A153435 -id:A153435 - cp's OEIS frontend

A043291 Every run length in base 2 is 2.

3, 12, 51, 204, 819, 3276, 13107, 52428, 209715, 838860, 3355443, 13421772, 53687091, 214748364, 858993459, 3435973836, 13743895347, 54975581388, 219902325555, 879609302220, 3518437208883, 14073748835532, 56294995342131, 225179981368524, 900719925474099

Offset: 1

Clark Kimberling

a(n) is the number whose binary representation is A153435(n). - Omar E. Pol, Jan 18 2009

See A033001 and following for the analog in other bases and the variant with runs of length >= 2. - M. F. Hasler, Feb 01 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (4,1,-4).

Cf. A153435 (binary).
Bisections: A108020, A182512. Bisection of A077854.
Cf. A000975, A033001, A033114.

[Floor(4^(n+1)/5): n in [1..30]]; // Vincenzo Librandi, Jun 26 2011

seq(floor(4^(n+1)/5),n=1..25); # Mircea Merca, Dec 26 2010

f[n_] := Floor[4^(n + 1)/5]; Array[f, 23] (* or *)
a[1] = 3; a[2] = 12; a[3] = 51; a[n_] := a[n] = 4 a[n - 1] + a[n - 2] - 4 a[n - 3]; Array[a, 23] (* or *)
3 LinearRecurrence[{4, 1, -4}, {1, 4, 17}, 23] (* Robert G. Wilson v, Jul 01 2014 *)

A043291 = n->4^(n+1)\5 \\ M. F. Hasler, Feb 01 2014

def a(n): return int(''.join([['11', '00'][i%2] for i in range(n)]), 2)
print([a(n) for n in range(1, 26)]) # Michael S. Branicky, Mar 12 2021

a(n) = 4*a(n-1)+a(n-2)-4*a(n-3), n>3. - John W. Layman, Feb 01 2000
a(n) = floor(4^(n+1)/5). - Mircea Merca, Dec 26 2010
G.f.: 3*x / ( (x-1)*(4*x-1)*(1+x) ). - Joerg Arndt, Jan 08 2011
a(n) = 3*A033114(n). - R. J. Mathar, Jan 08 2011

A152775 Numbers with 3n binary digits where every run length is 3, written in binary.

Original entry on oeis.org

111, 111000, 111000111, 111000111000, 111000111000111, 111000111000111000, 111000111000111000111, 111000111000111000111000, 111000111000111000111000111, 111000111000111000111000111000

Offset: 1

Omar E. Pol, Jan 18 2009

A152776 written in base 2.

			n ... a(n) .............. A152776(n)
1 ... 111 ............... 7
2 ... 111000 ............ 56
3 ... 111000111 ......... 455
4 ... 111000111000 ...... 3640
5 ... 111000111000111 ... 29127

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (1000,1,-1000).

Cf. A043291, A153435, A152776.

FromDigits/@Table[Flatten[PadRight[{},n,{a,b}]/.{a->{1,1,1},b->{0,0,0}}],{n,10}] (* Harvey P. Dale, Mar 23 2012 *)
CoefficientList[Series[111/((x - 1) (x + 1) (1000 x - 1)), {x, 0, 20}], x] (* Vincenzo Librandi, Apr 21 2014 *)

Vec(111*x / ((x-1)*(x+1)*(1000*x-1)) + O(x^100)) \\ Colin Barker, Apr 20 2014

From Colin Barker, Apr 20 2014: (Start)
a(n) = (-1001-999*(-1)^n+2^(4+3*n)*125^(1+n))/18018.
a(n) = 1000*a(n-1)+a(n-2)-1000*a(n-3).
G.f.: 111*x / ((x-1)*(x+1)*(1000*x-1)). (End).

A152776 Numbers such that every run length in base 2 is 3.

Original entry on oeis.org

7, 56, 455, 3640, 29127, 233016, 1864135, 14913080, 119304647, 954437176, 7635497415, 61083979320, 488671834567, 3909374676536, 31274997412295, 250199979298360, 2001599834386887, 16012798675095096, 128102389400760775

Offset: 1

Omar E. Pol, Jan 18 2009

a(n) is the number whose binary representation is A152775(n).

Zak Seidov, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (8, 1, -8).

Cf. A043291, A152775, A153435.

a(n)= 8*a(n-1) +a(n-2) -8*a(n-3). G.f.: 7x/((1-x)(1-8x)(1+x)). a(n)= (-7*(-1)^n-9+16*8^n)/18 = 7*A033118(n). [From R. J. Mathar, Jan 20 2009]

More terms from R. J. Mathar, Jan 20 2009

A154805 Numbers with 4n binary digits where every run length is 4, written in binary.

Original entry on oeis.org

1111, 11110000, 111100001111, 1111000011110000, 11110000111100001111, 111100001111000011110000, 1111000011110000111100001111, 11110000111100001111000011110000, 111100001111000011110000111100001111

Offset: 1

Omar E. Pol, Jan 25 2009

A154806 written in base 2.

			n ... a(n) ................... A154806(n)
1 ... 1111 ................... 15
2 ... 11110000 ............... 240
3 ... 111100001111 ........... 3855
4 ... 1111000011110000 ....... 61680
5 ... 11110000111100001111 ... 986895

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (10000,1,-10000).

Cf. A152775, A153435, A154806, A154807.

CoefficientList[Series[1111/((x - 1) (x + 1) (10000 x - 1)), {x, 0, 10}], x] (* Vincenzo Librandi, Apr 22 2014 *)
LinearRecurrence[{10000,1,-10000},{1111,11110000,111100001111},20] (* Harvey P. Dale, Jul 31 2017 *)

Vec(1111*x/((x-1)*(x+1)*(10000*x-1)) + O(x^100)) \\ Colin Barker, Apr 20 2014

From Colin Barker, Apr 20 2014: (Start)
a(n) = (-10001-9999*(-1)^n+2^(5+4*n)*625^(1+n))/180018.
a(n) = 10000*a(n-1)+a(n-2)-10000*a(n-3).
G.f.: 1111*x / ((x-1)*(x+1)*(10000*x-1)). (End)

A154807 Numbers with 5n binary digits where every run length is 5, written in binary.

Original entry on oeis.org

11111, 1111100000, 111110000011111, 11111000001111100000, 1111100000111110000011111, 111110000011111000001111100000, 11111000001111100000111110000011111, 1111100000111110000011111000001111100000, 111110000011111000001111100000111110000011111

Offset: 1

Omar E. Pol, Jan 25 2009

A154808 written in base 2.

			n ... a(n) ........................ A154808(n)
1 ... 11111 ....................... 31
2 ... 1111100000 .................. 992
3 ... 111110000011111 ............. 31775
4 ... 11111000001111100000 ........ 1016800
5 ... 1111100000111110000011111 ... 32537631

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (100000,1,-100000).

Cf. A152775, A153435, A154805, A154808.

CoefficientList[Series[11111/((x - 1) (x + 1) (100000 x - 1)), {x, 0, 10}], x] (* Vincenzo Librandi, Apr 22 2014 *)
LinearRecurrence[{100000,1,-100000},{11111,1111100000,111110000011111},20] (* Harvey P. Dale, Aug 08 2023 *)

Vec(11111*x/((x-1)*(x+1)*(100000*x-1)) + O(x^100)) \\ Colin Barker, Apr 20 2014

From Colin Barker, Apr 20 2014: (Start)
a(n) = (-100001-99999*(-1)^n+2^(6+5*n)*3125^(1+n))/1800018.
a(n) = 100000*a(n-1)+a(n-2)-100000*a(n-3).
G.f.: 11111*x / ((x-1)*(x+1)*(100000*x-1)). (End)

More terms from Colin Barker, Apr 20 2014

A044941 Number of runs of even length in base-10 representation of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 1

Clark Kimberling

Alois P. Heinz, Table of n, a(n) for n = 1..65536

Cf. A153435 (n where new high).

Incorrect formulas and programs (as pointed out by Kevin Ryde) removed by Joerg Arndt, Nov 08 2020

A153434 a(n) is the smallest number m such that all n numbers m+1, m.m+1, ..., m.m. ... .m+1 are prime, where dot means concatenation.

Original entry on oeis.org

1, 2, 2, 42, 42, 1179990, 243464796

Offset: 1

Farideh Firoozbakht, Apr 01 2009

			All 5 numbers 1+42,1+4242,1+424242,1+42424242 & 1+4242424242 are prime and 42
is the smallest such number so a(5)=42.

Cf. A153435.

cp's OEIS Frontend

A043291 Every run length in base 2 is 2.

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A152775 Numbers with 3n binary digits where every run length is 3, written in binary.

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A152776 Numbers such that every run length in base 2 is 3.

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A154805 Numbers with 4n binary digits where every run length is 4, written in binary.

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A154807 Numbers with 5n binary digits where every run length is 5, written in binary.

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A044941 Number of runs of even length in base-10 representation of n.

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A153434 a(n) is the smallest number m such that all n numbers m+1, m.m+1, ..., m.m. ... .m+1 are prime, where dot means concatenation.

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