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 1-2 of 2 results.

A076660 Numbers m such that A076644(m) = floor((2/3)*m*(sqrt(m)+1)).

Original entry on oeis.org

1, 2, 6, 17, 24, 26, 29, 43, 83, 88, 193, 207, 243, 357, 534, 806, 1082, 1197, 1377, 2246, 2362, 3148, 4342, 4570, 6463, 11687, 14305, 35681, 41369, 53410, 75292, 98348, 146608, 219393, 246121, 257663, 419767, 476246, 596919, 860060, 912227, 1225215, 2239828, 2727103, 2819396
Offset: 1

Views

Author

Benoit Cloitre, Oct 24 2002

Keywords

Comments

Sequence is probably infinite.

Links

Crossrefs

Cf. A076644.

Programs

  • PARI
    lista(nn) = {my(v=vector(nn)); v[1] = 1; for (n=2, nn, v[n] = n + v[n-sqrtint(n)]; ); for (m=1, nn, if (v[m] == floor((2/3)*m*(sqrt(m)+1)), print1(m, ", ")););} \\ Michel Marcus, Feb 16 2021

Formula

It seems that a(n) is asymptotic to c*exp(d*n) with d around 0.33...

Extensions

Incorrect term 4 removed and more terms added from Michel Marcus, Feb 16 2021

A122196 Fractal sequence: count down by 2's from successive integers.

Original entry on oeis.org

1, 2, 3, 1, 4, 2, 5, 3, 1, 6, 4, 2, 7, 5, 3, 1, 8, 6, 4, 2, 9, 7, 5, 3, 1, 10, 8, 6, 4, 2, 11, 9, 7, 5, 3, 1, 12, 10, 8, 6, 4, 2, 13, 11, 9, 7, 5, 3, 1, 14, 12, 10, 8, 6, 4, 2, 15, 13, 11, 9, 7, 5, 3, 1, 16, 14, 12, 10, 8, 6, 4, 2, 17, 15, 13, 11, 9, 7, 5, 3, 1, 18, 16, 14, 12, 10, 8, 6, 4, 2, 19, 17
Offset: 1

Views

Author

Franklin T. Adams-Watters, Aug 25 2006

Keywords

Comments

First differences of A076644. Fractal - deleting the first occurrence of each integer leaves the original sequence. Also, original sequence plus 1. 1's occur at square indices. New values occur at indices m^2+1 and m^2+m+1.
Ordinal transform of A122197.
Row sums give A002620. - Gary W. Adamson, Nov 29 2008
From Gary W. Adamson, Dec 05 2009: (Start)
A122196 considered as an infinite lower triangular matrix * [1,2,3,...] =
A006918 starting (1, 2, 5, 8, 14, 20, 30, 40, ...).
Let A122196 = an infinite lower triangular matrix M; then lim_{n->infinity} M^n = A171238, a left-shifted vector considered as a matrix. (End)
A122196 is the fractal sequence associated with the dispersion A082156; that is, A122196(n) is the number of the row of A082156 that contains n. - Clark Kimberling, Aug 12 2011
From Johannes W. Meijer, Sep 09 2013: (Start)
The alternating row sums lead to A004524(n+2).
The antidiagonal sums equal A001840(n). (End)

Examples

			The first few rows of the sequence a(n) as a triangle T(n, k):
n/k  1   2   3
1    1
2    2
3    3,  1
4    4,  2
5    5,  3,  1
6    6,  4,  2

Links

Crossrefs

Cf. A076644, A122197, A000290, A033638, A002620, A006918, A171238, A082156, A000267.

Programs

  • Haskell
    a122196 n = a122196_list !! (n-1)
a122196_list = concatMap (\x -> enumFromThenTo x (x - 2) 1) [1..]
-- Reinhard Zumkeller, Jul 19 2012
  • Maple
    From Johannes W. Meijer, Sep 09 2013: (Start)
a := proc(n) local t: t:=floor((sqrt(4*n-3)-1)/2): floor(sqrt(4*n-1))-2*((n-1) mod (t+1)) end: seq(a(n), n=1..92); # End first program.
T := (n, k) -> n-2*k+2: seq(seq(T(n, k), k=1..floor((n+1)/2)), n=1..18); # End second program. (End)
  • Mathematica
    Flatten@Range[Range[10], 1, -2] (* Birkas Gyorgy, Apr 07 2011 *)

Formula

From Boris Putievskiy, Sep 09 2013: (Start)
a(n) = 2*(1-A122197(n)) + A000267(n-1).
a(n) = floor(sqrt(4*n-1)) - 2*((n-1) mod (t+1)), where t = floor((sqrt(4*n-3)-1)/2). (End)
From Johannes W. Meijer, Sep 09 2013: (Start)
T(n, k) = n - 2*k + 2, for n >= 1 and 1 <= k <= floor((n+1)/2).
T(n, k) = A002260(n, n-2*k+2). (End)
Showing 1-2 of 2 results.

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