A028875 -id:A028875 - cp's OEIS frontend

A214870 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

1, 2, 3, 5, 4, 7, 10, 9, 8, 13, 17, 16, 6, 14, 21, 26, 25, 11, 12, 22, 31, 37, 36, 18, 15, 20, 32, 43, 50, 49, 27, 24, 23, 30, 44, 57, 65, 64, 38, 35, 19, 33, 42, 58, 73, 82, 81, 51, 48, 28, 29, 45, 56, 74, 91, 101, 100, 66, 63, 39, 34, 41, 59, 72, 92, 111

Offset: 1

Boris Putievskiy, Mar 11 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1).
Enumeration table T(n,k) layer by layer. The order of the list:
  T(1,1)=1;
  T(1,2), T(2,1), T(2,2);
  . . .
  T(1,n), T(3,n), ... T(n,3), T(n,1); T(n,2), T(n,4), ... T(4,n), T(2,n);
  . . .

			The start of the sequence as table:
   1   2   5  10  17  26 ...
   3   4   9  16  25  36 ...
   7   8   6  11  18  27 ...
  13  14  12  15  24  35 ...
  21  22  20  23  19  28 ...
  31  32  30  33  29  34 ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   2,  3;
   5,  4,  7;
  10,  9,  8, 13;
  17, 16,  6, 14, 21;
  26, 25, 11, 12, 22, 31;
  ...

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A060736, A185725, A213921, A213922; table T(n,k) contains: in rows A002522, A000290, A059100, A005563, A117950, A008865, A087475, A028872, A117951, A028347, A114949, A028875, A117619, A028878, A189833, A028881, A189834, A028884, A114948, A028560, A189836; in columns A002061, A014206, A002378, A027688, A028387, A027689, A028552, A027690, A014209, A027691, A027692, A082111, A027693, A028557, A027694, A108195, A187710, A048058, A048840.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i > j:
   result=i*i-i+(j%2)*(2-(j+1)/2)+((j+1)%2)*(j/2+1)
else:
   result=j*j-2*(i%2)*j + (i%2)*((i+1)/2+1) + ((i+1)%2)*(-i/2+1)

As table
T(n,k) = k*k-2*(n mod 2)*k+(n mod 2)*((n+1)/2+1)+((n+1) mod 2)*(-n/2+1), if n<=k;
T(n,k) = n*n-n+(k mod 2)*(2-(k+1)/2)+((k+1) mod 2)*(k/2+1), if n>k.
As linear sequence
a(n) = j*j-2*(i mod 2)*j+(i mod 2)*((i+1)/2+1)+((i+1) mod 2)*(-i/2+1), if i<=j;
a(n) = i*i-i+(j mod 2)*(2-(j+1)/2)+((j+1) mod 2)*(j/2+1), if i>j; where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A370759 Numbers expressible in the form km + 2(k+m) - 1, for positive k and m.

Original entry on oeis.org

4, 7, 10, 11, 13, 15, 16, 19, 20, 22, 23, 25, 27, 28, 30, 31, 34, 35, 37, 39, 40, 43, 44, 45, 46, 47, 49, 50, 51, 52, 55, 58, 59, 60, 61, 63, 64, 65, 67, 70, 71, 72, 73, 75, 76, 79, 80, 82, 83, 85, 86, 87, 88, 90, 91, 93, 94, 95, 97, 99, 100, 103, 105, 106, 107, 109, 110, 111, 112

Offset: 1

Nicolay Avilov, Mar 01 2024

nonn

All such numbers are answers to the question: How many plane regions result from partitioning by two sets of straight lines, such that:
Each of the k straight lines of the first bundle passes though a single point A, and intersects each of the m straight lines of the second bundle each of which passes through a different point B. There are no straight lines belonging to both bundles, i.e. the line AB is not involved.
Because k*m+2*(k+m)-1 = (k+2)*(m+2)-5, and k and m are both positive, a(n) = A264828(n+2) - 5. - Kevin Ryde, Mar 26 2024

			4 is a term: if each bundle consists of one straight line, the plane is divided into 4 regions.
7 is a term: if the first bundle consists of one line and the second consists of two lines, the plane is divided into 7 regions.
These and other examples are illustrated in the linked figures.

Nicolay Avilov, Explanatory drawing.
Nicolay Avilov, Illustration for terms a(1) - a(6).

Cf. A264828, A028875 (case when k=m).

print(Vec(setbinop((k,m)->k*m + 2*(k + m) - 1, [1..112]), 69)) \\ Michel Marcus, Mar 02 2024

maxval = 112
av = [[k*m+2*k+2*m-1 for k in range(1,maxval)] for m in range(1,maxval)]
flat = [n for row in av for n in row]
uniq = list(set(flat))
a370759 = list(filter(lambda x: x<=maxval, uniq))
print(a370759)
# Robert Munafo, Mar 25 2024

from itertools import count, islice
from sympy import isprime
def A370759_gen(startvalue=4): # generator of terms >= startvalue
    return filter(lambda n:not (isprime(n+5) or (n&1 and isprime((n>>1)+3))),count(max(startvalue,4)))
A370759_list = list(islice(A370759_gen(),20)) # Chai Wah Wu, Mar 26 2024

If there are k straight lines in the first bundle and m straight lines in the second bundle, then we get k*m + 2*(k + m) - 1 regions.

A386206 Triangle read by rows: T(n,k) = n^2 - k, with 0 <= k <= n.

Original entry on oeis.org

0, 1, 0, 4, 3, 2, 9, 8, 7, 6, 16, 15, 14, 13, 12, 25, 24, 23, 22, 21, 20, 36, 35, 34, 33, 32, 31, 30, 49, 48, 47, 46, 45, 44, 43, 42, 64, 63, 62, 61, 60, 59, 58, 57, 56, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 100, 99, 98, 97, 96, 95, 94, 93, 92, 91, 90

Offset: 0

Stefano Spezia, Jul 15 2025

			The triangle begins as:
   0;
   1,  0;
   4,  3,  2;
   9,  8,  7,  6;
  16, 15, 14, 13, 12;
  25, 24, 23, 22, 21, 20;
  36, 35, 34, 33, 32, 31, 30;
  49, 48, 47, 46, 45, 44, 43, 42;
  64, 63, 62, 61, 60, 59, 58, 57, 56;
  ...

Vincenzo Librandi, Table of n, a(n) for n = 0..3320 (rows 0..80)

Cf. A000290 (k=0), A002414 (row sums), A005563, A008865, A028347 (k=4), A028872 (k=3), A028875 (k=5), A279019 (diagonal).

[[n^2-k: k in [0..n]]: n in [0..9]]; // Vincenzo Librandi, Jul 17 2025

T[n_,k_]:=n^2-k; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten

G.f.: x*(1 + x + 2*x*y^2 + 5*x^3*y^2 - x^2*y*(4 + 5*y))/((1 - x)^3*(1 - x*y)^3).
T(n,1) = A005563(n-1) for n > 0.
T(n,2) = A008865(n) for n > 1.

cp's OEIS Frontend

A214870 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

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A370759 Numbers expressible in the form km + 2(k+m) - 1, for positive k and m.

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A386206 Triangle read by rows: T(n,k) = n^2 - k, with 0 <= k <= n.

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cp's OEIS Frontend

A214870 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Formula

A370759 Numbers expressible in the form k*m + 2*(k+m) - 1, for positive k and m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

Python

Formula

A386206 Triangle read by rows: T(n,k) = n^2 - k, with 0 <= k <= n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A370759 Numbers expressible in the form km + 2(k+m) - 1, for positive k and m.