A147539 -id:A147539 - cp's OEIS frontend

A138120 Concatenation of n digits 1, 2n-1 digits 0 and n digits 1.

101, 1100011, 11100000111, 111100000001111, 1111100000000011111, 11111100000000000111111, 111111100000000000001111111, 1111111100000000000000011111111, 11111111100000000000000000111111111, 111111111100000000000000000001111111111

Offset: 1

Omar E. Pol, Apr 06 2008

a(n) has 4n-1 digits.

a(n) is also A147539(n) written in base 2. [Omar E. Pol, Nov 08 2008]

			n ........... a(n)
1 ........... 101
2 ......... 1100011
3 ....... 11100000111
4 ..... 111100000001111
5 ... 1111100000000011111

Index entries for linear recurrences with constant coefficients, signature (11011,-10121010,110110000,-100000000).

Cf. A138144, A138145, A138146, A138148, A138720, A138721, A138722, A138826.

Cf. A147539. [Omar E. Pol, Nov 08 2008]

a:= n-> parse(cat(1$n,0$(2*n-1),1$n)):
seq(a(n), n=1..11);  # Alois P. Heinz, Mar 03 2022

Table[FromDigits[Join[PadRight[{},n,1],PadRight[{},2n-1,0], PadRight[ {},n,1]]],{n,10}] (* or *) LinearRecurrence[{11011,-10121010,110110000,-100000000},{101,1100011,11100000111,111100000001111},10] (* Harvey P. Dale, Mar 19 2016 *)

Vec(x*(10001000*x^2-12100*x+101)/((x-1)*(10*x-1)*(1000*x-1)*(10000*x-1)) + O(x^100)) \\ Colin Barker, Sep 16 2013

def a(n): return int("1"*n + "0"*(2*n-1) + "1"*n)
print([a(n) for n in range(1, 11)]) # Michael S. Branicky, Mar 03 2022

G.f.: x*(10001000*x^2-12100*x+101) / ((x-1)*(10*x-1)*(1000*x-1)*(10000*x-1)). [Colin Barker, Sep 16 2013]

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

A147818 Period 4: repeat [5, 9, 9, 5].

Original entry on oeis.org

5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 9, 5, 5

Offset: 1

Omar E. Pol, Nov 14 2008, Jan 25 2009

Last digit of the number whose binary representation is the concatenation of n 1's, 2n-1 0's and n 1's.

a(n) is the final digit of A147539(n).

Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Cf. A138120, A147539.

&cat [[5, 9, 9, 5]^^30]; // Wesley Ivan Hurt, Jul 09 2016

A010879 := proc(n) n mod 10; end:
A147539 := proc(n) 2^n-1+2^(4*n-1)-2^(3*n-1); end:
A147818 := proc(n) A010879(A147539(n)); end: # R. J. Mathar, Jan 22 2009
seq(op([5, 9, 9, 5]), n=1..40); # Wesley Ivan Hurt, Jul 09 2016

PadRight[{}, 100, {5, 9, 9, 5}] (* Wesley Ivan Hurt, Jul 09 2016 *)

a(n+1) = 7-2*cos(Pi*n/2)+2*sin(Pi*n/2). - R. J. Mathar, Oct 08 2011
a(n) = a(n-1)-a(n-2)+a(n-3) for n>3. G.f.: x*(5*x^2+4*x+5)/((1-x)*(x^2+1)). [Colin Barker, Nov 04 2012]
a(n) = a(n-4) for n>4. - Wesley Ivan Hurt, Jul 09 2016

More terms from R. J. Mathar, Jan 22 2009

cp's OEIS Frontend

A138120 Concatenation of n digits 1, 2n-1 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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A147818 Period 4: repeat [5, 9, 9, 5].

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