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