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 11 results. Next

A206450 Number of 0..3 arrays of length n avoiding the consecutive pattern 0..3.

Original entry on oeis.org

4, 16, 64, 255, 1016, 4048, 16128, 64257, 256012, 1020000, 4063872, 16191231, 64508912, 257015648, 1023998720, 4079803649, 16254705684, 64761807088, 258023229632, 1028013114879, 4095797753832, 16318429208240, 65015693603328
Offset: 1

Views

Author

R. H. Hardin, Feb 07 2012

Keywords

Comments

Column 3 of A206455.

Crossrefs

Cf. A206455.

Formula

a(n) = 4*a(n-1) -a(n-4).
Empirical: a(n) = sum{i in 0..floor(n/4)} ((-1)^i*4^(n-4*i)*binomial(n-3*i,i)).
g.f.: x*(4 - x^3) / (1 - 4*x + x^4). - Colin Barker, Feb 23 2018

A206451 Number of 0..4 arrays of length n avoiding the consecutive pattern 0..4.

Original entry on oeis.org

5, 25, 125, 625, 3124, 15615, 78050, 390125, 1950000, 9746876, 48718765, 243515775, 1217188750, 6083993750, 30410221874, 152002390605, 759768437250, 3797624997500, 18982040993750, 94879794746876, 474246971343775
Offset: 1

Views

Author

R. H. Hardin, Feb 07 2012

Keywords

Comments

Column 4 of A206455

Programs

  • Maple
    M:= <<4|3|3|3|3>,<1|1|1|1|1>,<0|1|0|0|0>,<0|0|1|0|0>,<0|0|0|1|0>>:
    seq(<1|1|1|1|1> . M^n . <1,0,0,0,0>, n=1..30); # Robert Israel, Jan 08 2016

Formula

a(n) = 5*a(n-1) -a(n-5)
Empirical: a(n) = sum{i in 0..floor(n/5)} ((-1)^i*5^(n-5*i)*binomial(n-4*i,i))
From Robert Israel, Jan 08 2016: (Start) The recursion can be proved using the matrix representation
a(n) = [ 1 1 1 1 1] M^n [ 1 0 0 0 0 ]^T, where
M = [ 4 3 3 3 3 ]
[ 1 1 1 1 1 ]
[ 0 1 0 0 0 ]
[ 0 0 1 0 0 ]
[ 0 0 0 1 0 ]
which satisfies M^5 = 5 M^4 - I.
G.f.: -x*(-5+x^4) / ( 1-5*x+x^5 ).. (End)

A206452 Number of 0..5 arrays of length n avoiding the consecutive pattern 0..5.

Original entry on oeis.org

6, 36, 216, 1296, 7776, 46655, 279924, 1679508, 10076832, 60459696, 362750400, 2176455745, 13058454546, 78349047768, 470084209776, 2820444798960, 16922306043360, 101531659804415, 609176900371944, 3654983053183896
Offset: 1

Views

Author

R. H. Hardin, Feb 07 2012

Keywords

Comments

Column 5 of A206455.

Crossrefs

Cf. A206455.

Formula

a(n) = 6*a(n-1) - a(n-6).
Empirical: a(n) = Sum_{i=0..floor(n/6)} ((-1)^i*6^(n-6*i)*binomial(n-5*i,i)).
g.f.: x*(6 - x^5) / (1 - 6*x + x^6). - Colin Barker, Jun 16 2018

A206453 Number of 0..6 arrays of length n avoiding the consecutive pattern 0..6.

Original entry on oeis.org

7, 49, 343, 2401, 16807, 117649, 823542, 5764787, 40353460, 282473877, 1977314738, 13841186359, 96888186864, 678216484506, 4747509626755, 33232527033825, 232627406762898, 1628389870025548, 11398715248992477, 79790909854760475
Offset: 1

Views

Author

R. H. Hardin, Feb 07 2012

Keywords

Comments

Column 6 of A206455.

Crossrefs

Cf. A206455.

Formula

a(n) = 7*a(n-1) - a(n-7).
Empirical: a(n) = Sum_{i=0..floor(n/7)} ((-1)^i*7^(n-7*i)*binomial(n-6*i,i)).
g.f.: x*(7 - x^6) / (1 - 7*x + x^7). - Colin Barker, Jun 16 2018

A206454 Number of 0..7 arrays of length n avoiding the consecutive pattern 0..7.

Original entry on oeis.org

8, 64, 512, 4096, 32768, 262144, 2097152, 16777215, 134217712, 1073741632, 8589932544, 68719456256, 549755617280, 4398044676096, 35184355311616, 281474825715713, 2251798471507992, 18014386698322304, 144115084996645888, 1152920611253710848, 9223364340274069504
Offset: 1

Views

Author

R. H. Hardin, Feb 07 2012

Keywords

Comments

Column 7 of A206455.

Crossrefs

Cf. A206455.

Programs

  • Mathematica
    Table[Sum[(-1)^i*8^(n-8*i)*Binomial[n-7*i,i],{i,0,Floor[n/8]}],{n,20}] (* Stefano Spezia, Feb 12 2023 *)
    LinearRecurrence[{8,0,0,0,0,0,0,-1},{8,64,512,4096,32768,262144,2097152,16777215},30] (* Harvey P. Dale, Jan 10 2025 *)

Formula

a(n) = 8*a(n-1) - a(n-8).
a(n) = Sum_{i=0..floor(n/8)} (-1)^i*8^(n-8*i)*binomial(n-7*i,i).
G.f.: x*(8 - x^7) / (1 - 8*x + x^8). - Colin Barker, Jun 16 2018

A206456 Number of 0..n arrays of length n+2 avoiding the consecutive pattern 0..n.

Original entry on oeis.org

4, 75, 1016, 15615, 279924, 5764787, 134217712, 3486784383, 99999999980, 3138428376699, 106993205379048, 3937376385699263, 155568095557812196, 6568408355712890595, 295147905179352825824, 14063084452067724990975, 708235345355337676357596, 37589973457545958193355563
Offset: 1

Views

Author

R. H. Hardin, Feb 07 2012

Keywords

Crossrefs

Subdiagonal 2 of A206455.

Formula

Empirical: a(n) = Sum_{i=0..floor((n+2)/(n+1))} ((-1)^i*(n+1)^((n+2) -(n+1)*i)*binomial((n+2) -n*i,i)).

A206457 Number of 0..n arrays of length n+3 avoiding the consecutive pattern 0..n.

Original entry on oeis.org

5, 216, 4048, 78050, 1679508, 40353460, 1073741632, 31381059366, 999999999700, 34522712143568, 1283918464548432, 51185893014090250, 2177953337809370548, 98526125335693358700, 4722366482869645212928, 239072435685151324846286
Offset: 1

Views

Author

R. H. Hardin, Feb 07 2012

Keywords

Comments

Subdiagonal 3 of A206455.

Crossrefs

Cf. A206455.

Formula

Empirical: a(n) = Sum_{i=0..floor((n+3)/(n+1))} ((-1)^i*(n+1)^((n+3) -(n+1)*i)*binomial((n+3)-n*i,i)).

A206458 Number of 0..n arrays of length n+4 avoiding the consecutive pattern 0..n.

Original entry on oeis.org

6, 622, 16128, 390125, 10076832, 282473877, 8589932544, 282429533565, 9999999996000, 379749833577917, 15407021574579456, 665416609183171053, 30491346729331184928, 1477891880035400377125, 75557863725914323402752
Offset: 1

Views

Author

R. H. Hardin Feb 07 2012

Keywords

Comments

Subdiagonal 4 of A206455

Formula

Empirical: a(n) = sum{i in 0..floor((n+4)/(n+1))} ((-1)^i*(n+1)^((n+4) -(n+1)*i)*binomial((n+4) -n*i,i))

A206459 Number of 0..n arrays of length n+5 avoiding the consecutive pattern 0..n.

Original entry on oeis.org

7, 1791, 64257, 1950000, 60459696, 1977314738, 68719456256, 2541865795524, 99999999950000, 4177248169342446, 184884258894932736, 8650415919381195128, 426878854210636550576, 22168378200531005606250, 1208925819614629174378496
Offset: 1

Views

Author

R. H. Hardin Feb 07 2012

Keywords

Comments

Subdiagonal 5 of A206455

Formula

Empirical: a(n) = sum{i in 0..floor((n+5)/(n+1))} ((-1)^i*(n+1)^((n+5) -(n+1)*i)*binomial((n+5) -n*i,i))

A206460 Number of 0..n arrays of length n+6 avoiding the consecutive pattern 0..n.

Original entry on oeis.org

8, 5157, 256012, 9746876, 362750400, 13841186359, 549755617280, 22876792100667, 999999999400000, 45949729862605855, 2218611106738944000, 112455406951955165371, 5976303958948911170240, 332525673007965083334375
Offset: 1

Views

Author

R. H. Hardin, Feb 07 2012

Keywords

Comments

Subdiagonal 6 of A206455.

Crossrefs

Cf. A206455.

Formula

Empirical: a(n) = Sum_{i=0..floor((n+6)/(n+1))} ((-1)^i*(n+1)^((n+6) -(n+1)*i)*binomial((n+6) -n*i,i)).
Showing 1-10 of 11 results. Next