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 1-3 of 3 results.

A269501 Subsequence immediately following the instances of n in the sequence is n, n-1, ..., 1, n+1, n+2, ....

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 3, 2, 3, 1, 4, 4, 3, 4, 2, 4, 1, 5, 5, 4, 5, 3, 5, 2, 5, 1, 6, 6, 5, 6, 4, 6, 3, 6, 2, 6, 1, 7, 7, 6, 7, 5, 7, 4, 7, 3, 7, 2, 7, 1, 8, 8, 7, 8, 6, 8, 5, 8, 4, 8, 3, 8, 2, 8, 1, 9, 9, 8, 9, 7, 9, 6, 9, 5, 9, 4, 9, 3, 9, 2, 9, 1, 10, 10, 9, 10, 8, 10, 7, 10, 6, 10, 5, 10, 4, 10, 3, 10, 2, 10, 1
Offset: 0

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Author

Franklin T. Adams-Watters, Mar 04 2016

Keywords

Comments

The sequence includes every ordered pair of positive integers exactly once as consecutive terms of the sequence. Through n = k^2, it has every pair i,j with 0 < i,j <= k.
Can be regarded as an irregular triangle where row k contains 1, k, k, k-1, k, k-2, ..., 2, k, with 2n-1 terms.
See A305615 for an essentially identical sequence: a(n) = A305615(n)+1. - N. J. A. Sloane, Jul 03 2018

Examples

			The first 3 occurs as a(5), so a(6) = 3, the first term of 3, 2, 1, 4, 5, 6, .... The second 3 is thus a(6), so a(7) = 2. The third 3 is a(8), so a(9) = 1. The fourth 3 is a(12), now we start incrementing, and a(13) = 4.
The triangle starts:
  1
  1, 2, 2
  1, 3, 3, 2, 3
  1, 4, 4, 3, 4, 2, 4
  1, 5, 5, 4, 5, 3, 5, 2, 5

Crossrefs

Cf. A003059, A060747 (row lengths), A000326 (row sums), A097291, A269780.
See also A315615.

Programs

  • PARI
    a(n) = my(r = if(n<=0, 0, sqrtint(n-1)+1));if((n-r)%2,r,(r^2-n)/2 + 1)

Formula

Let r = ceiling(sqrt(n)) = A003059(n). If n and r have the same parity, a(n) = (r^2-n)/2 + 1; otherwise a(n) = r.

A305615 Next term is the largest earlier term that would not create a repetition of an earlier subsequence of length 2, if such a number exists; otherwise it is the smallest nonnegative number not yet in the sequence.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 2, 1, 2, 0, 3, 3, 2, 3, 1, 3, 0, 4, 4, 3, 4, 2, 4, 1, 4, 0, 5, 5, 4, 5, 3, 5, 2, 5, 1, 5, 0, 6, 6, 5, 6, 4, 6, 3, 6, 2, 6, 1, 6, 0, 7, 7, 6, 7, 5, 7, 4, 7, 3, 7, 2, 7, 1, 7, 0, 8, 8, 7, 8, 6, 8, 5, 8, 4, 8, 3, 8, 2, 8, 1, 8, 0, 9, 9, 8, 9, 7, 9, 6, 9, 5, 9, 4, 9, 3, 9, 2, 9, 1, 9, 0
Offset: 0

Views

Author

Luc Rousseau, Jun 06 2018

Keywords

Comments

The map n |-> (a(n), a(n+1)) is a bijection between N and N X N: when drawn in a 2D array, this map makes progress by finishing the filling of a square gnomon before starting to fill the next one. This and the predictable zigzag way each gnomon is filled make it possible to deduce a closed formula for a(n).
A269501 is an essentially identical sequence: a(n) = A269501(n)-1. - N. J. A. Sloane, Jul 03 2018
For n > 3 indices for values = 1 are A008865(m), m > 2. - Bill McEachen, Oct 26 2023

Examples

			a(0): no already-used value exists, so one has to take the least nonnegative integer, so a(0) = 0;
a(1): reusing 0 is legal, so a(1) = 0. Repeating (0, 0) now becomes illegal;
a(2): reusing 0 is illegal since (a(1), a(2)) would repeat (0, 0). The smallest unused value is 1, so a(2) = 1. Repeating (0, 1) becomes illegal;
a(3): reusing 1 is legal. a(3) = 1. Repeating (1, 1) becomes illegal;
a(4): reusing 1 is illegal (would repeat (1, 1)) but reusing 0 is legal. a(4) = 0. Repeating (1, 0) becomes illegal;
and so on.
a(n) is also the x-coordinate of the cell that contains n in the following 2D infinite array:
  y
  ^
  |
  4 |... ... ... ... ...
    +---------------+
  3 | 9  14  12  10 |...
    +-----------+   |
  2 | 4   7   5 |11 |...
    +-------+   |   |
  1 | 1   2 | 6 |13 |...
    +---+   |   |   |
  0 | 0 | 3 | 8 |15 |...
    +---+---+---+---+---
      0   1   2   3   4 --->x

Links

Crossrefs

Cf. A000196, A053186, A269501, A008865.
For the 2D array shown in the EXAMPLE section, see A316323 and A269780. - N. J. A. Sloane, Jul 03 2018

Programs

  • Mathematica
    A[n_] := Module[{k, t}, k = Floor[Sqrt[n]]; t = n - k^2;
  Boole[t != 0]*k - Boole[OddQ[t]]*(t - 1)/2]; Table[A[n], {n, 0, 100}]
  • PARI
    a(n)=k=floor(sqrt(n));t=n-k^2;(t!=0)*k-(t%2)*(t-1)/2
for(n=0,100,print1(a(n),", "))
  • Prolog
    main :- a(100, A, , ), reverse(A, R), writeln(R).
a(0, [0], [0], []) :- !.
a(N, A, V, P) :-
  M is N - 1, a(M, AA, VV, PP), AA = [AM | _],
  findall(L, (member(L, VV), not(member([AM, L], PP))), Ls),
  (Ls = [L | _] -> V = VV ; (length(VV, L), V = [L | VV])),
  A = [L | AA], P = [[AM, L] | PP].

Formula

a(n) = [t!=0]*k-[t is odd]*(t-1)/2, where k = floor(sqrt(n)), t = n-k^2 and [] stands for the Iverson bracket.

A316323 The square array in A305615 read by antidiagonals.

Original entry on oeis.org

0, 1, 3, 4, 2, 8, 9, 7, 6, 15, 16, 14, 5, 13, 24, 25, 23, 12, 11, 22, 35, 36, 34, 21, 10, 20, 33, 48, 49, 47, 32, 19, 18, 31, 46, 63, 64, 62, 45, 30, 17, 29, 44, 61, 80, 81, 79, 60, 43, 28, 27, 42, 59, 78, 99, 100, 98, 77, 58, 41, 26, 40, 57, 76, 97, 120, 121, 119, 96, 75, 56, 39, 38, 55, 74, 95, 118, 143
Offset: 0

Views

Author

N. J. A. Sloane, Jul 03 2018

Keywords

Examples

			The array in A305615 begins:
  ^
  |
  4 |... ... ... ... ...
    +---------------+
  3 | 9  14  12  10 |...
    +-----------+   |
  2 | 4   7   5 |11 |...
    +-------+   |   |
  1 | 1   2 | 6 |13 |...
    +---+   |   |   |
  0 | 0 | 3 | 8 |15 |...
    +---+---+---+---+---
      0   1   2   3   4 ...
The first few antidiagonals are:L
0,
1, 3,
4, 2, 8,
9, 7, 6, 15,
16, 14, 5, 13, 24,
25, 23, 12, 11, 22, 35,
36, 34, 21, 10, 20, 33, 48,
...

Crossrefs

Cf. A305615, A269501, A269780.

Formula

If 1 is added to every term we get the array in A269780, which has an explicit formula for the (i,j)-th term.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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