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 1-3 of 3 results.

A099464 Trisection of tribonacci numbers.

Original entry on oeis.org

0, 1, 7, 44, 274, 1705, 10609, 66012, 410744, 2555757, 15902591, 98950096, 615693474, 3831006429, 23837527729, 148323355432, 922906855808, 5742568741225, 35731770264967, 222332455004452, 1383410902447554, 8607945812375585, 53560898629395777, 333269972246340068
Offset: 0

Views

Author

Paul Barry, Oct 16 2004

Keywords

Comments

Let A = [1,1,1; 2,4,3; 1,2,2]. a(n) is given by the (1,2) term in A^n.

Links

Crossrefs

Cf. A009943, A120987.

Programs

  • Maple
    a:= n-> (<<0|1|0>, <0|0|1>, <1|-5|7>>^n)[3, 1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Dec 11 2015
  • Mathematica
    LinearRecurrence[{7,-5,1},{0,1,7},30] (* Harvey P. Dale, Jan 14 2016 *)

Formula

G.f.: x/(1-7*x+5*x^2-x^3).
a(n) = 7a(n-1) -5a(n-2) +a(n-3).
a(n) = A000073(3n).
a(n) = Sum_{i>=n-1} A120987(i,n-1) for n>0. - Alois P. Heinz, Dec 11 2015

A157241 Expansion of x / ((1-x)*(4*x^2-2*x+1)).

Original entry on oeis.org

0, 1, 3, 3, -5, -21, -21, 43, 171, 171, -341, -1365, -1365, 2731, 10923, 10923, -21845, -87381, -87381, 174763, 699051, 699051, -1398101, -5592405, -5592405, 11184811, 44739243, 44739243, -89478485, -357913941
Offset: 0

Views

Author

Creighton Dement, Feb 25 2009

Keywords

Comments

Generating floretion is Y = .5('i + 'j + 'k + i' + j' + k') + ee. ("ibasek"). This is the same floretion which generates A157240.

Links

Crossrefs

Cf. A128018, A157240.

Programs

  • Mathematica
    CoefficientList[Series[x/((1-x)(4x^2-2x+1)),{x,0,40}],x] (* or *) LinearRecurrence[{3,-6,4},{0,1,3},40] (* Harvey P. Dale, Oct 27 2013 *)
Table[1/9 (3 + (-1)^Floor[1/3 (-2 + n)] 2^(4 + 3 Floor[1/3 (-2 + n)]) + (-1)^Floor[1/3 (-1 + n)] 2^(3 + 3 Floor[1/3 (-1 + n)])), {n, 0, 500}] (* John M. Campbell, Dec 23 2016 *)
  • PARI
    concat(0, Vec(x / ((1 - x)*(1 - 2*x + 4*x^2)) + O(x^40))) \\ Colin Barker, May 22 2019

Formula

a(n+1) - a(n) = A088138(n+1).
a(n+1) = Sum_{k=0..n} A120987(n,k)*(-1)^(n-k). - Philippe Deléham, Oct 25 2011
G.f.: 2*x-2*x/(G(0) + 1) where G(k)= 1 + 2*(2*k+3)*x/(2*k+1 - 2*x*(k+2)*(2*k+1)/(2*x*(k+2) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 23 2012
a(n) = 1/9*(3 + (-1)^floor((n-2)/3)*2^(4+3*floor((n-2)/3)) + (-1)^floor((n-1)/3)*2^(3+3*floor((n-1)/3))). - John M. Campbell, Dec 23 2016
From Colin Barker, May 22 2019: (Start)
a(n) = (2 - (1+i*sqrt(3))^(1+n) + i*(1-i*sqrt(3))^n*(i+sqrt(3))) / 6 where i=sqrt(-1).
a(n) = 3*a(n-1) - 6*a(n-2) + 4*a(n-3) for n>2.
(End)

A265644 Triangle read by rows: T(n,m) is the number of quaternary words of length n with m strictly increasing runs (0 <= m <= n).

Original entry on oeis.org

1, 0, 4, 0, 6, 10, 0, 4, 40, 20, 0, 1, 65, 155, 35, 0, 0, 56, 456, 456, 56, 0, 0, 28, 728, 2128, 1128, 84, 0, 0, 8, 728, 5328, 7728, 2472, 120, 0, 0, 1, 486, 8451, 27876, 23607, 4950, 165
Offset: 0

Views

Author

Giuliano Cabrele, Dec 13 2015

Keywords

Comments

In the following description the alphabet {0..r} is taken as a basis, with r = 3 in this case.
For example, the quaternary word 2|03|123|3 of length n=7, has m=4 strictly increasing runs.
The empty word has n = 0 and m = 0, and T(0, 0) = 1.
T(n, 0) = 0 for n >= 1.
T(n, m) <> 0 for m <= n <= m*(r+1). T(m*(r+1), m) = 1.
T(n,m) is a partition, based on m, of all the words of length n, so Sum_{k=0..n} T(n,k) = (r+1)^n.

Examples

			Triangle starts:
1;
0, 4;
0, 6, 10;
0, 4, 40,  20;
0, 1, 65, 155,  35;
0, 0, 56, 456, 456, 56;
.
T(3,2) = 40, which accounts for the following words:
[0 <= a <= 0, 1 |    0 <= b <= 1]  =   2
[0 <= a <= 1, 2 |    0 <= b <= 2]  =   6
[0 <= a <= 2, 3 |    0 <= b <= 3]  =  12
[0 <= a <= 3    | 0, 1 <= b <= 3]  =  12
[1 <= a <= 3    | 1, 2 <= b <= 3]  =   6
[2 <= a <= 3    | 2, 3 <= b <= 3]  =   2

References

  • R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, p. 24, p. 154.

Links

Crossrefs

Cf. A119900 (r=1, binary words), A120987 (r=2, ternary words), A008287 (quadrinomial coefficients).
Row sums give A000302.
Cf. A000292.

Programs

  • MuPAD
    T:=(n,m)->_plus((-1)^(m-j)*binomial(n+1, m-j)*binomial(4*j, n)$j=0..m):
  • PARI
    T(n, k) = sum(j=0, k, (-1)^(k-j)*binomial(n+1, k-j)*binomial(4*j, n));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print()); \\ Michel Marcus, Feb 09 2016

Formula

Refer to comment to A120987 concerning formulas for general values of r and considerations.
Therefrom we get
T(n, m) = Qsc(3, n, m) =
Nb(4*m-n, 3, n+1) = Nb(4*(n-m)+3, 3, n+1) =
Sum_{j=0..n+1} (-1)^j*Cb(n+1, j)*Cb(4*(m-j), 4*(m-j)-n) =
Sum_{j=0..m} (-1)^(m-j)*Cb(n+1, m-j)*Cb(4*j, n) =
(in this last version Cb(n,m) can be replaced by binomial(n,m))
Sum_{j=0..m} (-1)^(m-j)*binomial(n+1, m-j)*binomial(4*j, n) = [z^n, t^m](1-t)/(1-t(1+(1-t)z)^4) where [x^n]F(x) denotes the coefficient of x^n in the formal power series expansion of F(x),
Nb(s,r,n) denotes the (r+1)-nomial coefficient [x^s](1+x+..+x^r)^n,(Nb(s,3,n) = A008287(n,s)).
Cb(x,m) denotes the binomial coefficient in its extended falling factorial notation (Cb(x,m)= x^_m/m! iff m is a nonnegative integer, 0 otherwise), as defined in the Graham et al. reference.
The diagonal T(n, n) = Nb(3, 3, n+1) = Sum_{j=0..n} (-1)^(n-j)*Cb(n+1, n-j)*Cb(4*j, n) = Cb(n+3, 3) = binomial(n+3, 3) = A000292(n+1).
Showing 1-3 of 3 results.

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