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.

Previous Showing 11-15 of 15 results.

A188763 T(n,k)=Number of nXk binary arrays without the pattern 0 0 1 vertically or horizontally.

Original entry on oeis.org

2, 4, 4, 7, 16, 7, 12, 49, 49, 12, 20, 144, 240, 144, 20, 33, 400, 1112, 1112, 400, 33, 54, 1089, 4792, 8024, 4792, 1089, 54, 88, 2916, 20129, 53024, 53024, 20129, 2916, 88, 143, 7744, 82807, 339927, 532168, 339927, 82807, 7744, 143, 232, 20449, 337209, 2125134
Offset: 1

Views

Author

R. H. Hardin Apr 09 2011

Keywords

Comments

Table starts
...2.....4.......7........12..........20............33..............54
...4....16......49.......144.........400..........1089............2916
...7....49.....240......1112........4792.........20129...........82807
..12...144....1112......8024.......53024........339927.........2125134
..20...400....4792.....53024......532168.......5163989........48759352
..33..1089...20129....339927.....5163989......75643222......1076278662
..54..2916...82807...2125134....48759352....1076278662.....23051042448
..88..7744..337209..13128024...454436200...15104206828....486675087796
.143.20449.1363568..80418708..4196906454..209938880964..10173867472000
.232.53824.5492088.490332106.38563523452.2902462402332.211512633571598

Examples

			Some solutions for 5X3
..0..1..1....0..0..0....0..1..1....0..1..1....1..0..1....0..1..0....0..1..0
..0..0..0....1..1..1....1..1..0....1..1..1....0..1..1....1..1..1....1..0..1
..0..1..1....0..1..0....1..1..1....0..1..1....1..1..1....0..0..0....1..1..1
..0..1..0....1..1..1....0..0..0....0..0..0....0..1..1....0..0..0....1..1..0
..0..0..0....1..0..1....0..1..1....0..0..0....1..0..0....0..0..0....0..1..1

Links

Crossrefs

Column 1 is A000071(n+3)
Column 2 is A188516

A206785 T(n,k) = Number of n X k 0..1 arrays avoiding 0 0 1 horizontally and 0 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 7, 16, 7, 12, 49, 49, 12, 20, 144, 230, 144, 20, 33, 400, 1020, 1020, 400, 33, 54, 1089, 4120, 6752, 4120, 1089, 54, 88, 2916, 16109, 39276, 39276, 16109, 2916, 88, 143, 7744, 61003, 219061, 316744, 219061, 61003, 7744, 143, 232, 20449, 227197, 1165366
Offset: 1

Views

Author

R. H. Hardin Feb 12 2012

Keywords

Comments

Table starts
..2....4......7......12........20.........33..........54...........88
..4...16.....49.....144.......400.......1089........2916.........7744
..7...49....230....1020......4120......16109.......61003.......227197
.12..144...1020....6752.....39276.....219061.....1165366......6052170
.20..400...4120...39276....316744....2431277....17515482....122505076
.33.1089..16109..219061...2431277...25571618...248652016...2339185464
.54.2916..61003.1165366..17515482..248652016..3210775788..40009922008
.88.7744.227197.6052170.122505076.2339185464.40009922008.659966594476

Examples

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

Links

Crossrefs

Column 1 is A000071(n+3).
Column 2 is A188516.

A202451 Upper triangular Fibonacci matrix, by SW antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 1, 3, 0, 0, 1, 2, 5, 0, 0, 0, 1, 3, 8, 0, 0, 0, 1, 2, 5, 13, 0, 0, 0, 0, 1, 3, 8, 21, 0, 0, 0, 0, 1, 2, 5, 13, 34, 0, 0, 0, 0, 0, 1, 3, 8, 21, 55, 0, 0, 0, 0, 0, 1, 2, 5, 13, 34, 89, 0, 0, 0, 0, 0, 0, 1, 3, 8, 21, 55, 144
Offset: 1

Views

Author

Clark Kimberling, Dec 19 2011

Keywords

Examples

			Northwest corner:
1...1...2...3...5...8...13...21...34
0...1...1...2...3...5....8...13...21
0...0...1...1...2...3....5....8...13
0...0...0...1...1...2....3....5....8

Links

Crossrefs

Cf. A000045, A188516, A202452, A202453, A202462.

Programs

  • Mathematica
    n = 12;
Q = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[Fibonacci[k], {k, 1, n}]];
P = Transpose[Q]; F = P.Q;
Flatten[Table[P[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202451 as a sequence *)
Flatten[Table[Q[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202452 as a sequence *)
Flatten[Table[F[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202453 as a sequence *)
TableForm[Q]  (* A202451, upper triangular Fibonacci matrix *)
TableForm[P]  (* A202452, lower triangular Fibonacci matrix *)
TableForm[F]  (* A202453, Fibonacci self-fusion matrix *)
TableForm[FactorInteger[F]]

Formula

Row n consists of n-1 zeros followed by the Fibonacci sequence (1, 1, 2, 3, 5, 8, ...).

A202462 a(n) = Sum_{j=1..n} Sum_{i=1..n} F(i,j), where F is the Fibonacci fusion array of A202453.

Original entry on oeis.org

1, 5, 21, 70, 214, 614, 1703, 4619, 12363, 32812, 86636, 228012, 598893, 1571089, 4118305, 10790194, 28262594, 74014290, 193807315, 507451415, 1328617751, 3478516440, 9107117016, 23843134680, 62422772569, 163425968669, 427856404653
Offset: 1

Views

Author

Clark Kimberling, Dec 19 2011

Keywords

Comments

Partial sums of A188516.

Links

Crossrefs

Cf. A000045, A188616, A202451.

Programs

  • GAP
    F:=Fibonacci;; List([1..30], n-> F(n+2)*F(n+3) -2*F(n+4) +n+4); # G. C. Greubel, Jul 23 2019
  • Magma
    F:=Fibonacci; [F(n+2)*F(n+3) -2*F(n+4) +n+4: n in [1..30]]; // G. C. Greubel, Jul 23 2019
  • Mathematica
    (* First program *)
n = 28;
Q = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[
   Table[Fibonacci[k], {k, 1, n}]];
P = Transpose[Q]; F = P.Q;
a[m_] := Sum[F[[i]][[j]], {i, 1, m}, {j, 1, m}]
Table[a[m], {m, 1, n}]  (* A202462 *)
Table[a[m] - a[m - 1], {m, 1, n}]  (* A188516 *)
(* Additional programs *)
LinearRecurrence[{5,-6,-4,10,-2,-3,1},{1,5,21,70,214,614,1703},30] (* Harvey P. Dale, Jul 23 2015 *)
With[{F=Fibonacci}, Table[F[n+2]*F[n+3] -2*F[n+4] +n+4, {n,30}]] (* G. C. Greubel, Jul 23 2019 *)
  • PARI
    vector(30, n, f=fibonacci; f(n+2)*f(n+3) -2*f(n+4) +n+4) \\ G. C. Greubel, Jul 23 2019
  • Sage
    f=fibonacci; [f(n+2)*f(n+3)-2*f(n+4) +n+4 for n in (1..30)] # G. C. Greubel, Jul 23 2019

Formula

G.f.: x*(1+2*x^2-x^3)/((1+x)*(1-3*x+x^2)*(1-x-x^2)*(1-x)^2). - R. J. Mathar, Dec 20 2011
a(n) = Fibonacci(n+2)*Fibonacci(n+3) - 2*Fibonacci(n+4) + n + 4. - G. C. Greubel, Jul 23 2019

A202452 Lower triangular Fibonacci matrix, by SW antidiagonals.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 3, 1, 0, 0, 5, 2, 1, 0, 0, 8, 3, 1, 0, 0, 0, 13, 5, 2, 1, 0, 0, 0, 21, 8, 3, 1, 0, 0, 0, 0, 34, 13, 5, 2, 1, 0, 0, 0, 0, 55, 21, 8, 3, 1, 0, 0, 0, 0, 0, 89, 34, 13, 5, 2, 1, 0, 0, 0, 0, 0, 144, 55, 21, 8, 3, 1, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Clark Kimberling, Dec 19 2011

Keywords

Examples

			Northwest corner:
1...0...0...0...0...0...0...0...0
1...1...0...0...0...0...0...0...0
2...1...1...0...0...0...0...0...0
3...2...1...1...0...0...0...0...0
5...3...2...1...1...0...0...0...0

Links

Crossrefs

Cf. A202451, A202453, A202462, A188516, A000045.

Programs

  • Mathematica
    n = 12;
Q = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[Fibonacci[k], {k, 1, n}]];
P = Transpose[Q]; F = P.Q;
Flatten[Table[P[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202451 as a sequence *)
Flatten[Table[Q[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202452 as a sequence *)
Flatten[Table[F[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202453 as a sequence *)
TableForm[Q]  (* A202451, upper triangular Fibonacci array *)
TableForm[P]  (* A202452, lower triangular Fibonacci array *)
TableForm[F]  (* A202453, Fibonacci self-fusion matrix *)
TableForm[FactorInteger[F]]

Formula

Column n consists of n-1 zeros followed by the Fibonacci sequence (1,1,2,3,5,8,...).
Previous Showing 11-15 of 15 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.