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 1-4 of 4 results.

A210342 Row sums of triangle A210341.

Original entry on oeis.org

1, 2, 4, 9, 24, 78, 317, 1636, 10752, 89871, 955536, 12930172, 222618065, 4874855542, 135781292308, 4811103270053, 216847500834512, 12432143862756778, 906625645142897789, 84102571511631809864, 9923979699312024569440, 1489546408205976627946331
Offset: 0

Views

Author

Emanuele Munarini, Mar 20 2012

Keywords

Crossrefs

Cf. A210341.

Programs

  • Magma
    [&+[Fibonacci(n-k+2)^k: k in [0..n]]: n in [0..21]]; // Bruno Berselli, Mar 28 2012
  • Mathematica
    Table[Sum[Fibonacci[n - k + 2]^k, {k, 0, n}], {n, 0, 100}]
  • Maxima
    makelist(sum(fib(n-k+2)^k,k,0,n),n,0,12);
    

Formula

G.f.: Sum_{k>=0} x^k/(1-Fibonacci(k+2)*x).
a(n) = [x^n] Sum_{k=0..n} x^k/(1-Fibonacci(k+2)*x).
a(n) = A135961(n+2)-1. - Vaclav Kotesovec, Jan 05 2013

A067966 Number of binary arrangements without adjacent 1's on n X n array connected n-s.

Original entry on oeis.org

1, 2, 9, 125, 4096, 371293, 85766121, 52523350144, 83733937890625, 350356403707485209, 3833759992447475122176, 109879109551310452512114617, 8243206936713178643875538610721, 1619152874321527556575810000000000000
Offset: 0

Views

Author

R. H. Hardin, Feb 02 2002

Keywords

Comments

Central coefficients of triangle A210341.

Examples

			Neighbors for n=4:
o o o o
| | | |
| | | |
o o o o
| | | |
| | | |
o o o o
| | | |
| | | |
o o o o
		

Crossrefs

Cf. circle A000204, line A000045, arrays: ne-sw nw-se A067965, e-w ne-sw nw-se A067963, n-s nw-se A067964, e-w n-s nw-se A066864, e-w ne-sw n-s nw-se A063443, e-w n-s A006506, nw-se A067962, toruses: bare A002416, ne-sw nw-se A067960, ne-sw n-s nw-se A067959, e-w ne-sw n-s nw-se A067958, n-s A067961, e-w n-s A027683, e-w ne-sw n-s A066866.

Programs

  • Magma
    [Fibonacci(n+2)^n: n in [0..13]]; // Bruno Berselli, Mar 28 2012
  • Mathematica
    Table[Fibonacci[n+2]^n, {n, 0, 100}]
  • Maxima
    makelist(fib(n+2)^n, n, 0, 14);
    
  • PARI
    a(n)=fibonacci(n+2)^n \\ Charles R Greathouse IV, Mar 28 2012
    

Formula

a(n) = F(n+2)^n, where F(n) = A000045(n) is the n-th Fibonacci number.
a(n) ~ phi^2/sqrt(5) phi^n^2. [Charles R Greathouse IV, Mar 28 2012]

Extensions

Edited by Dean Hickerson, Feb 15 2002

A210343 a(n) = Fibonacci(n+1)^n.

Original entry on oeis.org

1, 1, 4, 27, 625, 32768, 4826809, 1801088541, 1785793904896, 4605366583984375, 31181719929966183601, 552061438912436417593344, 25601832525455335435322705761, 3107689015140868348741078056241817, 987683253336131809511244100000000000000
Offset: 0

Views

Author

Emanuele Munarini, Mar 20 2012

Keywords

Crossrefs

Programs

  • Magma
    [Fibonacci(n+1)^n: n in [0..14]]; // Bruno Berselli, Mar 28 2012
  • Maple
    a:= n-> (<<1|1>, <1|0>>^n)[1,1]^n:
    seq(a(n), n=0..15);  # Alois P. Heinz, Dec 05 2015
  • Mathematica
    Table[Fibonacci[n+1]^n,{n,0,100}]
  • Maxima
    makelist(fib(n+1)^n,n,0,14);
    

A210574 Lower triangular matrix in the LU-decomposition of the Vandermonde determinants generated by Fibonacci numbers.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 12, 1, 1, 15, 50, 264, 1, 1, 31, 180, 1920, 11970, 1, 1, 63, 602, 11760, 146160, 1689600, 1, 1, 127, 1932, 66024, 1477980, 34214400, 603233280, 1, 1, 255, 6050, 353304, 13556970, 568656000, 20043279360, 586244602944, 1
Offset: 0

Views

Author

Emanuele Munarini, Mar 22 2012

Keywords

Comments

If the Vandermonde matrix V = [F(i+2)^j]_(i,j=0)^n has LU-decomposition, then this triangle is given by L.

Examples

			The triangle begins:
1
1, 1
1, 3,   1
1, 7,   12,   1
1, 15,  50,   264,    1
1, 31,  180,  1920,   11970,    1
1, 63,  602,  11760,  146160,   1689600,   1
1, 127, 1932, 66024,  1477980,  34214400,  603233280,   1
1, 255, 6050, 353304, 13556970, 568656000, 20043279360, 586244602944, 1
		

Crossrefs

Programs

  • Mathematica
    n = 10; f = Fibonacci[Range[2, n + 1]]; m = Outer[ Power, f, Range[0, n - 1]]; mi = Transpose[LUDecomposition[m][[1]]]; Flatten[Table[Append[Take[mi[[i]], i - 1], 1], {i, n}]] (* T. D. Noe, Mar 22 2012 *)
Showing 1-4 of 4 results.