A206371 -id:A206371 - cp's OEIS frontend

A257548 a(1) = 1, a(2) = 2, a(3) = 5, a(4) = 8 and a(5) = 15, a(n) = Sum_{j=1..n-1} a(j).

1, 2, 5, 8, 15, 31, 62, 124, 248, 496, 992, 1984, 3968, 7936, 15872, 31744, 63488, 126976, 253952, 507904, 1015808, 2031616, 4063232, 8126464, 16252928, 32505856, 65011712, 130023424, 260046848, 520093696, 1040187392, 2080374784, 4160749568, 8321499136

Offset: 1

Giovanni Teofilatto, Apr 29 2015

31 is the only prime after 5 (the remaining terms are even).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A206371.

[1,2,5,8] cat [31*2^n div 64: n in [5..50]]; // Vincenzo Librandi, May 03 2015

Join[{1,2,5,8,15}, Table[31*2^(n-6), {n,6,50}]] (* Vincenzo Librandi, May 03 2015 *)
CoefficientList[ Series[(x^5 -x^4 -2x^3 +x^2 +1)/(1 -2x), {x, 0, 33}], x] (* Robert G. Wilson v, May 05 2015 *)
Join[{1,2,5,8,15},NestList[2#&,31,30]] (* Harvey P. Dale, Oct 09 2018 *)

def A257548(n): return (4*fibonacci(n+1) -3 -(-1)^n)/2 if (n<6) else 31*2^(n-6)
[A257548(n) for n in range(1,51)] # G. C. Greubel, Jan 05 2023

For n>=6, a(n) = 31*2^(n-6).
For n>=6, a(n) = A206371(n-6) - 1.
G.f.: x*(1+x^2-2*x^3-x^4+x^5)/(1-2*x). - Robert G. Wilson v, May 05 2015
E.g.f.: (31/64)*exp(2*x) + x/32 + x^2/32 + 3*x^3/16 + x^4/96 - x^5/240. - G. C. Greubel, Jan 05 2023

A224195 Ordered sequence of numbers of form (2^n - 1)*2^m + 1 where n >= 1, m >= 1.

Original entry on oeis.org

3, 5, 7, 9, 13, 15, 17, 25, 29, 31, 33, 49, 57, 61, 63, 65, 97, 113, 121, 125, 127, 129, 193, 225, 241, 249, 253, 255, 257, 385, 449, 481, 497, 505, 509, 511, 513, 769, 897, 961, 993, 1009, 1017, 1021, 1023, 1025, 1537, 1793, 1921, 1985, 2017, 2033, 2041, 2045, 2047

Offset: 1

Brad Clardy, Apr 01 2013

The table is constructed so that row labels are 2^n - 1, and column labels are 2^n. The body of the table is the row*col + 1. A MAGMA program is provided that generates the numbers in a table format. The sequence is read along the antidiagonals starting from the top left corner.
All of these numbers have the following property:
   let m be a member of A(n),
   if a sequence B(n) = all i such that i XOR (m - 1) = i - (m - 1), then
   the differences between successive members of B(n) is a repeating series
   of 1's with the last difference in the pattern m. The number of ones in
   the pattern is 2^j - 1, where j is the column index.
As an example consider A(4) which is 9,
   the sequence B(n) where i XOR 8 = i - 8 starts as:
   8, 9, 10, 11, 12, 13, 14, 15, 24... (A115419)
   with successive differences of:
   1, 1, 1, 1, 1, 1, 1, 9.
The main diagonal is the 6th cyclotomic polynomial evaluated at powers of two (A020515).
The formula for diagonals above the main diagonal
  2^(2*n+1) - 2^(n + (a+1)/2) + 1 n>=(a+1)/2 a=odd number above diagonal
  2^(2*n) - 2^(n  + (b/2)) + 1    n>=(b/2)+1 b=even number above diagonal
The formulas for diagonals below the main diagonal
  2^(2*n+1) - 2^(n + 1 -(a+1)/2) + 1 n>=(a+1)/2 a=odd number below diagonal
  2^(2*n) - 2^(n - (b/2)) + 1       n>=(b/2)+1 b=even number below diagonal
Primes of this sequence are in A152449.

			Using the lexicographic ordering of A057555 the sequence is:
A(n) = Table(i,j) with (i,j)=(1,1),(1,2),(2,1),(1,3),(2,2),(3,1)...
  +1  |    2    4     8    16    32     64    128    256     512    1024 ...
  ----|-----------------------------------------------------------------
  1   |    3    5     9    17    33     65    129    257     513    1025
  3   |    7   13    25    49    97    193    385    769    1537    3073
  7   |   15   29    57   113   225    449    897   1793    3585    7169
  15  |   31   61   121   241   481    961   1921   3841    7681   15361
  31  |   63  125   249   497   993   1985   3969   7937   15873   31745
  63  |  127  253   505  1009  2017   4033   8065  16129   32257   64513
  127 |  255  509  1017  2033  4065   8129  16257  32513   65025  130049
  255 |  511 1021  2041  4081  8161  16321  32641  65281  130561  261121
  511 | 1023 2045  4089  8177 16353  32705  65409 130817  261633  523265
  1023| 2047 4093  8185 16369 32737  65473 130945 261889  523777 1047553
  ...

Brad Clardy, Table of n, a(n) for n = 1..1000

Cf. A081118, A152449 (primes), A057555 (lexicographic ordering), A115419 (example).
Rows: A000051(i=1), A181565(2), A083686(3), A195744(4), A206371(5), A196657(6).
Cols: A000225(j=1), A036563(2), A048490(3), A176303 (7 offset of 8).
Diagonals: A020515 (main), A092440, A060867 (above), A134169 (below).

//program generates values in a table form
for i:=1 to 10 do
    m:=2^i - 1;
    m,[ m*2^n +1 : n in [1..10]];
end for;
//program generates sequence in lexicographic ordering of A057555, read
//along antidiagonals from top. Primes in the sequence are marked with *.
for i:=2 to 18 do
    for j:=1 to i-1 do
       m:=2^j -1;
       k:=m*2^(i-j) + 1;
       if IsPrime(k) then k,"*";
          else k;
       end if;;
    end for;
end for;

Table[(2^j-1)*2^(i-j+1) + 1, {i, 10}, {j, i}] (* Paolo Xausa, Apr 02 2024 *)

a(n) = (2^(A057555(2*n-1)) - 1)*2^(A057555(2*n)) + 1 for n>=1. [corrected by Jason Yuen, Feb 22 2025]

a(n) = A081118(n)+2; a(n)=(2^i-1)*2^j+1, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Apr 04 2013

A334164 a(n) is the number of ON-cells in the completed n-th level of a triangular wedge in the hexagonal grid of A151723 (i.e., after 2^k >= n generations of the automaton in A151723 have been computed).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 4, 8, 2, 10, 6, 10, 4, 14, 8, 16, 2, 18, 10, 16, 8, 20, 12, 22, 6, 26, 14, 22, 8, 30, 16, 32, 2, 34, 18, 28, 16, 34, 18, 32, 14, 40, 22, 34, 16, 42, 24, 44, 10, 50, 26, 40, 20, 48, 28, 50, 14, 58, 30, 46, 16, 62, 32, 64

Offset: 1

Hartmut F. W. Hoft, Apr 17 2020

nonn

Conjecture 1: Except for a(2^n + 1) = 2, n >= 1, for odd-numbered completed levels a lower bound of the ratio of ON-cells to the length of the level is (2^n + 2)/(3*2^(n+1) + 1) with limit 1/6, determined by the subsequence of levels starting with: 13, 25, 49, 97, 193, 385, 769, 1537, 3073, ..., and the associated ON-cell counts: 4, 6, 10, 18, 34, 66, 130, 258, 514, ..., as listed in the second column of each of the two triangles below.
The ON-cell counts for the indices in each row define a line of slope 1/2.  The formula for the indices of levels in row k >= 2 is  L(k, i) = 1 + Sum_{j = 0, ..., i} 2^(k-j), 0 <= i <= k - 2, and the formula for the associated numbers of ON-cells is C(k, i) = 2 + Sum_{j = 1..i} 2^(k-1-j), 0 <= i <= k - 2:
Index of the level: L(k, i) number of ON-cells: C(k, i)
k\i  0   1   2   3   4   5   6    k/i  0   1   2   3   4   5   6
2:   5                             2:  2
3:   9  13                         3:  2   4
4:  17  25  29                     4:  2   6   8
5:  33  49  57  61                 5:  2  10  14  16
6:  65  97 113 121 125             6:  2  18  26  30  32
7: 129 193 225 241 249 253         7:  2  34  50  58  62  64
8: 257 385 449 481 497 505 509     8:  2  66  98 114 122 126 128
...
The pairs ( L(k, i), C(k, i) ), for 0 <= i <= k-2, define a line of slope 1/2 for each k >= 3.
For triangle L(k, i): column 0 is A000051(n), n >= 2; column 1 is A181565(n), n >= 3; column 2 is A083686(n), n >= 2; columns 3 is A195744(n), n >= 1; column 4 is A206371(n), n >= 2; column 5 is A196657(n), n >= 1; the bounding diagonal is A036563(n), n >= 3.
For triangle C(k, i): column 1 is A052548(n), n >= 1; column 2 is A164094(n), n >= 1.
Conjecture 2: In an even-numbered completed level 2*n the fraction of ON-cells is bounded below by (23 * 2^n - 24)/(2^(n+5) - 36) with limit 23/32, determined by the subsequence of levels starting with: 28, 92, 220, 476, 988, 2012, 4060, ... .
There are 16 numbers less than 1000 that do not occur as the number of ON-cells in a completed level through level 16384: 136, 164, 330, 334, 402, 444, 526, 570, 598, 604, 614, 714, 740, 822, 832, 878.
Sequence A334169 of even-numbered completed levels in which all cells are ON-cells is a subsequence of this sequence.

Hartmut F. W. Hoft, Diagram of ON-cell counts

Cf. A000051, A036563, A052548, A083686, A151723, A164094, A181565, A195744, A196657, A206371, A334169.

(* a169781[] and support functions are defined in A169781 and create the list nTriangle *)
a334164[n_] := Module[{k, levels={}}, a169781[n]; For[k=1, k<=n, k++, AppendTo[levels, Count[nTriangle[[k]], 1] - 2]]; levels]/;(n>=3 && IntegerQ[Log[2,n]])
a334164[64] (* sequence data *)

cp's OEIS Frontend

A257548 a(1) = 1, a(2) = 2, a(3) = 5, a(4) = 8 and a(5) = 15, a(n) = Sum_{j=1..n-1} a(j).

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A224195 Ordered sequence of numbers of form (2^n - 1)*2^m + 1 where n >= 1, m >= 1.

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Crossrefs

Programs

Magma

Mathematica

Formula

A334164 a(n) is the number of ON-cells in the completed n-th level of a triangular wedge in the hexagonal grid of A151723 (i.e., after 2^k >= n generations of the automaton in A151723 have been computed).

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Mathematica