A052996 -id:A052996 - cp's OEIS frontend

A036044 BCR(n): write in binary, complement, reverse.

1, 0, 2, 0, 6, 2, 4, 0, 14, 6, 10, 2, 12, 4, 8, 0, 30, 14, 22, 6, 26, 10, 18, 2, 28, 12, 20, 4, 24, 8, 16, 0, 62, 30, 46, 14, 54, 22, 38, 6, 58, 26, 42, 10, 50, 18, 34, 2, 60, 28, 44, 12, 52, 20, 36, 4, 56, 24, 40, 8, 48, 16, 32, 0, 126, 62, 94, 30, 110, 46, 78, 14, 118, 54, 86

Offset: 0

N. J. A. Sloane

a(0) could be considered to be 0 if the binary representation of zero were chosen to be the empty string. - Jason Kimberley, Sep 19 2011
From Bernard Schott, Jun 15 2021: (Start)
Except for a(0) = 1, every term is even.
For each q >= 0, there is one and only one odd number h such that a(n) = 2*q iff n = h*2^m-1 for m >= 1 when q = 0, and for m >= 0 when q >= 1 (see A345401 and some examples below).
a(n) =  0 iff n =    2^m-1 for m >= 1 (Mersenne numbers) (A000225).
a(n) =  2 iff n =  3*2^m-1 for m >= 0 (A153893).
a(n) =  4 iff n =  7*2^m-1 for m >= 0 (A086224).
a(n) =  6 iff n =  5*2^m-1 for m >= 0 (A153894).
a(n) =  8 iff n = 15*2^m-1 for m >= 0 (A196305).
a(n) = 10 iff n = 11*2^m-1 for m >= 0 (A086225).
a(n) = 12 iff n = 13*2^m-1 for m >= 0 (A198274).
For k >= 1, a(n) = 2^k iff n = (2^(k+1)-1)*2^m - 1 for m >= 0.
Explanation for a(n) = 2:
   For m >= 0, A153893(m) = 3*2^m-1 -> 1011...11 -> 0100...00 -> 10 -> 2 where 1011...11_2 is 10 followed by m 1's. (End)

			4 -> 100 -> 011 -> 110 -> 6.

Indranil Ghosh, Table of n, a(n) for n = 0..10000 (first 1024 terms from T. D. Noe)

Cf. A030101, A056539, A329369, A345401.
Cf. A035928 (fixed points), A195063, A195064, A195065, A195066.
Indices of terms 0, 2, 4, 6, 8, 10, 12, 14, 18, 22, 26, 30: A000225 \ {0}, A153893, A086224, A153894, A196305, A086225, A198274, A052996\{1,3}, A291557, A198276, A171389, A198275.

import Data.List (unfoldr)
a036044 0 = 1
a036044 n = foldl (\v d -> 2 * v + d) 0 (unfoldr bc n) where
   bc 0 = Nothing
   bc x = Just (1 - m, x') where (x',m) = divMod x 2
-- Reinhard Zumkeller, Sep 16 2011

A036044:=func; // Jason Kimberley, Sep 19 2011

A036044 := proc(n)
    local bcr ;
    if n = 0 then
        return 1;
    end if;
    convert(n,base,2) ;
    bcr := [seq(1-i,i=%)] ;
    add(op(-k,bcr)*2^(k-1),k=1..nops(bcr)) ;
end proc:
seq(A036044(n),n=0..200) ; # R. J. Mathar, Nov 06 2017

dtn[ L_ ] := Fold[ 2#1+#2&, 0, L ]; f[ n_ ] := dtn[ Reverse[ 1-IntegerDigits[ n, 2 ] ] ]; Table[ f[ n ], {n, 0, 100} ]
Table[FromDigits[Reverse[IntegerDigits[n,2]/.{1->0,0->1}],2],{n,0,80}] (* Harvey P. Dale, Mar 08 2015 *)

a(n)=fromdigits(Vecrev(apply(n->1-n,binary(n))),2) \\ Charles R Greathouse IV, Apr 22 2015

def comp(s): z, o = ord('0'), ord('1'); return s.translate({z:o, o:z})
def BCR(n): return int(comp(bin(n)[2:])[::-1], 2)
print([BCR(n) for n in range(75)]) # Michael S. Branicky, Jun 14 2021

def A036044(n): return -int((s:=bin(n)[-1:1:-1]),2)-1+2**len(s) # Chai Wah Wu, Feb 04 2022

a(2n) = 2*A059894(n), a(2n+1) = a(2n) - 2^floor(log_2(n)+1). - Ralf Stephan, Aug 21 2003

Conjecture: a(n) = (-1)^A023416(n)*b(n) for n > 0 with a(0) = 1 where b(2^m) = (-1)^m*(2^(m+1) - 2) for m >= 0, b(2n+1) = b(n) for n > 0, b(2n) = b(n) + b(n - 2^f(n)) + b(2n - 2^f(n)) for n > 0 and where f(n) = A007814(n) (see A329369). - Mikhail Kurkov, Dec 13 2024

A086224 a(n) = 7*2^n - 1.

Original entry on oeis.org

6, 13, 27, 55, 111, 223, 447, 895, 1791, 3583, 7167, 14335, 28671, 57343, 114687, 229375, 458751, 917503, 1835007, 3670015, 7340031, 14680063, 29360127, 58720255, 117440511, 234881023, 469762047, 939524095, 1879048191, 3758096383, 7516192767, 15032385535, 30064771071

Offset: 0

Marco Matosic, Jul 27 2003

a(n) = A164874(n+2,2); subsequence of A030130. - Reinhard Zumkeller, Aug 29 2009

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-3, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^n*charpoly(A,-5). - Milan Janjic, Jan 27 2010

Michael De Vlieger, Table of n, a(n) for n = 0..3319
Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 3-5, 14.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Other sequences with recurrence a(n+1) = 2*a(n) + 1 are:
a(0) = 2 gives A153893, a(0)=3 essentially A126646.
a(0) = 4 gives A153894, a(0)=5 essentially A153893.
a(0) = 7 gives essentially A000225.
a(0) = 8 gives A052996 except for some initial terms,
a(0) = 9 is essentially A153894.
a(0) = 10 gives A086225,
a(0) = 11 is essentially A153893.
a(0) = 13 is essentially A086224.
Cf. A030130, A052940, A164874.

7*2^Range[0,30]-1 (* Harvey P. Dale, May 09 2018 *)

a(n)=7<Charles R Greathouse IV, Sep 24 2015

a(n+1) = 2*a(n) + 1.
G.f.: (6-5*x)/((1-x)*(1-2*x)). - Jaume Oliver Lafont, Sep 14 2009
a(n-1)^2 + a(n) = (7*2^(n-1))^2. - Vincenzo Librandi, Aug 08 2010
a(n) = (A052940(n+1) + A000225(n+3))/2. - Gennady Eremin, Aug 31 2023
From Elmo R. Oliveira, Apr 22 2025: (Start)
E.g.f.: exp(x)*(7*exp(x) - 1).
a(n) = 3*a(n-1) - 2*a(n-2). (End)

More terms from David Wasserman, Feb 22 2005

A195744 a(n) = 15*2^(n+1) + 1.

Original entry on oeis.org

31, 61, 121, 241, 481, 961, 1921, 3841, 7681, 15361, 30721, 61441, 122881, 245761, 491521, 983041, 1966081, 3932161, 7864321, 15728641, 31457281, 62914561, 125829121, 251658241, 503316481, 1006632961, 2013265921, 4026531841, 8053063681, 16106127361

Offset: 0

Brad Clardy, Sep 23 2011

Binary numbers of form 1111(0^n)1 where n is the index and number of 0's.
Base 10 numbers of this sequence always end in 1.
An Engel expansion of 1/15 to the base 2 as defined in A181565, with the associated series expansion 1/15 = 2/31 + 2^2/(31*61) + 2^3/(31*61*121) + 2^4/(31*61*121*241) + ... . - Peter Bala, Oct 29 2013
The only squares in this sequence are 121 = 11^2 and 961 = 31^2. - Antti Karttunen, Sep 24 2023

			First few terms in binary are 11111, 111101, 1111001, 11110001, 111100001.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A052996, A164094.
Cf. A020737, A083575, A083683, A083686, A083705, A168596, A181565.
Cf. A110286, A128470.
Cf. A005940, A030514.

[15*2^(n+1) + 1: n in [0..30]]; // Vincenzo Librandi, Sep 24 2011

15*2^Range[50] + 1 (* Paolo Xausa, Apr 02 2024 *)

a(n)=30*2^n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A052996(n+3) + A164094(n+2).
From Bruno Berselli, Sep 23 2011: (Start)
G.f.: (31-32*x)/(1-3*x+2*x^2).
a(n) = 2*a(n-1)-1.
a(n) = A110286(n+1)+1 = A128470(2^n). (End)
E.g.f.: exp(x)*(1 + 30*exp(x)). - Stefano Spezia, Oct 08 2022
For n >= 0, A005940(a(n)) = A030514(2+n). - Antti Karttunen, Sep 24 2023

Corrected by Arkadiusz Wesolowski, Sep 23 2011

A075300 Array A read by antidiagonals upwards: A(n, k) = array A054582(n,k) - 1 = 2^n(2k+1) - 1 with n,k >= 0.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 7, 11, 9, 6, 15, 23, 19, 13, 8, 31, 47, 39, 27, 17, 10, 63, 95, 79, 55, 35, 21, 12, 127, 191, 159, 111, 71, 43, 25, 14, 255, 383, 319, 223, 143, 87, 51, 29, 16, 511, 767, 639, 447, 287, 175, 103, 59, 33, 18, 1023, 1535, 1279, 895, 575, 351, 207, 119

Offset: 0

Antti Karttunen, Sep 12 2002

From Philippe Deléham, Feb 19 2014: (Start)
A(0,k)  = 2*k = A005843(k),
A(1,k)  = 4*k + 1 = A016813(k),
A(2,k)  = 8*k + 3 = A017101(k),
A(n,0)  = A000225(n),
A(n,1)  = A153893(n),
A(n,2)  = A153894(n),
A(n,3)  = A086224(n),
A(n,4)  = A052996(n+2),
A(n,5)  = A086225(n),
A(n,6)  = A198274(n),
A(n,7)  = A238087(n),
A(n,8)  = A198275(n),
A(n,9)  = A198276(n),
A(n,10) = A171389(n). (End)
A permutation of the nonnegative integers. - Alzhekeyev Ascar M, Jun 05 2016
The values in array row n, when expressed in binary, have n trailing 1-bits. - Ruud H.G. van Tol, Mar 18 2025

			The array A begins:
   0    2    4    6    8   10   12   14   16   18 ...
   1    5    9   13   17   21   25   29   33   37 ...
   3   11   19   27   35   43   51   59   67   75 ...
   7   23   39   55   71   87  103  119  135  151 ...
  15   47   79  111  143  175  207  239  271  303 ...
  31   95  159  223  287  351  415  479  543  607 ...
  ... - _Philippe Deléham_, Feb 19 2014
From _Wolfdieter Lang_, Jan 31 2019: (Start)
The triangle T begins:
   n\k   0    1    2   3   4   5   6   7  8  9 10 ...
   0:    0
   1:    1    2
   2:    3    5    4
   3:    7   11    9   6
   4:   15   23   19  13   8
   5    31   47   39  27  17  10
   6:   63   95   79  55  35  21  12
   7:  127  191  159 111  71  43  25  14
   8:  255  383  319 223 143  87  51  29 16
   9:  511  767  639 447 287 175 103  59 33 18
  10: 1023 1535 1279 895 575 351 207 119 67 37 20
  ...
T(3, 1) = 2^2*(2*1+1) - 1 = 12 - 1 = 11.  (End)

Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A075301. Transpose: A075302. The X-projection is given by A007814(n+1) and the Y-projection A025480.

Cf. A002262, A025581, A054582, A241957.

A075300bi := (x,y) -> (2^x * (2*y + 1))-1;
A075300 := n -> A075300bi(A025581(n), A002262(n));
A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);
A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))),2) - (n+1);

Table[(2^# (2 k + 1)) - 1 &[m - k], {m, 0, 10}, {k, 0, m}] (* Michael De Vlieger, Jun 05 2016 *)

From Wolfdieter Lang, Jan 31 2019: (Start)
Array A(n, k) = 2^n*(2*k+1) - 1, for n >= 0 and m >= 0.
The triangle is T(n, k) = A(n-k, k) = 2^(n-k)*(2*k+1) - 1, n >= 0, k=0..n.
See also A054582 after subtracting 1. (End)
From Ruud H.G. van Tol, Mar 17 2025: (Start)
A(0, k) is even. For n > 0, A(n, k) is odd and (3 * A(n, k) + 1) / 2 = A(n-1, 3*k+1).
A(n, k) = 2^n - 1 (mod 2^(n+1)) (equivalent to the comment about trailing 1-bits). (End)

A176449 a(n) = 9*2^n - 2.

Original entry on oeis.org

7, 16, 34, 70, 142, 286, 574, 1150, 2302, 4606, 9214, 18430, 36862, 73726, 147454, 294910, 589822, 1179646, 2359294, 4718590, 9437182, 18874366, 37748734, 75497470, 150994942, 301989886, 603979774, 1207959550, 2415919102

Offset: 0

Vincenzo Librandi, Apr 18 2010

			For n = 1, a(1) = 2*(7+1) = 16;
for n = 2, a(2) = 2*(16+1) = 34;
for n = 3, a(3) = 2*(34+1) = 70.

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Magma
```
[9*2^n-2: n in [0..100]]
```

A176449:=n->9*2^n-2; seq(A176449(n), n=0..100); # Wesley Ivan Hurt, Nov 08 2013

Table[9*2^n-2, {n,0,100}] (* Wesley Ivan Hurt, Nov 08 2013 *)
LinearRecurrence[{3,-2},{7,16},40] (* Harvey P. Dale, Jan 12 2024 *)

apply( A176449(n)=9<M. F. Hasler, Dec 11 2018

a(n) = 2*(a(n-1)+1) with a(0) = 7.
a(n) = 3*a(n-1) -2*a(n-2). G.f.: (7-5*x) / ((2*x-1)*(x-1)). - R. J. Mathar, May 02 2010
a(n) = 2*A052996(n+1) for n > 0. - Bruno Berselli and Vincenzo Librandi, Aug 27 2010
a(n) = A033484(n+2) - A007283(n). - M. F. Hasler, Dec 11 2018

Edited by M. F. Hasler, Dec 11 2018

A246347 Record values in A135141.

Original entry on oeis.org

1, 2, 4, 8, 9, 17, 19, 35, 39, 71, 79, 143, 159, 287, 319, 575, 639, 1151, 1279, 2303, 2559, 4607, 5119, 9215, 10239, 18431, 20479, 36863, 40959, 73727, 81919, 147455, 163839, 294911, 327679, 589823, 655359, 1179647, 1310719, 2359295, 2621439, 4718591, 5242879, 9437183, 10485759

Offset: 1

Antti Karttunen, Aug 27 2014 after Robert G. Wilson v's note in A135141

nonn

In binary, the terms of the sequence seem to follow a simple pattern:
          1 = a(1)
         10 = a(2)
        100 = a(3)
       1000 = a(4)
       1001 = a(5)
      10001 = a(6)
      10011 = a(7)
     100011 = a(8)
     100111 = a(9)
    1000111 = a(10)
    1001111 = a(11)
   10001111 = a(12)
   10011111 = a(13)
  100011111 = a(14)
  100111111 = a(15)
  ...
thus the sequence seems to consist of, after 1 and 2, an interleaving of sequence A153894: 4, 9, 19, 39, 79, 159, 319, ... with the sequence A052996 from its third term onward: 8, 17, 35, 71, 143, ...

Antti Karttunen, Table of n, a(n) for n = 1..112

A246346 gives the corresponding positions in A135141.

Cf. A052996, A153894, A246348, A246360.

Union[FromDigits[#,2]&/@Flatten[Table[{PadRight[{1,0,0},n,1],PadRight[ {1,0,0,0},n,1]},{n,30}],1]] (* Harvey P. Dale, May 03 2015 *)

PARI
```
\\ See code in A246348.
```

(define (A246347 n) (A135141 (A246346 n)))

a(n) = A135141(A246346(n)).

A360573 Odd numbers with exactly three zeros in their binary expansion.

Original entry on oeis.org

17, 35, 37, 41, 49, 71, 75, 77, 83, 85, 89, 99, 101, 105, 113, 143, 151, 155, 157, 167, 171, 173, 179, 181, 185, 199, 203, 205, 211, 213, 217, 227, 229, 233, 241, 287, 303, 311, 315, 317, 335, 343, 347, 349, 359, 363, 365, 371, 373, 377, 399, 407, 411, 413

Offset: 1

Bernard Schott, Feb 12 2023

If m is a term then 2*m+1 is another term, since if M is the binary expansion of m, then M.1 where . stands for concatenation is the binary expansion of 2*m+1.
A052996 \ {1,3,8} is a subsequence, since for m >= 3, A052996(m) = 9*2^(m-2) - 1 has 100011..11 with m-2 trailing 1 for binary expansion.
A171389 \ {20} is a subsequence, since for m >= 1, A171389(m) = 21*2^m - 1 has 1010011..11 with m trailing 1 for binary expansion.
A198276 \ {18} is a subsequence, since for m >= 1, A198276(m) = 19*2^m - 1 has 1001011..11 with m trailing 1 for binary expansion.
Binary expansion of a(n) is A360574(n).
{8*a(n), n>0} form a subsequence of A353654 (numbers with three trailing 0 bits and three other 0 bits).
Numbers of the form 2^(a+1) - 2^b - 2^c - 2^d - 1 where a > b > c > d > 0. - Robert Israel, Feb 13 2023

			35_10 = 100011_2, so 35 is a term.

Cf. A005408, A023416, A056557, A333516, A353654, A360010, A360574.
Subsequences: A052996 \ {1,3,8}, A171389 \ {20}, A198276 \ {18}.
Odd numbers with k zeros in their binary expansion: A000225 (k=0), A190620 (k=1), A357773 (k=2), this sequence (k=3).

q:= n-> n::odd and add(1-i, i=Bits[Split](n))=3:
select(q, [$1..575])[];  # Alois P. Heinz, Feb 12 2023
# Alternative:
[seq(seq(seq(seq(2^(a+1) - 2^b - 2^c - 2^d - 1, d = c-1..1,-1), c=b-1..2,-1),b=a-1..3,-1),a=4..12)]; # Robert Israel, Feb 13 2023

Select[Range[1, 500, 2], DigitCount[#, 2, 0] == 3 &] (* Amiram Eldar, Feb 12 2023 *)

isok(m) = (m%2) && #select(x->(x==0), binary(m)) == 3; \\ Michel Marcus, Feb 13 2023

def ok(n): return n&1 and bin(n)[2:].count("0") == 3
print([k for k in range(414) if ok(k)]) # Michael S. Branicky, Feb 12 2023

from itertools import count, islice
from sympy.utilities.iterables import multiset_permutations
def A360573_gen(): # generator of terms
    yield from (int('1'+''.join(d)+'1',2) for l in count(0) for d in  multiset_permutations('000'+'1'*l))
A360573_list = list(islice(A360573_gen(),54)) # Chai Wah Wu, Feb 18 2023

from itertools import combinations, count, islice
def agen(): yield from ((1<Michael S. Branicky, Feb 18 2023

from math import comb, isqrt
from sympy import integer_nthroot
def A056557(n): return (k:=isqrt(r:=n+1-comb((m:=integer_nthroot(6*(n+1), 3)[0])-(nA333516(n): return (r:=n-1-comb((m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2, 3))+1, 3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)), 2)+1
def A360010(n): return (m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2, 3))
def A360573(n):
    a = (a2:=integer_nthroot(24*n, 4)[0])+(n>comb(a2+2, 4))+3
    j = comb(a-1,4)-n
    b, c, d = A360010(j+1)+2, A056557(j)+2, A333516(j+1)
    return (1<Chai Wah Wu, Dec 18 2024

A023416(a(n)) = 3.

Let a = floor((24n)^(1/4))+4 if n>binomial(floor((24n)^(1/4))+2,4) and a = floor((24n)^(1/4))+3 otherwise. Let j = binomial(a-1,4)-n. Then a(n) = 2^a-1-2^(A360010(j+1)+2)-2^(A056557(j)+2)-2^(A333516(j+1)). - Chai Wah Wu, Dec 18 2024

A191664 Dispersion of A014601 (numbers >2, congruent to 0 or 3 mod 4), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 7, 4, 5, 15, 8, 11, 6, 31, 16, 23, 12, 9, 63, 32, 47, 24, 19, 10, 127, 64, 95, 48, 39, 20, 13, 255, 128, 191, 96, 79, 40, 27, 14, 511, 256, 383, 192, 159, 80, 55, 28, 17, 1023, 512, 767, 384, 319, 160, 111, 56, 35, 18, 2047, 1024, 1535, 768, 639

Offset: 1

Clark Kimberling, Jun 11 2011

Row 1:  A000225 (-1+2^n)
Row 2:  A000079 (2^n)
Row 3:  A055010
Row 4:  3*A000079
Row 5:  A153894
Row 6:  5*A000079
Row 7:  A086224
Row 8:  A005009
Row 9:  A052996
For a background discussion of dispersions, see A191426.
...
Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion.  Each complement (beginning with its first term >1) also generates a dispersion.  The six sequences and dispersions are listed here:
...
A191663=dispersion of A042948 (0 or 1 mod 4 and >1)
A054582=dispersion of A005843 (0 or 2 mod 4 and >1; evens)
A191664=dispersion of A014601 (0 or 3 mod 4 and >1)
A191665=dispersion of A042963 (1 or 2 mod 4 and >1)
A191448=dispersion of A005408 (1 or 3 mod 4 and >1, odds)
A191666=dispersion of A042964 (2 or 3 mod 4)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191663 has 1st col A042964, all else A042948
A054582 has 1st col A005408, all else A005843
A191664 has 1st col A042963, all else A014601
A191665 has 1st col A014601, all else A042963
A191448 has 1st col A005843, all else A005408
A191666 has 1st col A042948, all else A042964
...
There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):
   If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by
   a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by
   a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.
This sequence is a permutation of the natural numbers.  - L. Edson Jeffery, Aug 13 2014

			Northwest corner:
1...3...7....15...31
2...4...8....16...32
5...11..23...47...95
6...12..24...48...96
9...19..39...79...159

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A042963, A014601, A191665, A191426.

Cf. A241957.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a = 3; b = 4; m[n_] := If[Mod[n, 2] == 0, 1, 0];
f[n_] := a*m[n + 1] + b*m[n] + 4*Floor[(n - 1)/2]
Table[f[n], {n, 1, 30}]  (* A014601(n+2): (4+4k,5+4k) *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191664 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191664  *)
(* Clark Kimberling, Jun 11 2011 *)
Grid[Table[2^k*(2*Floor[(n + 1)/2] - 1) - Mod[n, 2], {n, 12}, {k, 12}]] (* L. Edson Jeffery, Aug 13 2014 *)

A241957 Rectangular array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = 2^n(2k - 1) - 1, n,k >= 1.

Original entry on oeis.org

1, 3, 5, 7, 11, 9, 15, 23, 19, 13, 31, 47, 39, 27, 17, 63, 95, 79, 55, 35, 21, 127, 191, 159, 111, 71, 43, 25, 255, 383, 319, 223, 143, 87, 51, 29, 511, 767, 639, 447, 287, 175, 103, 59, 33, 1023, 1535, 1279, 895, 575, 351, 207, 119, 67, 37

Offset: 1

L. Edson Jeffery, Aug 09 2014

The sequence is a permutation of the odd natural numbers, since A(n,k) = 2*A054582(n-1,k-1) - 1 and A054582 is a permutation of the natural numbers.
For j a natural number, 2*j - 1 appears in row A001511(j) of A.
This is the square array A075300 with the first row omitted. - Peter Bala, Feb 07 2017

			Array begins:
.      1     5     9    13    17     21     25     29     33     37
.      3    11    19    27    35     43     51     59     67     75
.      7    23    39    55    71     87    103    119    135    151
.     15    47    79   111   143    175    207    239    271    303
.     31    95   159   223   287    351    415    479    543    607
.     63   191   319   447   575    703    831    959   1087   1215
.    127   383   639   895  1151   1407   1663   1919   2175   2431
.    255   767  1279  1791  2303   2815   3327   3839   4351   4863
.    511  1535  2559  3583  4607   5631   6655   7679   8703   9727
.   1023  3071  5119  7167  9215  11263  13311  15359  17407  19455

Vincenzo Librandi, Rows n = 0..50, flattened
Index entries for sequences related to 3x+1 (or Collatz) problem.

Cf. A016813, A017101 (rows 1 and 2).
Cf. A000225, A083329, A153894, A086224, A052996, etc. (columns 1-5).
Cf. A005408 (odd natural numbers), A054582.
Cf. A075300.

(* Array: *)
Grid[Table[2^n*(2*k - 1) - 1, {n, 10}, {k, 10}]]
(* Array antidiagonals flattened: *)
Flatten[Table[2^(n - k + 1)*(2*k - 1) - 1, {n, 10}, {k, n}]]

A(n,k) = 2*A054582(n-1,k-1) - 1.

A106196 Triangle read by rows, generated from Pascal's triangle.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 5, 10, 8, 4, 1, 8, 20, 17, 11, 5, 1, 13, 38, 35, 24, 14, 6, 1, 21, 71, 68, 50, 31, 17, 7, 1

Offset: 0

Gary W. Adamson, Apr 24 2005

The array P =
  1, 0, 0, 0, 0, 0, ...
  0, 1, 0, 0, 0, 0, ...
  0, 1, 1, 0, 0, 0, ...
  0, 0, 2, 1, 0, 0, ...
  0, 0, 1, 3, 1, 0, ...
  0, 0, 0, 3, 4, 1, ...
  ...
... as shown on page 107 of "A Primer for the Fibonacci Numbers".
The array A is composed of arithmetic sequences, as a matrix.
  1, 1, 1,  1,  1, ...
  1, 2, 3,  4,  5, ...
  1, 3, 5,  7,  9, ...
  1, 4, 7, 10, 13, ...
  1, 5, 9, 13, 17, ...
  ...
Leftmost column = Fibonacci numbers, next column (1, 2, 5, 10, 20, ...) = Fibonacci numbers convolved with themselves.

			The operation P * A generates the array:
  1,  1,  1,  1,  1, ...
  1,  2,  3,  4,  5, ...
  2,  5,  8, 11, 14, ...
  3, 10, 17, 24, 31, ...
  5, 20, 35, 50, 65, ...
  ...
from which we extract antidiagonals, read by rows, become triangle A106196:
   1;
   1,  1;
   2,  2,  1;
   3,  5,  3,  1;
   5, 10,  8,  4,  1;
   8, 20, 17, 11,  5,  1;
  13, 38, 35, 24, 14,  6,  1;
  21, 71, 68, 50, 31, 17,  7,  1;
  ...

V. E. Hoggatt, Jr., editor; "A Primer for the Fibonacci Numbers", 1963, p. 107.

Cf. A052996, A007678, A106196.

Let P = an array with columns composed of Pascal's Triangle rows, offset, spaces filled in with zeros; A = an array composed of arithmetic sequences(n, k). Perform P * A and extract antidiagonals which become the rows of A106196.

cp's OEIS Frontend

A036044 BCR(n): write in binary, complement, reverse.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Python

Python

Formula

A086224 a(n) = 7*2^n - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A195744 a(n) = 15*2^(n+1) + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A075300 Array A read by antidiagonals upwards: A(n, k) = array A054582(n,k) - 1 = 2^n*(2*k+1) - 1 with n,k >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A176449 a(n) = 9*2^n - 2.

Views

Author

Keywords

Examples

Links

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A246347 Record values in A135141.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Scheme

A075300 Array A read by antidiagonals upwards: A(n, k) = array A054582(n,k) - 1 = 2^n(2k+1) - 1 with n,k >= 0.

A241957 Rectangular array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = 2^n(2k - 1) - 1, n,k >= 1.