A145641 -id:A145641 - cp's OEIS frontend

A138721 Concatenation of n digits 1, n digits 0 and n digits 1.

101, 110011, 111000111, 111100001111, 111110000011111, 111111000000111111, 111111100000001111111, 111111110000000011111111, 111111111000000000111111111, 111111111100000000001111111111, 111111111110000000000011111111111, 111111111111000000000000111111111111

Offset: 1

Omar E. Pol, Mar 29 2008

a(n) is also A145641(n) written in base 2. - Omar E. Pol, Oct 15 2008

a(n) has 3n digits. - Omar E. Pol, Nov 12 2008

			From _Omar E. Pol_, Nov 12 2008: (Start)
n         Successive digits of a(n)
1                 ( 1 0 1 )
2              ( 1 1 0 0 1 1 )
3           ( 1 1 1 0 0 0 1 1 1 )
4        ( 1 1 1 1 0 0 0 0 1 1 1 1 )
5     ( 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 )
(End)

Paolo Xausa, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (1111,-112110,1111000,-1000000).

Cf. A000533, A135577, A138120, A138144, A138145, A138146, A138826, A145641, A147757, A147759.

a:= n-> parse(cat(1$n,0$n,1$n)):
seq(a(n), n=1..14);  # Alois P. Heinz, Mar 03 2022

Table[(100^n + 1)*(10^n - 1)/9, {n, 15}] (* Paolo Xausa, Aug 02 2024 *)

Vec(x*(101000*x^2-2200*x+101)/((x-1)*(10*x-1)*(100*x-1)*(1000*x-1)) + O(x^100)) \\ Colin Barker, Sep 16 2013

G.f.: x*(101000*x^2 - 2200*x + 101) / ((x-1)*(10*x-1)*(100*x-1)*(1000*x-1)). - Colin Barker, Sep 16 2013

a(n) = (100^n+1)*(10^n-1)/9. - Paolo Xausa, Aug 02 2024

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

cp's OEIS Frontend

A138721 Concatenation of n digits 1, n digits 0 and n digits 1.

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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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Magma

Mathematica

PARI

SageMath

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Extensions