A319514 -id:A319514 - cp's OEIS frontend

A319289 The x coordinates of the shell enumeration of N X N where N = {0, 1, 2, ...} (A319514).

0, 0, 1, 1, 2, 2, 2, 1, 0, 0, 1, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 6, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5

Offset: 0

Peter Luschny, Sep 23 2018

nonn

See A319514 for comments and references.

Peter Luschny, Table of n, a(n) for n = 0..10000

Cf. A319514.

function A319289(n)
    m = x = isqrt(n)
    y = n - x^2
    x <= y && ((x, y) = (2x - y, x))
    isodd(m) ? y : x
end

A319290 The y coordinates of the shell enumeration of N X N where N = {0, 1, 2, ...}(A319514).

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Sep 23 2018

nonn

See A319514 for comments and references.

Peter Luschny, Table of n, a(n) for n = 0..10000

Cf. A319514.

function A319290(n)
    m = x = isqrt(n)
    y = n - x^2
    x <= y && ((x, y) = (2x - y, x))
    isodd(m) ? x : y
end

A319571 The stripe enumeration of N X N where N = {0, 1, 2, ...}, also called boustrophedonic Cantor enumeration. Terms are interleaved x and y coordinates.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 2, 0, 3, 1, 2, 2, 1, 3, 0, 4, 0, 3, 1, 2, 2, 1, 3, 0, 4, 0, 5, 1, 4, 2, 3, 3, 2, 4, 1, 5, 0, 6, 0, 5, 1, 4, 2, 3, 3, 2, 4, 1, 5, 0, 6, 0, 7, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 8, 0, 7, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1

Offset: 0

Peter Luschny, Sep 23 2018

nonn

If (x, y) and (x', y') are adjacent points on the trajectory of the map then max(|x - x'|, |y - y'|) is always 1 whereas for the Cantor enumeration this quantity can become arbitrarily large. In this sense our boustrophedonic variant is continuous whereas Cantor's realization is not.
We implemented the recursive enumeration as a state machine with two states to avoid the evaluation of the square root function.
The inverse function, computing n for given (x, y), is (x + y)*(x + y + 1)/2 + p where p = x if x - y is odd and y otherwise.

			The map starts, for n = 0, 1, 2, ...
(0, 0), (0, 1), (1, 0), (2, 0), (1, 1), (0, 2), (0, 3), (1, 2), (2, 1), (3, 0),
(4, 0), (3, 1), (2, 2), (1, 3), (0, 4), (0, 5), (1, 4), (2, 3), (3, 2), (4, 1),
(5, 0), (6, 0), (5, 1), (4, 2), (3, 3), (2, 4), (1, 5), (0, 6), (0, 7), (1, 6),
(2, 5), (3, 4), (4, 3), (5, 2), (6, 1), (7, 0), ...
The enumeration can be seen as diagonal stripes layering on the origin:
(0, 0),
(0, 1), (1, 0),
(2, 0), (1, 1), (0, 2),
(0, 3), (1, 2), (2, 1), (3, 0),
(4, 0), (3, 1), (2, 2), (1, 3), (0, 4),
(0, 5), (1, 4), (2, 3), (3, 2), (4, 1), (5, 0),
(6, 0), (5, 1), (4, 2), (3, 3), (2, 4), (1, 5), (0, 6),
(0, 7), (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1), (7, 0)

Peter Luschny, Table of n, a(n) for n = 0..10000
Georg Cantor, Ein Beitrag zur Mannigfaltigkeitslehre, Journal für die reine und angewandte Mathematik 84 (1878), 242-258.
Steven Pigeon, Mœud, 2018.
A. L. Rosenberg, Allocating storage for extendible Arrays, J. ACM, vol 21(4), 1974, p. 652-670.
Index entries for sequences related to coordinates of 2D curves

Cf. A319572 (stripe x), A319573 (stripe y).

Cf. A319514 (shell enumeration), A319289 (shell x), A319290 (shell y).

function A319571(n)
    k, r = divrem(n, 2)
    d = div(isqrt(8k+1) - 1, 2)
    s = k - div(d*(d + 1), 2)
    isodd(d) ? (s, d-s)[r+1] : (d-s, s)[r+1]
end
function stripe(x, y, state)
    x == 0 && !state && return x, y+1, !state
    y == 0 &&  state && return x+1, y, !state
    state && return x+1, y-1, state
    return x-1, y+1, state
end
function StripeEnumeration(len)
    x, y, state = 0, 0, false
    for n in 0:len
        println("$n -> ($x, $y)")
        x, y, state = stripe(x, y, state)
    end
end
function Pairing(x, y)
    p = isodd(x - y) ? x : y
    div((x + y)*(x + y + 1), 2) + p
end
StripeEnumeration(40)

from itertools import count, islice
def A319571_gen(): # generator of terms
    for n in count(0):
        for m in range(n+1):
             yield from (m,n-m) if n % 2 else (n-m,m)
A319571_list = list(islice(A319571_gen(),100)) # Chai Wah Wu, May 21 2022

A332662 Put-and-count: An enumeration of N X N where N = {0, 1, 2, ...}. The terms are interleaved x and y coordinates. Or: A row-wise storage scheme for sequences of regular triangles.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 2, 0, 0, 2, 1, 1, 2, 1, 3, 0, 4, 0, 5, 0, 0, 3, 1, 2, 2, 2, 3, 1, 4, 1, 5, 1, 6, 0, 7, 0, 8, 0, 9, 0, 0, 4, 1, 3, 2, 3, 3, 2, 4, 2, 5, 2, 6, 1, 7, 1, 8, 1, 9, 1, 10, 0, 11, 0, 12, 0, 13, 0, 14, 0, 0, 5, 1, 4, 2, 4, 3, 3, 4, 3, 5, 3, 6, 2, 7, 2

Offset: 0

Peter Luschny, Feb 18 2020

nonn

Other enumerations of N X N designed with storage allocation for extensible arrays in mind include A319514 and A319571.

			Illustrating the linear storage layout of a sequence of regular triangles.
(A) [ 0], [ 2,  3], [ 7,  8,  9], [16, 17, 18, 19], [30, 31, 32, 33, 34], ...
(B) [ 1], [ 5,  6], [13, 14, 15], [26, 27, 28, 29], ...
(C) [ 4], [11, 12], [23, 24, 25], ...
(D) [10], [21, 22], ...
(E) [20], ...
...
The first column is A000292.
The start values of all partial rows (in ascending order) are 0 plus A014370.
The start values of the partial rows in the first row are A005581 (without first 0).
The start values of the partial rows on the main diagonal are A331987.
The end values of all partial rows (in ascending order) are A332023.
The end values of the partial rows in the first row are A062748.
The end values of the partial rows on the main diagonal are A332698.

Peter Luschny, Table of n, a(n) for n = 0..10000

A332663 (x-coordinates), A056559 (y-coordinates).
Cf. A000292, A014370, A002260, A005581, A319514, A319571, A331987.
Cf. A332023, A062748, A332698.

function a_list(N)
    a = Int[]
    for n in 1:N
        i = 0
        for j in ((k:-1:1) for k in 1:n)
            t = n - j[1]
            for m in j
                push!(a, i, t)
                i += 1
end end end; a end
a_list(5) |> println

count := (k, A) -> ListTools:-Occurrences(k, A): t := n -> n*(n+1)/2:
PutAndCount := proc(N) local L, n, v, c, seq; L := NULL; seq := NULL;
for n from 1 to N do
   for v from 0 to t(n)-1 do
     # How often did you see v in this sequence before?
     c := count(v, [seq]);
     L := L, v, c; seq := seq, v;
od od; L end:  PutAndCount(6);
# Returning 'seq' instead of 'L' gives the x-coordinates (A332663).

t[n_] := n*(n+1)/2;
PutAndCount[N_] := Module[{L, n, v, c, seq},
L = {}; seq = {};
For[n = 1, n <= N, n++,
   For[v = 0, v <= t[n]-1, v++,
      c = Count[seq, v];
      L = Join[L, {v, c}]; seq = Append[seq, v]
]]; L];
PutAndCount[6] (* Jean-François Alcover, Oct 13 2024, after Maple program *)

A319572 The x coordinates of the stripe enumeration of N X N where N = {0, 1, 2, ...}.

Original entry on oeis.org

0, 0, 1, 2, 1, 0, 0, 1, 2, 3, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 11, 10, 9, 8, 7, 6

Offset: 0

Peter Luschny, Sep 23 2018

nonn

See A319571 for comments and references.

Index entries for sequences related to coordinates of 2D curves

Cf. A319571 (stripe enumeration), this sequence (stripe x), A319573 (stripe y).

Cf. A319514 (shell enumeration), A319289 (shell x), A319290 (shell y).

# Function stripe is defined in A319571.
function StripeEnumerationX(len)
    x, y, state = 0, 0, false
    for n in 0:len
        print("$x, ")
        x, y, state = stripe(x, y, state)
    end
end
StripeEnumerationX(84)

A319573 The y coordinates of the stripe enumeration of N X N where N = {0, 1, 2, ...}.

Original entry on oeis.org

0, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 6

Offset: 0

Peter Luschny, Sep 23 2018

nonn

See A319571 for comments and references.

Index entries for sequences related to coordinates of 2D curves

Cf. A319571 (stripe enumeration), A319572 (stripe x), this sequence (stripe y).

Cf. A319514 (shell enumeration), A319289 (shell x), A319290 (shell y).

# Function stripe is defined in A319571.
function StripeEnumerationY(len)
    x, y, state = 0, 0, false
    for n in 0:len
        print("$y, ")
        x, y, state = stripe(x, y, state)
    end
end
StripeEnumerationY(84)

A319513 The boustrophedonic Rosenberg-Strong function maps N onto N X N where N = {0, 1, 2, ...} and n -> factor(a(n)) = 2^x*3^y -> (x, y).

Original entry on oeis.org

1, 3, 6, 2, 4, 12, 36, 18, 9, 27, 54, 108, 216, 72, 24, 8, 16, 48, 144, 432, 1296, 648, 324, 162, 81, 243, 486, 972, 1944, 3888, 7776, 2592, 864, 288, 96, 32, 64, 192, 576, 1728, 5184, 15552, 46656, 23328, 11664, 5832, 2916, 1458, 729, 2187, 4374, 8748, 17496

Offset: 0

Peter Luschny, Sep 21 2018

nonn

If (x, y) and (x', y') are adjacent points on the trajectory of the map then for the boustrophedonic Rosenberg-Strong function max(|x - x'|, |y - y'|) is always 1 whereas for the Rosenberg-Strong function this quantity can become arbitrarily large. In this sense the boustrophedonic variant is continuous in contrast to the original Rosenberg-Strong function.

A. L. Rosenberg, H. R. Strong, Addressing arrays by shells, IBM Technical Disclosure Bulletin, vol 14(10), 1972, p. 3026-3028.

Georg Cantor, Ein Beitrag zur Mannigfaltigkeitslehre, Journal für die reine und angewandte Mathematik 84 (1878), 242-258.
Steven Pigeon, Mœud deux, 2018.
A. L. Rosenberg, Allocating storage for extendible Arrays, J. ACM, vol 21(4), 1974, p. 652-670.
M. P. Szudzik, The Rosenberg-Strong Pairing Function", arXiv:1706.04129 [cs.DM], 2017.

See A319514 for a non-decoded variant with interleaved x and y coordinates.

Cf. A038566/A038567, A157807/A157813.

function bRS(n)
    m = x = isqrt(n)
    y = n - x^2
    x <= y && ((x, y) = (2x - y, x))
    isodd(m) ? (y, x) : (x, y)
end
A319513(n) = ((x, y) = bRS(n); 2^x * 3^y)
[A319513(n) for n in 0:52] |> println

A319513 := proc(n) local b, r, p, m;
    b := floor(sqrt(n)); r := n - b^2;
    p := `if`(r < b, [b, r], [2*b-r, b]);
    m := `if`(p[1] > p[2], p[1], p[2]);
    `if`(irem(m,2) = 0, 2^p[1]*3^p[2], 2^p[2]*3^p[1]) end:
seq(A319513(n), n=0..52);

a[n_] := Module[{b, r, p1, p2, m}, b = Floor[Sqrt[n]]; r = n-b^2; {p1, p2} = If[rp2, p1, p2]; If[EvenQ[m], 2^p1 3^p2, 2^p2 3^p1]]; Table[a[n], {n, 0, 52}] (* Jean-François Alcover, Feb 14 2019, from Maple *)

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A319289 The x coordinates of the shell enumeration of N X N where N = {0, 1, 2, ...} (A319514).

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A319290 The y coordinates of the shell enumeration of N X N where N = {0, 1, 2, ...}(A319514).

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A319571 The stripe enumeration of N X N where N = {0, 1, 2, ...}, also called boustrophedonic Cantor enumeration. Terms are interleaved x and y coordinates.

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A332662 Put-and-count: An enumeration of N X N where N = {0, 1, 2, ...}. The terms are interleaved x and y coordinates. Or: A row-wise storage scheme for sequences of regular triangles.

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A319572 The x coordinates of the stripe enumeration of N X N where N = {0, 1, 2, ...}.

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A319573 The y coordinates of the stripe enumeration of N X N where N = {0, 1, 2, ...}.

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A319513 The boustrophedonic Rosenberg-Strong function maps N onto N X N where N = {0, 1, 2, ...} and n -> factor(a(n)) = 2^x*3^y -> (x, y).

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