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A007401 Add n-1 to n-th term of 'n appears n times' sequence (A002024).

%I A007401 M2316 #99 Oct 19 2022 06:44:28
%S A007401 1,3,4,6,7,8,10,11,12,13,15,16,17,18,19,21,22,23,24,25,26,28,29,30,31,
%T A007401 32,33,34,36,37,38,39,40,41,42,43,45,46,47,48,49,50,51,52,53,55,56,57,
%U A007401 58,59,60,61,62,63,64,66,67,68,69,70,71,72,73,74,75,76
%N A007401 Add n-1 to n-th term of 'n appears n times' sequence (A002024).
%C A007401 Complement of A000096 = increasing sequence of positive integers excluding n*(n+3)/2. - _Jonathan Vos Post_, Aug 13 2005
%C A007401 As a triangle: (1; 3,4; 6,7,8; 10,11,12,13; ...), Row sums = A127736: (1, 7, 21, 46, 85, 141, 217, ...). - _Gary W. Adamson_, Oct 25 2007
%C A007401 Odd primes are a subsequence except 5, cf. A004139. - _Reinhard Zumkeller_, Jul 18 2011
%C A007401 A023532(a(n)) = 1. - _Reinhard Zumkeller_, Dec 04 2012
%C A007401 T(n,k) = ((n+k)^2+n-k)/2 - 1, n,k > 0, read by antidiagonals. - _Boris Putievskiy_, Jan 14 2013
%C A007401 A023531(a(n)) = 0. - _Reinhard Zumkeller_, Feb 14 2015
%D A007401 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007401 Reinhard Zumkeller, <a href="/A007401/b007401.txt">Table of n, a(n) for n = 1..10000</a>
%H A007401 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%H A007401 J. Riordan, <a href="/A007401/a007401_8.pdf">The enumeration of trees by height and diameter</a>, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
%H A007401 T. R. S. Walsh & N. J. A. Sloane, <a href="/A007401/a007401_5.pdf">Correspondence, 1991</a>
%H A007401 T. R. S. Walsh, <a href="/A007401/a007401_2.pdf">Data (Preprint 1, Part 1)</a>
%H A007401 T. R. S. Walsh, <a href="/A007401/a007401_3.pdf">Data (Preprint 1, Part 2)</a>
%H A007401 T. R. S. Walsh, <a href="/A007401/a007401_4.pdf">Data (Preprint 1, Part 3)</a>
%H A007401 T. R. S. Walsh, <a href="/A007401/a007401_1.pdf">Notes</a>
%H A007401 T. R. S. Walsh, <a href="/A007401/a007401.pdf">Number of sensed planar maps with n edges and m vertices</a>
%H A007401 N. C. Wormald, <a href="/A007401/a007401_6.pdf">On the number of planar maps</a>, Can. J. Math. 33.1 (1981), 1-11. (Annotated scanned copy)
%F A007401 From _Boris Putievskiy_, Jan 14 2013: (Start)
%F A007401 a(n) = A014132(n) - 1.
%F A007401 a(n) = A003057(n)^2 - A114327(n) - 1.
%F A007401 a(n) = ((t+2)^2 + i - j)/2-1, where
%F A007401 i = n-t*(t+1)/2,
%F A007401 j = (t*t+3*t+4)/2-n,
%F A007401 t = floor((-1+sqrt(8*n-7))/2). (End)
%e A007401 From _Boris Putievskiy_, Jan 14 2013: (Start)
%e A007401 The start of the sequence as table:
%e A007401    1,  3,  6, 10, 15, 21, 28, ...
%e A007401    4,  7, 11, 16, 22, 29, 37, ...
%e A007401    8, 12, 17, 23, 30, 38, 47, ...
%e A007401   13, 18, 24, 31, 39, 48, 58, ...
%e A007401   19, 25, 32, 40, 49, 59, 70, ...
%e A007401   26, 33, 41, 50, 60, 71, 83, ...
%e A007401   34, 42, 51, 61, 72, 84, 97, ...
%e A007401   ...
%e A007401 The start of the sequence as triangle array read by rows:
%e A007401    1;
%e A007401    3,  4;
%e A007401    6,  7,  8;
%e A007401   10, 11, 12, 13;
%e A007401   15, 16, 17, 18, 19;
%e A007401   21, 22, 23, 24, 25, 26;
%e A007401   28, 29, 30, 31, 32, 33, 34;
%e A007401   ...
%e A007401 Row number r contains r numbers r*(r+1)/2, r*(r+1)/2+1, ..., r*(r+1)/2+r-1. (End)
%t A007401 f[n_] := n + Floor[ Sqrt[2n] - 1/2]; Array[f, 66]; (* _Robert G. Wilson v_, Feb 13 2011 *)
%o A007401 (PARI) a(n)=n+floor(sqrt(n+n)-1/2) \\ _Charles R Greathouse IV_, Feb 13 2011
%o A007401 (PARI) for(m=1,9, for(n=m*(m+1)/2,(m^2+3*m-2)/2, print1(n", "))) \\ _Charles R Greathouse IV_, Feb 13 2011
%o A007401 (Haskell)
%o A007401 a007401 n = a007401_list !! n
%o A007401 a007701_list = [x | x <- [0..], a023531 x == 0]
%o A007401 -- _Reinhard Zumkeller_, Feb 14 2015, Dec 04 2012
%o A007401 (Python)
%o A007401 from math import isqrt
%o A007401 def A007401(n): return n-1+(isqrt(n<<3)+1>>1) # _Chai Wah Wu_, Oct 18 2022
%Y A007401 Cf. A002024, A000096, A127736, A014132, A002260, A004736, A003057, A114327, A023531.
%K A007401 nonn,easy
%O A007401 1,2
%A A007401 _N. J. A. Sloane_, _Mira Bernstein_