A007607 Skip 1, take 2, skip 3, etc. (Formerly M0821)

2, 3, 7, 8, 9, 10, 16, 17, 18, 19, 20, 21, 29, 30, 31, 32, 33, 34, 35, 36, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130

Offset

1,1

Comments

Numbers k with the property that the smallest Dyck path of the symmetric representation of sigma(k) has a central peak. (Cf. A237593.) - Omar E. Pol, Aug 28 2018

Union of A317303 and A014105. - Omar E. Pol, Aug 29 2018

References

R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 177.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

G.f.: 1/(1-x) * (1/(1-x) + x*Sum_{k>=1} (2k+1)*x^(k*(k+1))). - Ralf Stephan, Mar 03 2004

a(A000290(n)) = A001105(n). - Reinhard Zumkeller, Feb 12 2011

A057211(a(n)) = 0. - Reinhard Zumkeller, Dec 30 2011

a(n) = floor(sqrt(n) + 1/2)^2 + n = A053187(n) + n. - Ridouane Oudra, May 04 2019

Example

From Omar E. Pol, Aug 29 2018: (Start)
Written as an irregular triangle in which the row lengths are the nonzero even numbers the sequence begins:
    2,   3;
    7,   8,   9,  10;
   16,  17,  18,  19,  20,  21;
   29,  30,  31,  32,  33,  34,  35,  36;
   46,  47,  48,  49,  50,  51,  52,  53,  54,  55;
   67,  68,  69,  70,  71,  72,  73,  74,  75,  76,  77,  78;
   92,  93,  94,  95,  96,  97,  98,  99, 100, 101, 102, 103, 104, 105;
  121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136;
...
Row sums give the nonzero terms of A317297.
Column 1 gives A130883, n >= 1.
Right border gives A014105, n >= 1.
(End)

Mathematica

Flatten[ Table[i, {j, 2, 16, 2}, {i, j(j - 1)/2 + 1, j(j + 1)/2}]] (* Robert G. Wilson v, Mar 11 2004 *)
With[{t=20}, Flatten[Take[TakeList[Range[(t(t+1))/2], Range[t]], {2, -1, 2}]]] (* Harvey P. Dale, Sep 26 2021 *)

Prog

(Haskell)
a007607 n = a007607_list !! (n-1)
a007607_list = skipTake 1 [1..] where
   skipTake k xs = take (k + 1) (drop k xs)
                   ++ skipTake (k + 2) (drop (2*k + 1) xs)
-- Reinhard Zumkeller, Feb 12 2011
(PARI) for(m=0, 10, for(n=2*m^2+3*m+2, 2*m^2+5*m+3, print1(n", "))) \\ Charles R Greathouse IV, Feb 12 2011
(Haskell)
a007607_list' = f $ tail $ scanl (+) 0 [1..] where
   f (t:t':t'':ts) = [t+1..t'] ++ f (t'':ts)
-- Reinhard Zumkeller, Feb 12 2011

Crossrefs

Cf. A063656, A004202, A063657, A007606, A064801, A004202.

Complement of A007606.

Cf. A014105, A130883, A317297.

Similar to A360418.

Sequence in context: A284514 A268398 A249587 * A370902 A076682 A327224

Adjacent sequences: A007604 A007605 A007606 * A007608 A007609 A007610

Keyword

nonn,easy,tabf

Author

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Status

approved