A133523 -id:A133523 - cp's OEIS frontend

A213118 T(n,k)=Number of binary arrays of length n+2*k-1 with fewer than k ones in any length 2k subsequence (=less than 50% duty cycle).

1, 5, 1, 22, 7, 1, 93, 34, 10, 1, 386, 151, 54, 14, 1, 1586, 646, 252, 86, 19, 1, 6476, 2710, 1110, 424, 136, 26, 1, 26333, 11236, 4748, 1926, 714, 212, 36, 1, 106762, 46231, 19964, 8404, 3354, 1198, 324, 50, 1, 431910, 189214, 83024, 35836, 14946, 5842, 1996, 498

Offset: 1

R. H. Hardin Jun 05 2012

Table starts
.1..5..22...93...386..1586...6476...26333..106762...431910...1744436...7036530
.1..7..34..151...646..2710..11236...46231..189214...771442...3136156..12720982
.1.10..54..252..1110..4748..19964...83024..342678..1406748...5751636..23443240
.1.14..86..424..1926..8404..35836..150604..626726..2589844..10646676..43594464
.1.19.136..714..3354.14946..64664..274676.1152494..4793874..19813536..81495084
.1.26.212.1198..5842.26630.116992..502492.2126238..8903350..36998056.152862180
.1.36.324.1996.10154.47448.211888..920744.3930286.16570608..69240296.287379592
.1.50.498.3292.17578.84424.383728.1688200.7272622.30880672.129768616.541108840

			Some solutions for n=3 k=4
..0....0....1....0....1....1....1....0....0....1....1....0....0....0....1....0
..1....0....1....0....1....0....0....0....0....0....0....0....0....1....0....0
..1....0....0....1....1....1....0....0....1....1....1....0....0....0....0....0
..1....0....0....0....0....0....0....1....1....1....0....1....0....0....1....1
..0....0....0....0....0....1....1....0....0....0....1....0....0....0....0....0
..0....1....0....0....0....0....1....0....1....0....0....0....1....1....0....1
..0....1....0....1....0....0....0....1....0....0....0....1....0....1....0....0
..0....0....0....1....0....0....0....0....0....0....0....0....0....0....1....0
..0....0....0....0....1....0....1....0....0....0....0....1....1....0....1....1
..1....1....1....0....0....0....0....0....0....0....1....0....1....1....0....0

R. H. Hardin, Table of n, a(n) for n = 1..4938

Column 2 is A003269(n+7)
Column 3 is A133523(n+5)
Row 1 is A000346(n-1)

A120118 a(n) is the number of binary strings of length n such that no subsequence of length 5 or less contains 3 or more ones.

Original entry on oeis.org

1, 2, 4, 7, 11, 16, 26, 43, 71, 116, 186, 300, 487, 792, 1287, 2087, 3382, 5484, 8898, 14438, 23423, 37993, 61625, 99965, 162165, 263065, 426736, 692229, 1122903, 1821538, 2954849, 4793266, 7775472, 12613097, 20460538, 33190414, 53840404

Offset: 0

Tanya Khovanova, Aug 15 2006, Oct 11 2006

			This sequence is similar to A118647 - where no subsequence of length 4 contains 3 ones. It is obvious that the first 4 terms of these two sequences are the same. There are only 3 sequences of length 5 that contain 3 ones such that no subsequence of length 4 contains 3 ones: 10101, 11001, 10011. Hence the fifth term for this sequence is 3 less than the corresponding term of A118647.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kees Immink and Kui Cai, Properties and constructions of energy-harvesting sliding-window constrained codes, IEEE Comm. Lett., May 2020.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,2,0,0,-1,0,-1).

Cf. A118647, A133523, A133551, A334251, A335247.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1 +x*(1 +x+x^2)*(2+x^2+x^3-x^4-x^5-x^7)/(1-x-x^3-2*x^5+x^8+x^10) )); // G. C. Greubel, May 05 2023

LinearRecurrence[{1,0,1,0,2,0,0,-1,0,-1}, {1,2,4,7,11,16,26,43,71,116, 186}, 50] (* Harvey P. Dale, Nov 27 2013 *)

def A120118_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1 +x*(1+x+x^2)*(2+x^2+x^3-x^4-x^5-x^7)/(1-x-x^3-2*x^5 +
     x^8+x^10) ).list()
A120118_list(40) # G. C. Greubel, May 05 2023

a(n) = a(n-1) + a(n-3) + 2*a(n-5) - a(n-8) - a(n-10).

G.f.: 1 + x*(1+x+x^2)*(2+x^2+x^3-x^4-x^5-x^7)/(1-x-x^3-2*x^5+x^8+x^10). - R. J. Mathar, Nov 28 2011

A334251 a(n) is the number of binary (0,1) sequences of length n that have at most two zeros in a window of seven consecutive symbols.

Original entry on oeis.org

1, 2, 4, 7, 11, 16, 22, 29, 43, 66, 102, 157, 239, 358, 526, 777, 1159, 1740, 2619, 3942, 5923, 8870, 13259, 19822, 29667, 44451, 66641, 99912, 149745, 224338, 335993, 503199, 753720, 1129164, 1691796, 2534807, 3797721, 5689507, 8523275, 12768309, 19127928, 28655867, 42930562

Offset: 0

Kees Immink, Apr 20 2020

Application: Not all electronic devices connected to the Internet of Things (IoT) have batteries or are connected to the power cable. These self-contained devices must rely on the harvesting of energy of the signals sent by a transmitter. We investigate binary systems emitting 0's and 1's signals where it is assumed that the 1's carry the energy. A minimal number of 1's in transmitted sequences is required so as to carry sufficient energy within a prescribed time span. A binary sequence is said to obey the sliding-window (ell,t)-constraint if the number of 1's within any window of ell consecutive bits of that sequence is at least t, t

			a(3) = 7 as there are 8 possible binary (0,1) sequences of length 3 but exactly one of them has more than 2 zero's in a window of seven consecutive symbols (the sequence (000)) leaving 8-1 = 7 such sequences. - _David A. Corneth_, Apr 20 2020

Kees Immink, Table of n, a(n) for n = 0..50

Cf. A118647, A120118, A133523.

G.f.: (x^20 +x^19 +x^18 +2*x^17 +2*x^16 +x^15 -3*x^13 -4*x^12 -5*x^11 -7*x^10 -5*x^9 -3*x^8 -3*x^7 +2*x^6 +3*x^5 +3*x^4 +3*x^3 +2*x^2 +x +1) / (-x^21 -x^18 +x^15 +3*x^14 +x^12 +2*x^11 -3*x^7 -x^4 -x +1).
From David A. Corneth, Apr 21 2020: (Start)
a(n) ~ c*r^n where c = 1.81880731105 and r = 1.498122533939865577.
a(n) = a(n - 1) + a(n - 4) + 3*a(n - 6) - 2*a(n - 10) - a(n - 12) - 3*a(n - 13) - a(n - 15) + a(n - 18) + a(n - 21), n >= 21. (End)

A335247 a(n) is the number of binary (0,1) sequences of length n that have at least two ones in each window of eight consecutive symbols.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 127, 247, 487, 961, 1897, 3745, 7393, 14593, 28801, 56833, 112156, 221341, 436825, 862094, 1701380, 3357739, 6626611, 13077820, 25809478, 50935832, 100523529, 198386490, 391522260, 772682018, 1524913233, 3009466064, 5939279536, 11721362180

Offset: 0

Kees Immink, May 28 2020

Application: Not all electronic devices connected to the Internet of Things (IoT) have batteries or are connected to the power cable. These self-contained devices must rely on the harvesting of energy of the signals sent by a transmitter. A minimal number of 1's in transmitted sequences is required so as to carry sufficient energy within a prescribed time span. A binary sequence is said to obey the sliding-window (ell,t)-constraint if the number of 1's within any window of ell consecutive bits of that sequence is at least t, t

Georg Fischer, Table of n, a(n) for n = 0..1000
Kees Immink and Kui Cai, Properties and constructions of energy-harvesting sliding-window constrained code, IEEE Communications Letters, May 2020.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,0,1,2,3,0,-3,-4,-3,0,0,-2,-3,0,0,3,3,0,0,0,1,0,0,0,-1).

Cf. A118647, A120118, A133523, A334251, A133551.

CoefficientList[Series[-(x^27 +x^26 -x^23 -x^22 -3*x^19 -5*x^18 -3*x^17 +3*x^15 +4*x^14 +2*x^13 +3*x^11 +5*x^10 +5*x^9 +3*x^8 -3*x^7 -3*x^6 -2*x^5 -x^4 -x^3 -x^2 -x -1) / (x^28 -x^24 -3*x^20 -3*x^19 +3*x^16 +2*x^15 +3*x^12 +4*x^11 +3*x^10 -3*x^8 -2*x^7 -x^6 -x^4 -x^3 -x^2 -x +1),{x,0,100}],x] (* Georg Fischer, Oct 26 2020 *)
LinearRecurrence[{1,1,1,1,0,1,2,3,0,-3,-4,-3,0,0,-2,-3,0,0,3,3,0,0,0,1,0,0,0,-1},{1,2,4,8,16,32,64,127,247,487,961,1897,3745,7393,14593,28801,56833,112156,221341,436825,862094,1701380,3357739,6626611,13077820,25809478,50935832,100523529},40] (* Harvey P. Dale, Feb 21 2022 *)

G.f.: -(x^27+x^26-x^23-x^22-3*x^19-5*x^18-3*x^17+3*x^15+4*x^14+2*x^13 +3*x^11 +5*x^10+5*x^9+3*x^8-3*x^7-3*x^6-2*x^5-x^4-x^3-x^2-x-1) / (x^28-x^24-3*x^20 -3*x^19 +3*x^16 +2*x^15+3*x^12+4*x^11+3*x^10-3*x^8-2*x^7-x^6-x^4-x^3-x^2-x+1).
a(n) = a(n-1)+a(n-2)+a(n-3)+a(n-4)+a(n-6)+2*a(n-7)+3*a(n-8)-3*a(n-10) -4*a(n-11) -3*a(n-12) -2*a(n-15)-3*a(n-16)+3*a(n-19)+3*a(n-20)+a(n-24)-a(n-28), n>28.
a(n) ~ c*r^n where c = 1.07317641333 and r = 1.9735326811117101072.

A131348 Sum of squares of prime quadruplets.

Original entry on oeis.org

364, 940, 44140, 152140, 2722540, 8820940, 14062540, 17388940, 42380140, 48024940, 127916140, 356076940, 676520140, 979064140, 990360940, 1032336940, 1302488140, 1431108940, 1509322540, 1766520940, 1984702540, 2561372140

Offset: 1

Jonathan Vos Post, Sep 29 2007

This is to prime quadruplets A007530 as sums of squares of twin primes A063533 are to twin primes. This is to prime quadruplets A007530 as A133524 is to four consecutive primes. Note that prime quadruplets are not the same as four consecutive primes. After a(1) these are always multiples of 20, because after A007530(1) = 5, all A007530(n) == 1 mod 10. a(n) is a prime times 20 for an = 1, 2, 3, 12, 16, 21.

			a(1) = 364 = 5^2 + 7^2 + 11^2 + 13^2.
a(2) = 940 = 11^2 + 13^2 + 17^2 + 19^2.
a(3) = 44140 = 101^2 + (103)^2 + (107)^2 + (109)^2 because 101, 103, 107, 109 are a prime quadruplet.

Cf. A000040, A007530, A063533, A133523, A133524.

Total[#^2]&/@Select[Partition[Prime[Range[3000]],4,1],MatchQ[#,{#[[1]],#[[1]]+2,#[[1]]+6,#[[1]]+8}]&]  (* Harvey P. Dale, Feb 03 2011 *)

a(n) = p^2 + (p+2)^2 + (p+6)^2 + (p+8)^2 for p in A007530.

Corrected and extended by Harvey P. Dale, Feb 03 2011

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A213118 T(n,k)=Number of binary arrays of length n+2*k-1 with fewer than k ones in any length 2k subsequence (=less than 50% duty cycle).

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A120118 a(n) is the number of binary strings of length n such that no subsequence of length 5 or less contains 3 or more ones.

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A334251 a(n) is the number of binary (0,1) sequences of length n that have at most two zeros in a window of seven consecutive symbols.

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A335247 a(n) is the number of binary (0,1) sequences of length n that have at least two ones in each window of eight consecutive symbols.

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A131348 Sum of squares of prime quadruplets.

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