login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A204201 Triangle based on (0,1/3,1) averaging array. 7
1, 1, 4, 1, 5, 10, 1, 6, 15, 22, 1, 7, 21, 37, 46, 1, 8, 28, 58, 83, 94, 1, 9, 36, 86, 141, 177, 190, 1, 10, 45, 122, 227, 318, 367, 382, 1, 11, 55, 167, 349, 545, 685, 749, 766, 1, 12, 66, 222, 516, 894, 1230, 1434, 1515, 1534, 1, 13, 78, 288, 738, 1410 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

For a<r<b, let t(1,1)=r, and for n>1, let

t(n,1)=[a+t(n-1,1)]/2,

t(n,n)=[b+t(n-1,n-1)]/2,

t(n,k)=[t(n-1,k-1)+t(n-1,k)]/2 for 2<=k<=n-1.

We call (t(n,k)) the (a,r,b) averaging array.  If a and b

are integers and r is a rational number, then multiplying

row n of (t(n,k)) by the LCM of its denominators yields a

triangle of integers; A204201 arises in this manner from

(a,r,b)=(0,1/3,1).

...

Guide to related arrays:

(a,r,b).........triangle

(0,1/2,1).......A054143

(0,1/3,1).......A204201

(0,2/3,1).......A204202

(0,1/4,1).......A204203

(0,3/4,1).......A204204

(0,1/5,1).......A204205

(1,3/2,2).......A204206

(1,2,3).........A204207

LINKS

Table of n, a(n) for n=1..61.

FORMULA

From Philippe Deléham, Dec 24 2013: (Start)

T(n,n) = A033484(n-1).

Sum{k=1..n} T(n,k) = A053220(n).

T(n,k) = T(n-1,k)+3*T(n-1,k-1)-2*T(n-2,k-1)-2*T(n-2,k-2), T(1,1)=1, T(2,1)=1, T(2,2)=4, T(n,k)=0 if k<1 or if k>n. (End)

EXAMPLE

The (0,1/3,1) averaging array has these first four rows:

1/3

1/6....2/3

1/12...5/12...5/6

1/24...1/4....5/8...11/12.

Multiplying those rows by 3,6,12,24, respectively:

1

1...4

1...5...10

1...6...15...22

The first nine rows:

1

1...4

1...5...10

1...6...15...22

1...7...21...37...46

1...8...28...58...83...94

1...9...36...86...141..177..190

1...10..45...122..227..318..367..382

1...11..55...167..349..545..685..749..766

MATHEMATICA

a = 0; r = 1/3; b = 1;

t[1, 1] = r;

t[n_, 1] := (a + t[n - 1, 1])/2;

t[n_, n_] := (b + t[n - 1, n - 1])/2;

t[n_, k_] := (t[n - 1, k - 1] + t[n - 1, k])/2;

u[n_] := Table[t[n, k], {k, 1, n}]

Table[u[n], {n, 1, 5}]   (* averaging array *)

u = Table[(1/2) (1/r) 2^n*u[n], {n, 1, 12}];

TableForm[u]             (* A204102 triangle *)

Flatten[u]               (* A204201 sequence *)

CROSSREFS

Cf. A204202.

Sequence in context: A153426 A261720 A080852 * A090842 A120868 A100279

Adjacent sequences:  A204198 A204199 A204200 * A204202 A204203 A204204

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jan 12 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 22 05:47 EDT 2019. Contains 327287 sequences. (Running on oeis4.)