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User:Daniel Forgues/Contributions/Figurate numbers
User:Daniel Forgues/Contributions/Number triangles

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Primes pairs between ((n / 2)  −  ( log n) 2 ) and ((n / 2) + ( log n) 2 ) adding to even n

Main article page: Primes pairs between ( (n/2) - ( log(n) )^2 ) and ( (n/2) + ( log(n) )^2 ) adding to even n

Primes pairs between
((n / 2)  −  ( log n) 2 )
and
((n / 2) + ( log n) 2 )
adding to even
n

n
Primes less than
n / 2
are listed in decreasing order, primes from
n / 2
are listed in increasing order.
Distinct primes in pairs adding to
n
are surrounded by ■ (black squares); prime equal to
n / 2
surrounded by □ (white squares).
10 1
(4 primes)
(1.5 pairs)
Green tickY
2     ■3■

  □5□ ■7■
10 2
(11 primes)
(3 pairs)
Green tickY
■47■ 43 ■41■ 37 31 ■29■

■53■    ■59■ 61 67 ■71■
10 3
(14 primes)
(2 pairs)
Green tickY
499 ■491■ 487 ■479■ 467 463 461 457
 
503 ■509■     ■521■ 523 541     547
10 4
(23 primes)
(1 pair)
Green tickY
4999 4993 4987      4973 4969 4967 4957 4951 4943 4937 4933 4931      ■4919■

5003 5009 5011 5021 5023           5039 5051 5059                5077 ■5081■
10 5
(29 primes)
(2 pairs)
Green tickY
49999 49993 49991                   49957       49943 49939 49937 49927 49921 49919                   49891       ■49877■ ■49871■ 

                  50021 50023 50033 50047 50051 50053             50069 50077       50087 50093 50101 50111 50119 ■50123■ ■50129■ 50131 
10 6
(27 primes)
(1 pair)
Green tickY
       499979 499973 499969 499957 ■499943■ 499927 499903 499897               499883 499879 499853                      499819

500009        500029        500041 ■500057■ 500069 500083 500107 500111 500113 500119        500153 500167 500173 500177 500179
10 9
(41 primes)
(2 pairs)
Green tickY
  
          499999993                     ■499999931■                     499999909                     499999897 499999873 499999853  499999847  499999831 
                                         499999751  499999723 499999697 499999693                                                   ■499999613■
 

500000003 500000009 500000041 500000057 ■500000069■ 500000071 500000077 500000089 500000093 500000099 500000101 500000117                       500000183 
500000201 500000227 500000231 500000233  500000261  500000273 500000299 500000317 500000321 500000323 500000353 500000359 500000377 ■500000387■ 500000393
10 12
(52 primes)
(2 pairs)
Green tickY
499999999979 499999999943 499999999901 499999999897 499999999847 499999999819              499999999799 ■499999999769■              499999999739 499999999699 499999999661  499999999643  499999999571 
499999999559 499999999511 499999999507 499999999501 499999999487 499999999451 499999999427 499999999403  499999999391                            499999999357              ■499999999277■

500000000023 500000000033 500000000089 500000000131 500000000147 500000000173 500000000191 500000000209 ■500000000231■ 500000000243 500000000263 500000000273 500000000333  500000000387  500000000413 
500000000471                           500000000509 500000000537 500000000551              500000000609  500000000611  500000000623 500000000627 500000000651 500000000677 ■500000000723■ 500000000737 500000000761

A280172

Studying the sequence (created by Peter Kagey, Dec 27 2016):

A280172 Lexicographically earliest table of positive integers read by antidiagonals such that no row or column contains a repeated term.

{1, 2, 2, 3, 1, 3, 4, 4, 4, 4, 5, 3, 1, 3, 5, 6, 6, 2, 2, 6, 6, 7, 5, 7, 1, 7, 5, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 7, 5, 7, 1, 7, 5, 7, 9, 10, 10, 6, 6, 2, 2, 6, 6, 10, 10, 11, 9, 11, 5, 3, 1, 3, 5, 11, 9, 11, 12, 12, 12, 12, 4, 4, 4, 4, 12, 12, 12, 12, ...}

A?????? t(A280172(n)), n >= 1.

{1, 3, 3, 6, 1, 6, 10, 10, 10, 10, 15, 6, 1, 6, 15, 21, 21, 3, 3, 21, 21, ...}
A280172 Concatenated rows of equilateral triangle.
n
       
n

k  = 1
T  (n, k )

1   1  
1
2   2 2  
4
3   3 1 3  
7
4   4 4 4 4  
16
5   5 3 1 3 5  
17
6 6 6 2 2 6 6  
28
7   7 5 7 1 7 5 7  
39
8   8 8 8 8 8 8 8 8  
64
9   9 7 5 7 1 7 5 7 9  
57
10   10 10 6 6 2 2 6 6 10 10  
68
11 11 9 11 5 3 1 3 5 11 9 11  
79
12   12 12 12 12 4 4 4 4 12 12 12 12  
112
13   13 11 9 11 13 3 1 3 13 11 9 11 13  
121
14   14 14 10 10 14 14 2 2 14 14 10 10 14 14  
156
15 15 13 15 9 15 13 15 1 15 13 15 9 15 13 15  
191
16   16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16  
256

k = 1

2
3
4
5
6
7
8
9
10
11
12
13
14
15
16  
    
The equilateral version of Pascal’s triangle (Figurate Number Triangle)[1]
n
       
n

d  = 0
(  nd  )
= 2n

0   1  
1
1   1 1  
2
2   1 2 1  
4
3   1 3 3 1  
8
4   1 4 6 4 1  
16
5 1 5 10 10 5 1  
32
6   1 6 15 20 15 6 1  
64
7   1 7 21 35 35 21 7 1  
128
8   1 8 28 56 70 56 28 8 1  
256
9   1 9 36 84 126 126 84 36 9 1  
512
10 1 10 45 120 210 252 210 120 45 10 1  
1024
11   1 11 55 165 330 462 462 330 165 55 11 1  
2048
12   1 12 66 220 495 792 924 792 495 220 66 12 1  
4096
13   1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1  
8192
14 1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1  
16384
15   1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1  
32768

k = 0

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15  
Sierpiński’s triangle (Pascal’s triangle mod 2)
n
[row (n)]2  =  [  
n
i  = 0
  
((
n
i
) mod 2) 2i  ]2  =  [  
⌊  log2 n ⌋
i  = 0
  
Fi  ( 
⌊  n  / 2i
mod 2)
]2  = 
[  
⌊  log2 n ⌋
i  = 0
  
(2 2i + 1) ( 
⌊  n  / 2i
mod 2)
]2

A006943 Rows of Sierpiński’s triangle (Pascal’s triangle mod 2).
A047999 Concatenated rows of Sierpiński’s triangle (Pascal’s triangle mod 2).
[row (n)]10


(A001317)
0 1 1
1 1 1 3
2 1 0 1 5
3 1 1 1 1 15
4 1 0 0 0 1 17
5 1 1 0 0 1 1 51
6 1 0 1 0 1 0 1 85
7 1 1 1 1 1 1 1 1 255
8 1 0 0 0 0 0 0 0 1 257
9 1 1 0 0 0 0 0 0 1 1 771
10 1 0 1 0 0 0 0 0 1 0 1 1285
11 1 1 1 1 0 0 0 0 1 1 1 1 3855
12 1 0 0 0 1 0 0 0 1 0 0 0 1 4369
13 1 1 0 0 1 1 0 0 1 1 0 0 1 1 13107
14 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 21845
15 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 65535
16 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 65537
17 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 196611
18 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 327685
19 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 983055
20 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1114129
21 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 3342387
22 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 5570645
23 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 16711935
24 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 16843009
25 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 50529027
26 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 84215045
27 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 252645135
28 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 286331153
29 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 858993459
30 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1431655765
31 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4294967295
32 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4294967297
i  = 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

Algorithm:

  • T  (1, 1)  =  1
    ;
  • Left and right triangles (copied from above, terms incremented by
    2k  
    ): For
    k   ≥   0, 1   ≤   i   ≤   2k, 0   ≤    j   ≤   i  −  1
    :
T  (2k + i, 1 +  j)  =  T  (2k + i, 2k + i  −  j)  =  T  (i, 1 +  j) + 2k;
  • Central triangle (reflected from above, for
    i = 2k
    the row reflects on itself): For
    k   ≥   0, 1   ≤   i   ≤   2k  −  1, 0   ≤    j   ≤   i  −  1
    :
T  (2k  +1  −  i, 2k  −  i + 1 +  j)  =  T  (i, 1 +  j).
Algorithm (with
1   ≤    j   ≤   i
instead, better):
  • T  (1, 1)  =  1
    ;
  • Left and right triangles (copied from above, terms incremented by
    2k  
    ): for
    k   ≥   0, 1   ≤   i   ≤   2k, 1   ≤    j   ≤   i
T  (2k + i,  j)  =  T  (2k + i, 2k + i + 1  −  j)  =  T  (i,  j) + 2k;
  • Central triangle (reflected from above, for
    i = 2k
    the row reflects on itself): for
    k   ≥   0, 1   ≤   i   ≤   2k  −  1, 1   ≤    j   ≤   i
T  (2k  +1  −  i, 2k  −  i +  j)  =  T  (i,  j).

Observations:

  • Row
    n
    :
  • First and last terms are
    n
    ;
  • Row
    2 n
    :
  • The
    2 n
    terms of row
    2 n
    are twice the terms of row
    n
    , each term repeated twice.
  • Row
    2n
    :
  • The
    2n
    terms are
    2n
    ,
  • The triangle of terms above is reflected below (this implies that all central terms of odd-indexed rows are 1).

A?????? Row sums of equilateral triangle for A280172.

{1, 4, 7, 16, 17, 28, 39, 64, 57, 68, 79, 112, 121, 156, 191, 256, ...}
A?????? Row sums (for odd-indexed rows) of equilateral triangle for A280172. (The row sum for even-indexed row
2 n
is 4 times the row sum for row
n
.)
{1, 7, 17, 39, 57, 79, 121, 191, ...}

Future

Category:Transformations
Transformations
Category:Euler transformation
Euler transformation


Category:Program code
Program code
Category:Maple code
Maple code
Maple Programs to Format Sequences
Transformations of Integer Sequences
Category:Mathematica code
Mathematica code
Transformations of Integer Sequences (Mathematica Version)
Category:PARI/GP code
PARI/GP code

See also

Notes

  1. Weisstein, Eric W., Figurate Number Triangle, from MathWorld—A Wolfram Web Resource..