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.

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A090888 Matrix defined by a(n,k) = 3^n*Fibonacci(k) - 2^n*Fibonacci(k-2), read by antidiagonals.

Original entry on oeis.org

1, 2, 0, 4, 1, 1, 8, 5, 3, 1, 16, 19, 9, 4, 2, 32, 65, 27, 14, 7, 3, 64, 211, 81, 46, 23, 11, 5, 128, 665, 243, 146, 73, 37, 18, 8, 256, 2059, 729, 454, 227, 119, 60, 29, 13, 512, 6305, 2187, 1394, 697, 373, 192, 97, 47, 21, 1024, 19171, 6561, 4246, 2123, 1151, 600, 311
Offset: 0

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Author

Ross La Haye, Feb 12 2004; revised Sep 24 2004, Sep 10 2005

Keywords

Comments

a(0,k) = A000045(k-1); a(1,k) = A000032(k); a(2,k) = A000285(k+1).
a(n,1) = a(n-1,1) + a(n-1,3) for n > 0; a(n,1) = A001047(n) = 2^(2n) - A083324(n); a(n,2) = A000244(n) = 2^(2n) - A005061(n); a(n,3) = 2a(n-1,4) for n > 0; a(n,3) = A027649(n); a(n,4) = A083313(n+1); a(n,5) = A084171(n+1).
Sum[a(n-k,k), {k,0,n}] = A098703(n+1), antidiagonal sums.
Let R, S and T be binary relations on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xRy if x is a subset of y or y is a subset of x, xSy if x is a subset of y and xTy if x is a proper subset of y. Then a(n,3) = |R|, a(n,2) = |S| and a(n,1) = |T|. Note that a binary relation W on P(A) can be defined also such that for every element x, y of P(A) xWy if x is a proper subset of y and there are no z in P(A) such that x is a proper subset of z and z is a proper subset of y. A090802(n,1) = |W|. Also, a(n,0) = |P(A)|.

Examples

			   1    0    1    1    2    3    5    8    13    21    34
   2    1    3    4    7   11   18   29    47    76   123
   4    5    9   14   23   37   60   97   157   254   411
   8   19   27   46   73  119  192  311   503   814  1317
  16   65   81  146  227  373  600  973  1573  2546  4119
  32  211  243  454  697 1151 1848 2999  4847  7846 12693
  64  665  729 1394 2123 3517 5640 9157 14797 23954 38751
a(5,3) = 454 because Fibonacci(3) = 2, Fibonacci(1) = 1 and (2 * 3^5) - (1 * 2^5) = 454.

Links

Programs

  • Mathematica
    Table[3^(n - k) Fibonacci@ k - 2^(n - k) Fibonacci[k - 2], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Nov 28 2015 *)

Formula

a(n, k) = 3^n*Fibonacci(k) - 2^n*Fibonacci(k-2).
a(n, 0) = 2^n, a(n, 1) = 3^n - 2^n, a(n, k) = a(n, k-1) + a(n, k-2) for k > 1.
a(0, k) = Fibonacci(k-1), a(1, k) = Lucas(k), a(n, k) = 5a(n-1, k) - 6a(n-2, k) for n > 1.
O.g.f. (by rows) = (-2^n + (2^(n+1) - 3^n)x)/(-1+x+x^2). - Ross La Haye, Mar 30 2006
a(n,1) - a(n,0) = A003063(n+1). - Ross La Haye, Jun 22 2007
Binomial transform (by columns) of A118654. - Ross La Haye, Jun 22 2007

Extensions

More terms from Ray Chandler, Oct 27 2004

A250167 T(n,k)=Number of length n+1 0..k arrays with the sum of adjacent differences multiplied by some arrangement of +-1 equal to zero.

Original entry on oeis.org

2, 3, 4, 4, 11, 8, 5, 20, 37, 16, 6, 33, 96, 119, 32, 7, 48, 211, 436, 373, 64, 8, 67, 380, 1269, 1880, 1151, 128, 9, 88, 639, 2860, 7109, 7836, 3517, 256, 10, 113, 976, 5831, 19896, 37881, 32032, 10679, 512, 11, 140, 1437, 10460, 49037, 129648, 195927
Offset: 1

Views

Author

R. H. Hardin, Nov 13 2014

Keywords

Comments

Table starts
....2.....3.......4........5.........6..........7..........8...........9
....4....11......20.......33........48.........67.........88.........113
....8....37......96......211.......380........639........976........1437
...16...119.....436.....1269......2860.......5831......10460.......17765
...32...373....1880.....7109.....19896......49037.....103556......203615
...64..1151....7836....37881....129648.....380939.....938128.....2121089
..128..3517...32032...195927....810964....2810751....7989940....20567199
..256.10679..129572...996933...4962056...20169871...65768448...191480917
..512.32293..521256..5029417..30034672..142786013..532548628..1748028901
.1024.97391.2091052.25262121.180893724.1004527983.4281269376.15822382297

Examples

			Some solutions for n=5 k=4
..3....0....3....4....0....3....4....4....2....4....4....2....0....4....3....1
..2....0....4....2....0....4....1....4....4....1....3....1....1....2....1....1
..4....4....0....4....4....2....2....2....4....3....4....3....1....2....3....3
..0....2....0....1....2....1....3....2....1....0....3....2....3....1....0....4
..1....2....4....1....3....1....3....3....3....0....0....2....0....4....3....3
..1....0....3....2....0....1....2....4....0....4....0....2....0....2....3....1

Links

Crossrefs

Column 1 is A000079
Column 2 is A084171
Row 2 is A212959

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 5*a(n-1) -6*a(n-2)
k=3: a(n) = 8*a(n-1) -21*a(n-2) +22*a(n-3) -8*a(n-4)
k=4: [order 8]
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4); also a quadratic polynomial plus a constant quasipolynomial with period 2
n=3: a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +a(n-4) +2*a(n-5) -a(n-6); also a cubic polynomial plus a linear quasipolynomial with period 2
n=4: [order 12; also a quartic polynomial plus a quadratic quasipolynomial with period 12]
n=5: [order 24; also a polynomial of degree 5 plus a cubic quasipolynomialwith period 60]
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