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A007401 Add n-1 to n-th term of 'n appears n times' sequence (A002024).
(Formerly M2316)
45
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, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Complement of A000096 = increasing sequence of positive integers excluding n*(n+3)/2. - Jonathan Vos Post, Aug 13 2005
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
Odd primes are a subsequence except 5, cf. A004139. - Reinhard Zumkeller, Jul 18 2011
A023532(a(n)) = 1. - Reinhard Zumkeller, Dec 04 2012
T(n,k) = ((n+k)^2+n-k)/2 - 1, n,k > 0, read by antidiagonals. - Boris Putievskiy, Jan 14 2013
A023531(a(n)) = 0. - Reinhard Zumkeller, Feb 14 2015
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
T. R. S. Walsh & N. J. A. Sloane, Correspondence, 1991
T. R. S. Walsh, Notes
N. C. Wormald, On the number of planar maps, Can. J. Math. 33.1 (1981), 1-11. (Annotated scanned copy)
FORMULA
From Boris Putievskiy, Jan 14 2013: (Start)
a(n) = A014132(n) - 1.
a(n) = A003057(n)^2 - A114327(n) - 1.
a(n) = ((t+2)^2 + i - j)/2-1, where
i = n-t*(t+1)/2,
j = (t*t+3*t+4)/2-n,
t = floor((-1+sqrt(8*n-7))/2). (End)
EXAMPLE
From Boris Putievskiy, Jan 14 2013: (Start)
The start of the sequence as table:
1, 3, 6, 10, 15, 21, 28, ...
4, 7, 11, 16, 22, 29, 37, ...
8, 12, 17, 23, 30, 38, 47, ...
13, 18, 24, 31, 39, 48, 58, ...
19, 25, 32, 40, 49, 59, 70, ...
26, 33, 41, 50, 60, 71, 83, ...
34, 42, 51, 61, 72, 84, 97, ...
...
The start of the sequence as triangle array read by rows:
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, 32, 33, 34;
...
Row number r contains r numbers r*(r+1)/2, r*(r+1)/2+1, ..., r*(r+1)/2+r-1. (End)
MATHEMATICA
f[n_] := n + Floor[ Sqrt[2n] - 1/2]; Array[f, 66]; (* Robert G. Wilson v, Feb 13 2011 *)
PROG
(PARI) a(n)=n+floor(sqrt(n+n)-1/2) \\ Charles R Greathouse IV, Feb 13 2011
(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
(Haskell)
a007401 n = a007401_list !! n
a007701_list = [x | x <- [0..], a023531 x == 0]
-- Reinhard Zumkeller, Feb 14 2015, Dec 04 2012
(Python)
from math import isqrt
def A007401(n): return n-1+(isqrt(n<<3)+1>>1) # Chai Wah Wu, Oct 18 2022
CROSSREFS
Sequence in context: A298644 A075747 A143344 * A275481 A234349 A042954
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)