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%I M0821 #72 Feb 10 2023 12:54:13
%S 2,3,7,8,9,10,16,17,18,19,20,21,29,30,31,32,33,34,35,36,46,47,48,49,
%T 50,51,52,53,54,55,67,68,69,70,71,72,73,74,75,76,77,78,92,93,94,95,96,
%U 97,98,99,100,101,102,103,104,105,121,122,123,124,125,126,127,128,129,130
%N Skip 1, take 2, skip 3, etc.
%C 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
%C Union of A317303 and A014105. - _Omar E. Pol_, Aug 29 2018
%D R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 177.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Reinhard Zumkeller, <a href="/A007607/b007607.txt">Table of n, a(n) for n = 1..10000</a>
%F G.f.: 1/(1-x) * (1/(1-x) + x*Sum_{k>=1} (2k+1)*x^(k*(k+1))). - _Ralf Stephan_, Mar 03 2004
%F a(A000290(n)) = A001105(n). - _Reinhard Zumkeller_, Feb 12 2011
%F A057211(a(n)) = 0. - _Reinhard Zumkeller_, Dec 30 2011
%F a(n) = floor(sqrt(n) + 1/2)^2 + n = A053187(n) + n. - _Ridouane Oudra_, May 04 2019
%e From _Omar E. Pol_, Aug 29 2018: (Start)
%e Written as an irregular triangle in which the row lengths are the nonzero even numbers the sequence begins:
%e 2, 3;
%e 7, 8, 9, 10;
%e 16, 17, 18, 19, 20, 21;
%e 29, 30, 31, 32, 33, 34, 35, 36;
%e 46, 47, 48, 49, 50, 51, 52, 53, 54, 55;
%e 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78;
%e 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105;
%e 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136;
%e ...
%e Row sums give the nonzero terms of A317297.
%e Column 1 gives A130883, n >= 1.
%e Right border gives A014105, n >= 1.
%e (End)
%t Flatten[ Table[i, {j, 2, 16, 2}, {i, j(j - 1)/2 + 1, j(j + 1)/2}]] (* _Robert G. Wilson v_, Mar 11 2004 *)
%t With[{t=20},Flatten[Take[TakeList[Range[(t(t+1))/2],Range[t]],{2,-1,2}]]] (* _Harvey P. Dale_, Sep 26 2021 *)
%o (Haskell)
%o a007607 n = a007607_list !! (n-1)
%o a007607_list = skipTake 1 [1..] where
%o skipTake k xs = take (k + 1) (drop k xs)
%o ++ skipTake (k + 2) (drop (2*k + 1) xs)
%o -- _Reinhard Zumkeller_, Feb 12 2011
%o (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
%o (Haskell)
%o a007607_list' = f $ tail $ scanl (+) 0 [1..] where
%o f (t:t':t'':ts) = [t+1..t'] ++ f (t'':ts)
%o -- _Reinhard Zumkeller_, Feb 12 2011
%Y Cf. A063656, A004202, A063657, A007606, A064801, A004202.
%Y Complement of A007606.
%Y Cf. A014105, A130883, A317297.
%Y Similar to A360418.
%K nonn,easy,tabf
%O 1,1
%A _N. J. A. Sloane_, _Robert G. Wilson v_, _Mira Bernstein_
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