A133551 -id:A133551 - cp's OEIS frontend

A242144 T(n,k)=Number of length n+5 0..k arrays with no consecutive six elements summing to more than 3*k.

42, 435, 74, 2338, 1113, 132, 8688, 7862, 2902, 236, 25494, 36224, 27024, 7596, 421, 63490, 126894, 154647, 93308, 19834, 747, 140148, 367358, 647404, 663395, 321320, 51440, 1314, 282051, 924300, 2180310, 3319500, 2837837, 1098260, 131950, 2318

Offset: 1

R. H. Hardin, May 05 2014

Table starts
...42....435....2338.....8688.....25494......63490......140148......282051
...74...1113....7862....36224....126894.....367358......924300.....2088459
..132...2902...27024...154647....647404....2180310.....6256170....15876783
..236...7596...93308...663395...3319500...13006484....42564898...121330981
..421..19834..321320..2837837..16970962...77357343...288712815...924335053
..747..51440.1098260.12043599..86052208..456215409..1941492045..6980495147
.1314.131950.3708268.50455611.430518585.2653766000.12874102578.51971761446

			Some solutions for n=3 k=4
..2....1....0....0....0....2....0....0....0....0....2....0....1....0....1....2
..1....0....0....3....3....3....3....0....3....2....2....4....3....3....0....1
..1....1....4....2....1....0....0....4....1....0....1....3....1....2....1....0
..4....3....0....2....1....1....3....2....0....2....1....0....0....0....0....1
..1....0....3....2....3....1....1....0....4....0....1....0....1....1....1....1
..1....0....1....0....0....2....4....1....1....0....4....2....4....3....2....4
..1....1....0....2....0....4....0....1....1....2....3....1....2....0....1....0
..1....2....3....3....2....1....1....3....3....1....1....2....0....1....2....3

R. H. Hardin, Table of n, a(n) for n = 1..532

Column 1 is A133551(n+5)
Column 2 is A212227
Column 3 is A212466

Empirical for column k:
k=1: [linear recurrence of order 20]
Empirical for row n:
n=1: [polynomial of degree 6]
n=2: [polynomial of degree 7]
n=3: [polynomial of degree 8]
n=4: [polynomial of degree 9]
n=5: [polynomial of degree 10]
n=6: [polynomial of degree 11]
n=7: [polynomial of degree 12]

A212402 T(n,k)=Number of binary arrays of length n+2*k-1 with no more than k ones in any length 2k subsequence (=50% duty cycle).

Original entry on oeis.org

3, 11, 5, 42, 19, 8, 163, 74, 33, 13, 638, 291, 132, 57, 21, 2510, 1150, 527, 236, 97, 34, 9908, 4558, 2104, 959, 421, 166, 55, 39203, 18100, 8402, 3872, 1747, 747, 285, 89, 155382, 71971, 33560, 15586, 7143, 3179, 1314, 489, 144, 616666, 286454, 134075

Offset: 1

R. H. Hardin May 14 2012

Table starts
..3..11...42...163...638...2510...9908...39203...155382...616666...2449868
..5..19...74...291..1150...4558..18100...71971...286454..1140954...4547020
..8..33..132...527..2104...8402..33560..134075...535728..2140910...8556568
.13..57..236...959..3872..15586..62632..251419..1008536..4043582..16206152
.21..97..421..1747..7143..29002.117290..473171..1905675..7665886..30810054
.34.166..747..3179.13185..54042.220054..892387..3609005.14567294..58714842
.55.285.1314..5769.24322.100736.413220.1685039..6844362.27724036.112072540
.89.489.2318.10425.44794.187696.776116.3183631.12990818.52815156.214150732

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

R. H. Hardin, Table of n, a(n) for n = 1..4360

Column 1 is A000045(n+3)
Column 2 is A118647(n+3)
Column 3 is A133551(n+5)
Row 1 is A032443

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

A133548 a(n) = sum of cubes of first n odd primes.

Original entry on oeis.org

27, 152, 495, 1826, 4023, 8936, 15795, 27962, 52351, 82142, 132795, 201716, 281223, 385046, 533923, 739302, 966283, 1267046, 1624957, 2013974, 2507013, 3078800, 3783769, 4696442, 5726743, 6819470, 8044513, 9339542, 10782439, 12830822, 15078913, 17650266

Offset: 1

Artur Jasinski, Sep 16 2007, corrected Jun 08 2008

			a(3)=495 because 3^3+5^3+7^3=495.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A007148, A007504, A024450, A098999, A133547, A133549, A133550, A133551.

c = 3; a = {}; b = 0; Do[b = b + Prime[n]^c; AppendTo[a, b], {n, 2, 1000}]; a
Accumulate[Prime[Range[2,40]]^3] (* Harvey P. Dale, Oct 31 2024 *)

a(n) = sum(i=2, n+1, prime(i)^3); \\ Michel Marcus, Nov 05 2013

a(n) = A098999(n+1) - 8.

More terms from Michel Marcus, Nov 05 2013

A133550 Sum of fifth powers of n odd primes.

Original entry on oeis.org

243, 3368, 20175, 181226, 552519, 1972376, 4448475, 10884818, 31395967, 60025118, 129369075, 245225276, 392233719, 621578726, 1039774219, 1754698518, 2599294819, 3949419926, 5753649277, 7826720870, 10903777269, 14842817912

Offset: 1

Artur Jasinski, Sep 16 2007

			a(2)=3368 because 3^5+5^5 = 3368.

Cf. A007148, A007504, A024450, A098999, A122102, A122103, A133547, A133548, A133550, A133551.

c = 5; a = {}; b = 0; Do[b = b + Prime[n]^c; AppendTo[a, b], {n, 2, 1000}]; a

a(n) = A122103(n+1)-32.

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.

A133549 Sum of the fourth powers of the first n odd primes.

Original entry on oeis.org

81, 706, 3107, 17748, 46309, 129830, 260151, 539992, 1247273, 2170794, 4044955, 6870716, 10289517, 15169198, 23059679, 35177040, 49022881, 69174002, 94585683, 122983924, 161934005, 209392326, 272134567, 360663848, 464724249, 577275130

Offset: 1

Artur Jasinski, Sep 16 2007

			a(2)=706 because 3^4 + 5^4 = 706.

Cf. A007148, A007504, A024450, A098999, A122102, A133547, A133548, A133550, A133551.

a:=proc (n) options operator, arrow: add(ithprime(j)^4, j=2..n+1) end proc: seq(a(n),n=1..26); # Emeric Deutsch, Oct 02 2007

c = 4; a = {}; b = 0; Do[b = b + Prime[n]^c; AppendTo[a, b], {n, 2, 1000}]; a

a(n) = A122102(n+1) - 16. - Michel Marcus, Nov 05 2013

Comment corrected by Michel Marcus, Nov 05 2013

Showing 1-7 of 7 results.

cp's OEIS Frontend

A242144 T(n,k)=Number of length n+5 0..k arrays with no consecutive six elements summing to more than 3*k.

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A212402 T(n,k)=Number of binary arrays of length n+2*k-1 with no more than k ones in any length 2k subsequence (=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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A133548 a(n) = sum of cubes of first n odd primes.

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A133550 Sum of fifth powers of n odd primes.

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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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A133549 Sum of the fourth powers of the first n odd primes.

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