cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A007606 Take 1, skip 2, take 3, etc.

Original entry on oeis.org

1, 4, 5, 6, 11, 12, 13, 14, 15, 22, 23, 24, 25, 26, 27, 28, 37, 38, 39, 40, 41, 42, 43, 44, 45, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 137, 138
Offset: 1

Views

Author

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

Keywords

Comments

List the natural numbers: 1, 2, 3, 4, 5, 6, 7, ... . Keep the first number (1), delete the next two numbers (2, 3), keep the next three numbers (4, 5, 6), delete the next four numbers (7, 8, 9, 10) and so on.
a(A000290(n)) = A000384(n). - Reinhard Zumkeller, Feb 12 2011
A057211(a(n)) = 1. - Reinhard Zumkeller, Dec 30 2011
Numbers k with the property that the smallest Dyck path of the symmetric representation of sigma(k) has a central valley. (Cf. A237593.) - Omar E. Pol, Aug 28 2018
Union of nonzero terms of A000384 and A317304. - Omar E. Pol, Aug 29 2018
The values of k such that, in a listing of all congruence classes of positive integers, the k-th congruence class contains k. Here the class r mod m (with r in {1,...,m}) precedes the class r' mod m' (with r' in {1,...,m'}) iff mA360418. - James Propp, Feb 10 2023

Examples

			From _Omar E. Pol_, Aug 29 2018: (Start)
Written as an irregular triangle in which the row lengths are the odd numbers the sequence begins:
    1;
    4,   5,   6;
   11,  12,  13,  14,  15;
   22,  23,  24,  25,  26,  27,  28;
   37,  38,  39,  40,  41,  42,  43,  44,  45;
   56,  57,  58,  59,  60,  61,  62 , 63,  64,  65,  66;
   79,  80,  81,  82 , 83,  84,  85,  86,  87,  88,  89,  90,  91;
  106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120;
...
Row sums give A005917.
Column 1 gives A084849.
Column 2 gives A096376, n >= 1.
Right border gives A000384, n >= 1.
(End)

References

  • C. Dumitrescu & V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I, Erhus Publ., Glendale, 1994.
  • 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).
  • F. Smarandache, Properties of Numbers, 1972.

Links

Crossrefs

Complement of A007607.
Cf. A007950, A007951, A007952, A048859, A004201.
Cf. A000384, A005917, A084849, A096376.

Programs

  • Haskell
    a007606 n = a007606_list !! (n-1)
a007606_list = takeSkip 1 [1..] where
   takeSkip k xs = take k xs ++ takeSkip (k + 2) (drop (2*k + 1) xs)
-- Reinhard Zumkeller, Feb 12 2011
  • Mathematica
    Flatten[ Table[i, {j, 1, 17, 2}, {i, j(j - 1)/2 + 1, j(j + 1)/2}]] (* Robert G. Wilson v, Mar 11 2004 *)
Join[{1},Flatten[With[{nn=20},Range[#[[1]],Total[#]]&/@Take[Thread[ {Accumulate[ Range[nn]]+1,Range[nn]}],{2,-1,2}]]]] (* Harvey P. Dale, Jun 23 2013 *)
With[{nn=20},Take[TakeList[Range[(nn(nn+1))/2],Range[nn]],{1,nn,2}]]//Flatten (* Harvey P. Dale, Feb 10 2023 *)
  • PARI
    for(n=1,66,m=sqrtint(n-1);print1(n+m*(m+1),","))

Formula

a(n) = n + m*(m+1) where m = floor(sqrt(n-1)). - Klaus Brockhaus, Mar 26 2004
a(n+1) = a(n) + if n=k^2 then 2*k+1 else 1; a(1) = 1. - Reinhard Zumkeller, May 13 2009

A007607 Skip 1, take 2, skip 3, etc.

Original entry on oeis.org

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

Views

Author

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

Keywords

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

Examples

			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)

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

Crossrefs

Cf. A063656, A004202, A063657, A007606, A064801, A004202.
Complement of A007606.
Cf. A014105, A130883, A317297.
Similar to A360418.

Programs

  • 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
  • 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
  • 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 *)
  • 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

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

A360416 a(n) = 8*n^2 - 9*n + 3.

Original entry on oeis.org

3, 2, 17, 48, 95, 158, 237, 332, 443, 570, 713, 872, 1047, 1238, 1445, 1668, 1907, 2162, 2433, 2720, 3023, 3342, 3677, 4028, 4395, 4778, 5177, 5592, 6023, 6470, 6933, 7412, 7907, 8418, 8945, 9488, 10047, 10622, 11213, 11820, 12443, 13082, 13737, 14408, 15095
Offset: 0

Views

Author

James Propp, Feb 06 2023

Keywords

Comments

This is one of the four quadratic sequences that, interleaved, yield A360418.

Links

Crossrefs

Cf. A360417, A360418.

Programs

  • Mathematica
    Table[8*n^2 - 9*n + 3, {n, 0, 100}]

Formula

From Elmo R. Oliveira, Jan 28 2025: (Start)
G.f.: (3 - 7*x + 20*x^2)/(1 - x)^3.
E.g.f.: (3 - x + 8*x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3.
a(n) = A360418(4*n-2) for n >= 1. (End)

Extensions

a(0)=3 prepended and more terms from Elmo R. Oliveira, Jan 28 2025

A360417 a(n) = 8*n^2 - 7*n + 2.

Original entry on oeis.org

2, 3, 20, 53, 102, 167, 248, 345, 458, 587, 732, 893, 1070, 1263, 1472, 1697, 1938, 2195, 2468, 2757, 3062, 3383, 3720, 4073, 4442, 4827, 5228, 5645, 6078, 6527, 6992, 7473, 7970, 8483, 9012, 9557, 10118, 10695, 11288, 11897, 12522, 13163, 13820, 14493, 15182
Offset: 0

Views

Author

James Propp, Feb 06 2023

Keywords

Comments

This is one of the four quadratic sequences that, interleaved, yield A360418.

Links

Crossrefs

Cf. A051870, A125201, A360416, A360418.

Programs

  • Mathematica
    Table[8*n^2 - 7*n + 2, {n, 0, 100}]

Formula

From Elmo R. Oliveira, Jan 28 2025: (Start)
G.f.: (17*x^2-3*x+2)/(1-x)^3.
E.g.f.: exp(x)*(2 + x + 8*x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3.
a(n) = A360418(4*n-1) for n>=1. (End)
a(n) = A125201(n) + 1, n >= 1. - Omar E. Pol, Jan 28 2025

Extensions

a(0)=2 prepended and more terms from Alois P. Heinz, Jan 28 2025
Showing 1-4 of 4 results.

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