cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A212401 Number of binary arrays of length n+13 with no more than 7 ones in any length 14 subsequence (=50% duty cycle).

Original entry on oeis.org

9908, 18100, 33560, 62632, 117290, 220054, 413220, 776116, 1457281, 2734307, 5124772, 9591128, 17917498, 33399134, 62096428, 115525340, 215173819, 401141813, 748337734, 1396704062, 2607652294, 4869423980, 9093738783
Offset: 1

Views

Author

R. H. Hardin May 14 2012

Keywords

Comments

Column 7 of A212402

Examples

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

Links

A212398 Number of binary arrays of length n+7 with no more than 4 ones in any length 8 subsequence (=50% duty cycle).

Original entry on oeis.org

163, 291, 527, 959, 1747, 3179, 5769, 10425, 18729, 33706, 60797, 109800, 198415, 358592, 647959, 1170415, 2113372, 3815438, 6888722, 12439093, 22463681, 40568913, 73266801, 132315810, 238948339, 431506179, 779231793, 1407175435
Offset: 1

Views

Author

R. H. Hardin May 14 2012

Keywords

Comments

Column 4 of A212402

Examples

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

Links

Formula

Empirical: a(n) = a(n-1) +a(n-2) +a(n-4) -a(n-6) +3*a(n-7) +8*a(n-8) -8*a(n-10) -8*a(n-11) -10*a(n-12) -5*a(n-13) +8*a(n-14) -2*a(n-15) -28*a(n-16) -15*a(n-17) +25*a(n-18) +24*a(n-19) +28*a(n-20) +24*a(n-21) -19*a(n-22) -18*a(n-23) +51*a(n-24) +40*a(n-25) -55*a(n-26) -16*a(n-27) -55*a(n-28) -45*a(n-29) +51*a(n-30) +36*a(n-31) -61*a(n-32) -45*a(n-33) +70*a(n-34) -16*a(n-35) +67*a(n-36) +40*a(n-37) -70*a(n-38) -19*a(n-39) +56*a(n-40) +24*a(n-41) -58*a(n-42) +24*a(n-43) -56*a(n-44) -15*a(n-45) +56*a(n-46) -2*a(n-47) -28*a(n-48) -5*a(n-49) +28*a(n-50) -8*a(n-51) +28*a(n-52) -28*a(n-54) +3*a(n-55) +8*a(n-56) -8*a(n-58) -8*a(n-60) +a(n-61) +8*a(n-62) -a(n-64) +a(n-66) +a(n-68) -a(n-70)

A212399 Number of binary arrays of length n+9 with no more than 5 ones in any length 10 subsequence (=50% duty cycle).

Original entry on oeis.org

638, 1150, 2104, 3872, 7143, 13185, 24322, 44794, 82294, 150686, 274744, 501524, 917116, 1679070, 3076254, 5638155, 10334814, 18942534, 34712677, 63595283, 116481088, 213319856, 390670798, 715523414, 1310607210, 2400758159
Offset: 1

Views

Author

R. H. Hardin May 14 2012

Keywords

Comments

Column 5 of A212402

Examples

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

Links

A212400 Number of binary arrays of length n+11 with no more than 6 ones in any length 12 subsequence (=50% duty cycle).

Original entry on oeis.org

2510, 4558, 8402, 15586, 29002, 54042, 100736, 187696, 349360, 649232, 1203940, 2226644, 4104676, 7573112, 13992148, 25879040, 47898464, 88693997, 164277882, 304304060, 563672655, 1043992249, 1933271058, 3579337110, 6625751995
Offset: 1

Views

Author

R. H. Hardin May 14 2012

Keywords

Comments

Column 6 of A212402

Examples

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

Links

A212403 Number of binary arrays of length 2*n+1 with no more than n ones in any length 2n subsequence (=50% duty cycle).

Original entry on oeis.org

5, 19, 74, 291, 1150, 4558, 18100, 71971, 286454, 1140954, 4547020, 18129294, 72309164, 288493756, 1151300584, 4595507491, 18346672294, 73257044386, 292550538844, 1168434892186, 4667175448324, 18644235526276, 74485459541464
Offset: 1

Views

Author

R. H. Hardin, May 14 2012

Keywords

Comments

Row 2 of A212402.

Examples

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

Links

Programs

  • Mathematica
    Rest[CoefficientList[Series[1/(1-4*x)+1/(2*Sqrt[1-4*x]), {x, 0, 20}], x]] (* Vaclav Kotesovec, Oct 21 2012 *)
Table[4^n + Binomial[2*n-1, n],{n,1,20}] (* Vaclav Kotesovec, Oct 28 2012 *)

Formula

Recurrence: n*a(n) = 2*(4*n-3)*a(n-1) - 8*(2*n-3)*a(n-2). - Vaclav Kotesovec, Oct 19 2012
G.f.: 1/(1-4*x)+1/(2*sqrt(1-4*x)). - Vaclav Kotesovec, Oct 21 2012
a(n) = 4^n + C(2*n-1, n). - Vaclav Kotesovec, Oct 28 2012

A212404 Number of binary arrays of length 2*n+2 with no more than n ones in any length 2n subsequence (=50% duty cycle).

Original entry on oeis.org

8, 33, 132, 527, 2104, 8402, 33560, 134075, 535728, 2140910, 8556568, 34201078, 136713872, 546528612, 2184925808, 8735357267, 34925461088, 139642914902, 558353310488, 2232601256162, 8927375430608, 35698163696252, 142750104755408
Offset: 1

Views

Author

R. H. Hardin May 14 2012

Keywords

Comments

Row 3 of A212402

Examples

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

Links

Programs

  • Mathematica
    Flatten[{8,33,RecurrenceTable[{(n-4)*n*a[n]==2*(n-1)*(4*n-15)*a[n-1]-8*(n-3)*(2*n-5)*a[n-2],a[3]==132,a[4]==527},a,{n,3,20}]}] (* Vaclav Kotesovec, Oct 19 2012 *)
Table[2^(2*n+1)+Binomial[2*n-2,n],{n,1,20}] (* Vaclav Kotesovec, Oct 28 2012 *)

Formula

Recurrence (for n>4): (n-4)*n*a(n) = 2*(n-1)*(4*n-15)*a(n-1) - 8*(n-3)*(2*n-5)*a(n-2). - Vaclav Kotesovec, Oct 19 2012
a(n) = 2^(2*n+1) + C(2*n-2,n). - Vaclav Kotesovec, Oct 28 2012

A212405 Number of binary arrays of length 2*n+3 with no more than n ones in any length 2n subsequence (=50% duty cycle).

Original entry on oeis.org

13, 57, 236, 959, 3872, 15586, 62632, 251419, 1008536, 4043582, 16206152, 64933782, 260114976, 1041797124, 4171943056, 16704821779, 66880877896, 267747443494, 1071808583176, 4290243456514, 17172082337536, 68729504287324
Offset: 1

Views

Author

R. H. Hardin May 14 2012

Keywords

Comments

Row 4 of A212402

Examples

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

Links

Programs

  • Mathematica
    Flatten[{13,Table[2^(2*n+2)-(3*n+1)/n*Binomial[2*n-2,n-1],{n,2,20}]}] (* Vaclav Kotesovec, Oct 28 2012 *)

Formula

Recurrence (for n>3): n^2*a(n) = 2*(4*n^2-3*n-5)*a(n-1) - 8*(n+1)*(2*n-5)*a(n-2). - Vaclav Kotesovec, Oct 19 2012
a(n) = 2^(2*n+2) - (3*n+1)/n * C(2*n-2,n-1), for n>1. - Vaclav Kotesovec, Oct 28 2012

A212406 Number of binary arrays of length 2*n+4 with no more than n ones in any length 2n subsequence (=50% duty cycle).

Original entry on oeis.org

21, 97, 421, 1747, 7143, 29002, 117290, 473171, 1905675, 7665886, 30810054, 123745422, 496747206, 1993227892, 7995168852, 32060722883, 128532812627, 515187798518, 2064622548782, 8272744298618, 33143688036722, 132770436380108
Offset: 1

Views

Author

R. H. Hardin, May 14 2012

Keywords

Examples

			Some solutions for n=3:
  0  1  1  0  0  0  1  0  0  0  1  0  1  0  0  1
  1  0  0  0  1  0  1  0  1  1  1  0  1  1  1  0
  1  1  0  0  0  0  1  0  1  0  0  1  0  0  0  0
  0  0  0  1  0  0  0  1  0  1  0  0  0  0  0  1
  0  1  0  0  0  0  0  0  1  0  0  1  1  1  0  1
  0  0  1  0  0  0  0  0  0  0  0  1  0  0  1  0
  0  1  0  0  0  1  1  1  0  0  0  0  0  1  0  0
  1  0  0  0  1  0  1  1  1  1  1  0  0  0  1  1
  0  0  0  0  0  0  0  0  1  1  0  0  0  0  0  0
  1  1  1  1  1  1  1  1  0  1  1  0  1  0  0  1

Links

Crossrefs

Row 5 of A212402.

Programs

  • Maple
    #verified first terms (holds for all n<=210).
with(gfun): A212406:= rectoproc({a(2)=97, a(3)=421, n*(59*n^2-252*n+163)*a(n) = 2*(236*n^3-1185*n^2+1204*n+210)*a(n-1) - 8*(2*n-7)*(59*n^2-134*n-30)*a(n-2)},a(n),remember): 21,seq(A212406(n),n=2..20); A212406(210); # Vaclav Kotesovec, Nov 20 2012

Formula

Empirical (for n>=4): n*(59*n^2 - 252*n + 163)*a(n) = 2*(236*n^3 - 1185*n^2 + 1204*n + 210)*a(n-1) - 8*(2*n-7)*(59*n^2 - 134*n - 30)*a(n-2). - Vaclav Kotesovec, Nov 20 2012
Empirical (for n>=3): a(n) = 2^(2*n+3) - 2*(59*n^2 - 84*n - 6) * C(2*n - 5, n - 3) / (n*(n-1)). - Vaclav Kotesovec, Nov 20 2012

A212407 Number of binary arrays of length 2*n+5 with no more than n ones in any length 2n subsequence (=50% duty cycle).

Original entry on oeis.org

34, 166, 747, 3179, 13185, 54042, 220054, 892387, 3609005, 14567294, 58714842, 236397086, 950965002, 3822869204, 15359318444, 61681353571, 247609729669, 993662549686, 3986465243314, 15989373858826, 64118439206974
Offset: 1

Views

Author

R. H. Hardin, May 14 2012

Keywords

Examples

			Some solutions for n=3:
  1  1  1  0  0  0  1  1  1  0  0  0  0  0  1  1
  0  1  0  1  0  1  1  1  0  0  1  0  1  1  1  0
  1  0  1  0  0  1  0  0  0  0  0  1  1  0  0  0
  0  1  0  1  1  0  1  1  0  1  0  0  0  0  1  0
  0  0  0  1  0  0  0  0  1  0  1  0  0  1  0  0
  1  0  0  0  0  0  0  0  1  0  0  0  0  0  0  1
  1  1  0  0  1  0  0  1  0  0  0  1  1  1  0  0
  0  0  0  1  0  0  1  1  0  1  1  1  0  1  0  0
  0  0  1  0  0  0  1  0  0  1  0  0  0  0  0  1
  0  1  0  0  1  1  1  0  0  0  1  1  0  0  1  0
  1  0  1  0  0  1  0  1  1  1  1  0  0  1  1  1

Links

Crossrefs

Row 6 of A212402.

Programs

  • Maple
    #verified first terms (holds for all n<=210).
with(gfun): A212407:= rectoproc({a(3)=747, a(4)=3179, n*(181*n^2-792*n+581)*a(n) = 2*(724*n^3-3711*n^2+4112*n+210)*a(n-1) - 8*(2*n-7)*(181*n^2-430*n-30)*a(n-2)},a(n),remember): 34,166,seq(A212407(n),n=3..20); A212407(210); # Vaclav Kotesovec, Nov 20 2012

Formula

Empirical (for n>=5): n*(181*n^2 - 792*n + 581)*a(n) = 2*(724*n^3 - 3711*n^2 + 4112*n + 210)*a(n-1) - 8*(2*n-7)*(181*n^2 - 430*n - 30)*a(n-2). - Vaclav Kotesovec, Nov 20 2012
Empirical (for n>=3): a(n) = 4^(n+2) - 2*(181*n^2 - 264*n - 6) * C(2*n - 5, n - 3) / (n*(n-1)). - Vaclav Kotesovec, Nov 20 2012

A212408 Number of binary arrays of length 2*n+6 with no more than n ones in any length 2n subsequence (=50% duty cycle).

Original entry on oeis.org

55, 285, 1314, 5769, 24322, 100736, 413220, 1685039, 6844362, 27724036, 112072540, 452348578, 1823583124, 7344493104, 29556979016, 118871913787, 477820811258, 1919788147772, 7710323488748, 30956089143902, 124248950086268
Offset: 1

Views

Author

R. H. Hardin, May 14 2012

Keywords

Examples

			Some solutions for n=3:
  0  0  0  1  1  0  1  0  0  1  0  1  1  0  1  1
  0  0  1  0  0  0  0  0  1  0  1  0  0  0  0  0
  0  1  0  0  0  0  1  1  1  0  1  0  1  0  1  0
  1  0  1  1  1  0  0  1  0  0  0  1  0  1  0  1
  0  0  0  0  1  0  1  0  0  1  0  0  0  0  0  1
  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0
  0  0  0  1  1  1  0  1  1  0  0  0  1  1  0  0
  0  1  0  0  0  0  0  0  0  0  1  1  1  0  0  0
  1  0  0  0  0  1  1  1  0  0  1  0  0  0  1  0
  1  0  0  1  1  0  1  1  0  0  0  1  0  1  0  0
  0  1  0  0  0  0  0  0  1  1  0  0  1  0  1  0
  1  0  1  1  1  0  1  0  1  1  1  1  0  0  0  0

Links

Crossrefs

Row 7 of A212402.

Programs

  • Maple
    #verified first terms (holds for all n<=210).
with(gfun): A212408:= rectoproc({a(3)=1314, a(4)=5769, n*(955*n^3-8481*n^2+21998*n-14262)*a(n) = 2*(3820*n^4-36789*n^3+110342*n^2-99213*n-1890)*a(n-1) - 8*(2*n-9)*(955*n^3-5616*n^2+7901*n+210)*a(n-2)},a(n),remember): 55,285,seq(A212408(n),n=3..20); A212408(210); # Vaclav Kotesovec, Nov 20 2012

Formula

Empirical (for n>=5): n*(955*n^3 - 8481*n^2 + 21998*n - 14262)*a(n) = 2*(3820*n^4 - 36789*n^3 + 110342*n^2 - 99213*n - 1890)*a(n-1) - 8*(2*n-9)*(955*n^3 - 5616*n^2 + 7901*n + 210)*a(n-2). - Vaclav Kotesovec, Nov 20 2012
Empirical (for n>=4): a(n) = 2^(2*n+5) - 4*(955*n^3 - 3782*n^2 + 3475*n + 30) * C(2*n-7, n-4) / ((n-2)*(n-1)*n). - Vaclav Kotesovec, Nov 20 2012
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