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A083479 The natural numbers with all terms of A033638 inserted.

%I A083479 #40 Feb 18 2024 01:36:39
%S A083479 0,1,1,1,2,2,3,3,4,5,5,6,7,7,8,9,10,10,11,12,13,13,14,15,16,17,17,18,
%T A083479 19,20,21,21,22,23,24,25,26,26,27,28,29,30,31,31,32,33,34,35,36,37,37,
%U A083479 38,39,40,41,42,43,43,44,45,46,47,48,49,50,50,51,52,53,54,55,56,57,57
%N A083479 The natural numbers with all terms of A033638 inserted.
%C A083479 Row n of A049597 has a(n+1) nonzero values.
%C A083479 When considering the set of nested parabolas defined by -(x^2) + p*x for integer values of p, a(n) tells us how many parabolas are intersected by the line from (1,n) to (n,n). - _Gregory R. Bryant_, Apr 01 2013
%C A083479 Number of distinct perimeters for polyominoes with n square cells. - _Wesley Prosser_, Sep 06 2017
%H A083479 Reinhard Zumkeller, <a href="/A083479/b083479.txt">Table of n, a(n) for n = 0..10000</a>
%F A083479 a(n) = (n+2) - ceiling(sqrt(4*n)), for n > 0. - _Gregory R. Bryant_, Apr 01 2013
%F A083479 From _Wesley Prosser_, Sep 06 2017: (Start)
%F A083479 a(n) = (n+2) - A027709(n)/2.
%F A083479 a(n) = (n+2) - A027434(n).
%F A083479 a(n) = (2n+2) - A049068(n).
%F A083479 a(n) = (2n+3) - A080037(n).
%F A083479 (End)
%e A083479 There are three 1's, one from the natural numbers and two from A033638.
%e A083479 When viewed as an array the sequence begins:
%e A083479    0
%e A083479    1
%e A083479    1  1
%e A083479    2  2
%e A083479    3  3  4
%e A083479    5  5  6
%e A083479    7  7  8  9
%e A083479   10 10 11 12
%e A083479   13 13 14 15 16
%e A083479   17 17 18 19 20
%e A083479   21 21 22 23 24 25
%e A083479   26 26 27 28 29 30
%e A083479   ...
%t A083479 Table[(n + 2) - Ceiling@ Sqrt[4 n] - 2 Boole[n == 0], {n, 0, 73}] (* _Michael De Vlieger_, Sep 05 2017 *)
%o A083479 (Haskell)
%o A083479 a083479 n = a083479_list !! n
%o A083479 a083479_list = m [0..] a033638_list where
%o A083479    m xs'@(x:xs) ys'@(y:ys) | x <= y    = x : m xs ys'
%o A083479                            | otherwise = y : m xs' ys
%o A083479 -- _Reinhard Zumkeller_, Apr 06 2012
%o A083479 (Maxima)
%o A083479 a(n):=((n+2)-ceiling(sqrt(4*n))); /* _Gregory R. Bryant_, Apr 01 2013 */
%o A083479 (Python)
%o A083479 from math import isqrt
%o A083479 def A083479(n): return n+1-isqrt((n<<2)-1) if n else 0 # _Chai Wah Wu_, Jul 28 2022
%o A083479 (Magma) [n eq 0 select 0 else (n+2)-Ceiling(Sqrt(4*n)): n in [0..100]]; // _G. C. Greubel_, Feb 17 2024
%o A083479 (SageMath) [(n+2)-ceil(sqrt(4*n)) -2*int(n==0) for n in range(101)] # _G. C. Greubel_, Feb 17 2024
%Y A083479 Cf. A002620, A027434, A027709, A033638, A049068, A049597.
%Y A083479 Cf. A054243, A060510, A080037, A083480, A083906.
%K A083479 easy,nonn,tabf
%O A083479 0,5
%A A083479 _Alford Arnold_, Jun 08 2003
%E A083479 Edited and extended by _David Wasserman_, Nov 16 2004