A147537 -id:A147537 - cp's OEIS frontend

A138118 Concatenation of 2n-1 digits 1 and n digits 0.

10, 11100, 11111000, 11111110000, 11111111100000, 11111111111000000, 11111111111110000000, 11111111111111100000000, 11111111111111111000000000, 11111111111111111110000000000

Offset: 1

Omar E. Pol, Mar 29 2008

a(n) is also A147537(n) written in base 2. [From Omar E. Pol, Nov 08 2008]

			n .......... a(n)
1 ........... 10
2 ......... 11100
3 ....... 11111000
4 ..... 11111110000
5 ... 11111111100000

Index entries for linear recurrences with constant coefficients, signature (1010,-10000)

Cf. A138147, A138721.

Cf. A147537.

FromDigits/@Table[Join[PadRight[{},2n-1,1],PadRight[{},n,0]],{n,15}] (* Harvey P. Dale, Dec 09 2011 *)

O.g.f.: 10*(1+100x)/[(-1+1000x)*(-1+10x)]. a(n)=A100706(n)*10^n = 10*a(n-1)+11*1000^n. - R. J. Mathar, Apr 03 2008

A147590 Numbers whose binary representation is the concatenation of 2n-1 digits 1 and n-1 digits 0.

Original entry on oeis.org

1, 14, 124, 1016, 8176, 65504, 524224, 4194176, 33554176, 268434944, 2147482624, 17179867136, 137438949376, 1099511619584, 8796093005824, 70368744144896, 562949953355776, 4503599627239424, 36028797018701824, 288230376151187456

Offset: 1

Omar E. Pol, Nov 08 2008

a(n) is the number whose binary representation is A147589(n).

			     1_10 is 1_2;
    14_10 is 1110_2;
   124_10 is 1111100_2;
  1016_10 is 1111111000_2.

Nathaniel Johnston, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (10,-16).

Cf. A020540, A138118, A147537, A147589.

List([1..25], n-> 2^(n-2)*(4^n-2)); # G. C. Greubel, Jul 27 2019

[8^n/4-2^(n-1): n in [1..25]]; // Vincenzo Librandi, Jul 27 2019

seq(8^n/4-2^(n-1),n=1..25); # Nathaniel Johnston, Apr 30 2011

LinearRecurrence[{10,-16},{1,14},30] (* Harvey P. Dale, Oct 10 2014 *)
Table[8^n / 4 - 2^(n - 1), {n, 25}] (* Vincenzo Librandi, Jul 27 2019 *)

vector(25, n, 2^(n-2)*(4^n-2)) \\ G. C. Greubel, Jul 27 2019

[2^(n-2)*(4^n-2) for n in (1..25)] # G. C. Greubel, Jul 27 2019

a(n) = A147537(n)/2.
From R. J. Mathar, Jul 13 2009: (Start)
a(n) = 8^n/4 - 2^(n-1) = A083332(2n-2).
a(n) = 10*a(n-1) - 16*a(n-2).
G.f.: x*(1+4*x)/((1-2*x)*(1-8*x)). (End)
From César Aguilera, Jul 26 2019: (Start)
Lim_{n->infinity} a(n)/a(n-1) = 8;
a(n)/a(n-1) = 8 + 6/A083420(n). (End)
E.g.f.: (1/4)*(exp(2*x)*(-2 + exp(6*x)) + 1). - Stefano Spezia, Aug 05 2019
a(n) = A020540(n - 1)/4. - Jon Maiga, Aug 05 2019

More terms from R. J. Mathar, Jul 13 2009

Typo in a(12) corrected by Omar E. Pol, Jul 20 2009

A147595 a(n) is the number whose binary representation is A138144(n).

Original entry on oeis.org

1, 3, 7, 15, 27, 51, 99, 195, 387, 771, 1539, 3075, 6147, 12291, 24579, 49155, 98307, 196611, 393219, 786435, 1572867, 3145731, 6291459, 12582915, 25165827, 50331651, 100663299, 201326595, 402653187, 805306371, 1610612739, 3221225475

Offset: 1

Omar E. Pol, Nov 08 2008

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A138144, A145641, A147537, A147538, A147539, A147540, A147590, A147596, A147597.

[1,3,7] cat [3*(1+2^(n-2)): n in [4..40]]; // G. C. Greubel, Oct 25 2022

LinearRecurrence[{3,-2},{1,3,7,15,27},40] (* Harvey P. Dale, Nov 30 2020 *)

Vec(-x*(2*x^2-1)*(2*x^2+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Sep 15 2013

[1,3,7]+[3*(1+2^(n-2)) for n in range(4,40)] # G. C. Greubel, Oct 25 2022

a(n) = A060013(n+2), n > 3. - R. J. Mathar, Feb 05 2010
a(n+4) = 3*(2^(n+2) + 1), n >= 0. - Brad Clardy, Apr 03 2013
From Colin Barker, Sep 15 2013: (Start)
a(n) = 3*(4 + 2^n)/4 for n>3.
a(n) = 3*a(n-1) - 2*a(n-2).
G.f.: x*(1-2*x^2)*(1+2*x^2) / ((1-x)*(1-2*x)). (End)
E.g.f.: (3/4)*(4*exp(x) + exp(2*x)) - (15/4) - 7*x/2 - 3*x^2/2 - x^3/3. - G. C. Greubel, Oct 25 2022

Extended by R. J. Mathar, Feb 05 2010

A147596 a(n) is the number whose binary representation is A138145(n).

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 119, 231, 455, 903, 1799, 3591, 7175, 14343, 28679, 57351, 114695, 229383, 458759, 917511, 1835015, 3670023, 7340039, 14680071, 29360135, 58720263, 117440519, 234881031, 469762055, 939524103, 1879048199, 3758096391

Offset: 1

Omar E. Pol, Nov 08 2008

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A138145, A145641, A147537, A147538, A147539.

Cf. A147540, A147590, A147595, A147597.

[1,3,7,15,31] cat [7*(1+2^(n-3)): n in [6..40]]; // G. C. Greubel, Oct 25 2022

Join[{1,3,7,15,31}, 7*(1+2^(Range[6, 40] -3))] (* G. C. Greubel, Oct 25 2022 *)

Vec(-x*(2*x^2-1)*(4*x^4+2*x^2+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Sep 15 2013

def A147596(n): return 7*(1+2^(n-3)) -(1/8)*(63*int(n==0) +62*int(n==1) +60*int(n ==2)) -(7*int(n==3) +6*int(n==4) +4*int(n==5))
[A147596(n) for n in range(1,40)] # G. C. Greubel, Oct 25 2022

a(n) = 7*(2^(n-3) + 1) if n >= 6. - Hagen von Eitzen, Jun 02 2009
From Colin Barker, Sep 15 2013: (Start)
a(n) = 3*a(n-1) - 2*a(n-2), for n >= 8.
G.f.: x*(1-2*x^2)*(1+2*x^2+4*x^4) / ((1-x)*(1-2*x)). (End)
E.g.f.: (7/8)*(8*exp(x) + exp(2*x)) - (1/8)*(63 + 62*x + 30*x^2) - 7*x^3/6 - x^4/4 - x^5/30. - G. C. Greubel, Oct 25 2022

More terms from Hagen von Eitzen, Jun 02 2009

A147597 a(n) is the number whose binary representation is A138146(n).

Original entry on oeis.org

1, 7, 31, 119, 455, 1799, 7175, 28679, 114695, 458759, 1835015, 7340039, 29360135, 117440519, 469762055, 1879048199, 7516192775, 30064771079, 120259084295, 481036337159, 1924145348615, 7696581394439, 30786325577735, 123145302310919, 492581209243655

Offset: 1

Omar E. Pol, Nov 08 2008

Bisection of A147596.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A138146, A145641, A147537, A147538, A147539.

Cf. A147540, A147590, A147595, A147596.

[1,7,31] cat [7*(1+4^(n-2)): n in [4..40]]; // G. C. Greubel, Oct 25 2022

Table[FromDigits[#, 2] &@ If[n < 4, ConstantArray[1, 2 n - 1], Join[#, ConstantArray[0, 2 n - 7], #]] &@ ConstantArray[1, 3], {n, 25}] (* or *)
Rest@ CoefficientList[Series[x (2 x + 1) (2 x - 1) (4 x^2 + 2 x + 1)/((4 x - 1) (1 - x)), {x, 0, 25}], x] (* Michael De Vlieger, Nov 25 2016 *)
LinearRecurrence[{5,-4},{1,7,31,119,455},30] (* Harvey P. Dale, Aug 04 2025 *)

Vec(x*(2*x+1)*(2*x-1)*(4*x^2+2*x+1)/((4*x-1)*(1-x)) + O(x^30)) \\ Colin Barker, Nov 25 2016

def A147597(n): return 7*(1+4^(n-2)) -(119/16)*int(n==0) -(31/4)*int(n==1) -7*int(n==2) -4*int(n==3)
[A147597(n) for n in range(1,41)] # G. C. Greubel, Oct 25 2022

From R. J. Mathar, Feb 05 2010: (Start)
a(n) = 5*a(n-1) - 4*a(n-2) for n>5.
G.f.: x*(2*x+1)*(2*x-1)*(4*x^2+2*x+1)/((4*x-1)*(1-x)). (End)
a(n) = 7*4^(n-2) + 7 for n>3. - Colin Barker, Nov 25 2016
E.g.f.: (7/16)*(16*exp(x) + exp(4*x)) -(119/16) -31*x/4 -7*x^2/2 -2*x^3/3. - G. C. Greubel, Oct 25 2022

More terms from R. J. Mathar, Feb 05 2010

A147589 Concatenation of 2n-1 digits 1 and n-1 digits 0.

Original entry on oeis.org

1, 1110, 1111100, 1111111000, 1111111110000, 1111111111100000, 1111111111111000000, 1111111111111110000000, 1111111111111111100000000, 1111111111111111111000000000, 1111111111111111111110000000000

Offset: 1

Omar E. Pol, Nov 08 2008

a(n) is also A147590(n) written in base 2.

			n .......... a(n)
1 ........... 1
2 ......... 1110
3 ....... 1111100
4 ..... 1111111000
5 ... 1111111110000

Index entries for linear recurrences with constant coefficients, signature (1010,-10000).

Cf. A138118, A147537, A147590.

vector(100, n, 10^(-2+n)*(-10+100^n)/9) \\ Colin Barker, Jul 08 2014

Vec(x*(100*x+1)/((10*x-1)*(1000*x-1)) + O(x^100)) \\ Colin Barker, Jul 08 2014

a(n) = A138118(n)/10.
a(n) = {[10^(2*n-1)-1]*10^(n-1)}/9, with n>=1. - Paolo P. Lava, Nov 26 2008
G.f.: x*(100*x+1) / ((10*x-1)*(1000*x-1)). - Colin Barker, Jul 08 2014

Keyword:base added by Charles R Greathouse IV, Apr 28 2010

A358167 Irregular triangle read by rows: T(n, k) = k-th fixed point in Zhegalkin permutation n (row n of A197819).

Original entry on oeis.org

0, 1, 0, 2, 0, 6, 8, 14, 0, 30, 40, 54, 72, 86, 96, 126, 128, 158, 168, 182, 200, 214, 224, 254, 0, 510, 680, 854, 1224, 1334, 1632, 1950, 2176, 2430, 2600, 3030, 3144, 3510, 3808, 3870, 4320, 4382, 4680, 5046, 5160

Offset: 0

Tilman Piesk, Nov 01 2022

Let R = A197819(n, ...) and F = a(n, ...). Then F are the fixed points of R.
But there is a second relationship between F and R:
  Let X(i) = R(i) XOR i. Then X(i) is an element of F.
  Let I_k = {i | X(i) = F(k)}. Let Q = A197819(n-1, ...).
  Then I_k = {Q(k) XOR f | f in F}.
Row lengths are 2, 2, 4, 16, 256, 65536, ..., i.e., A001146(n-1) for n > 0.
Row sums are 1, 2, 28, 2032, 8388352, ..., i.e., A147537(A000225) for n > 0.

			Triangle begins:
     k  0    1    2   3    4   5   6    7    8    9   10   11   12   13   14   15
  n
  0     0,   1
  1     0,   2
  2     0,   6,   8, 14
  3     0,  30,  40, 54,  72, 86, 96, 126, 128, 158, 168, 182, 200, 214, 224, 254
  4     0, 510, 680...
A197819(3, 168) = a(3, 10) = 168.
How to calculate the term for n=3, k=10:
  p = A197819(n-1, k) = A197819(2, 10) = 2
  p XOR k = 2 XOR 10 = 8
  shifted_k = 2^(2^(n-1)) * k = 2^(2^2) * 10 = 160
  (p XOR k) + shifted_k = 8 + 160 = 168
168 in little-endian binary is 00010101. The corresponding algebraic normal form is XOR(AND(x0, x1), AND(x0, x2), AND(x0, x1, x2)). (Its ANDs correspond to the 3 binary 1s.) The truth table of this Boolean function is again 00010101.
  (With x0 = 01010101, x1 = 00110011, x2 = 00001111.)
Example for the second relationship with A197819, as described in COMMENTS:
  Let R = A197819(3, 0..255), F = a(3, 0..15), Q = A197819(2, 0..15).
  I_3 = {i | R(i) XOR i = F(3)}
      = {Q(3) XOR f | f in F} = {5 XOR f | f in F}
      = {5, 27, 45, 51, 77, 83, 101, 123, 133, 155, 173, 179, 205, 211, 229, 251}
  R(5) XOR 5  =  R(27) XOR 27  =  R(45) XOR 45  =  R(51) XOR 51  =  ...  =  F(3)
   51  XOR 5  =    45  XOR 27  =    27  XOR 45  =     5  XOR 51  =  ...  =   54

Tilman Piesk, Rows 0..4 of the triangle, flattened
Tilman Piesk, row 3 and row 4 as binary matrices, row 3 as part of the permutation
Tilman Piesk, Zhegalkin matrix (Wikiversity)

Cf. A197819, A001146, A147537, A000225, A058891.

def a(n, k):
    if n == 0:
        assert k < 2
        return k
    else:
        row_length = 1 << (1 << (n-1))  # 2 ** 2 ** (n-1)
        assert k < row_length
    p = a197819(n-1, k)
    p_xor_k = p ^ k
    shifted_k = row_length * k
    return p_xor_k + shifted_k

For n>0: T(n, k) = [A197819(n-1, k) XOR k] + [2^(2^(n-1)) * k].
  (On this page "XOR" always is the bitwise exclusive or.)
For n>0: T(n, A058891(n)) = A058891(n+1) is the unique power of 2 in row n.

cp's OEIS Frontend

A138118 Concatenation of 2n-1 digits 1 and n digits 0.

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A147590 Numbers whose binary representation is the concatenation of 2n-1 digits 1 and n-1 digits 0.

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A147595 a(n) is the number whose binary representation is A138144(n).

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A147596 a(n) is the number whose binary representation is A138145(n).

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A147597 a(n) is the number whose binary representation is A138146(n).

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A147589 Concatenation of 2n-1 digits 1 and n-1 digits 0.

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