abs, {n, 0, max}]; Table[Quotient[A000111[[n + 1]], A000111[[n]]], {n, 1, max}] (* _Jean-François Alcover_, Jan 08 2015 *)
%o (PARI) Vec(x*(x^14-x^13+x^12-x^11+x^10+x^9+x^7+x^6+x^4+x^2+1)/((x-1)^2*(x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)) + O(x^100)) \\ _Colin Barker_, Jan 22 2015
%o (Python)
%o # requires python 3.2 or higher
%o from itertools import accumulate
%o A253671_list, blist, l1, l2 = [1], [1], 1, 1
%o for n in range(10**2):
%o ....blist = list(reversed(list(accumulate(reversed(blist))))) + [0] if n % 2 else [0]+list(accumulate(blist))
%o ....l2, l1 = l1, sum(blist)
%o ....A253671_list.append(l1//l2) # _Chai Wah Wu_, Jan 29 2015
%Y Cf. A000111, A010727, A017005, A017029, A017053.
%K nonn,easy
%O 1,3
%A _Paul Curtz_, Jan 08 2015, with the help of _Jean-François Alcover_.
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