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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A228570 Triangle read by rows, formed from antidiagonals of triangle A102541. T(n, k) = A034851(n-2*k, k), n>= 0 and 0 <= k <= floor(n/3). 28
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 2, 1, 3, 4, 1, 4, 6, 1, 1, 4, 9, 2, 1, 5, 12, 6, 1, 5, 16, 10, 1, 1, 6, 20, 19, 3, 1, 6, 25, 28, 9, 1, 7, 30, 44, 19, 1, 1, 7, 36, 60, 38, 3, 1, 8, 42, 85, 66, 12, 1, 8, 49, 110, 110, 28, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

The row sums of this triangle are A102543. The antidiagonal sums are given by A192928 and the backwards antidiagonal sums are given by A228571.

Moving the terms in each column of this triangle, see the example, upwards to row 0 gives Losanitsch’s triangle A034851 as a square array.

Also the number of equivalence classes of ways of placing k 3 X 3 tiles in an n X 3 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Feb 16 2014

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..1000

FORMULA

T(n, k) = A034851(n-2*k, k), n >= 0 and 0 <= k <= floor(n/3).

T(n, k) = T(n-1, k) + T(n-3, k-1) - C((n-4)/2 - 2*(k-1)/2, (k-1)/2) where the last term is present only if n even and k odd; T(0, 0) = 1, T(1, 0) = 1, T(2, 0) = 1, T(n, k) = 0 for n < 0 and T(n, k) = 0 for k < 0 and k  > floor(n/3).

EXAMPLE

The first few rows of triangle T(n, k) are:

   n/k: 0,  1,  2,  3

   0:   1

   1:   1

   2:   1

   3:   1,  1

   4:   1,  1

   5:   1,  2

   6:   1,  2,  1

   7:   1,  3,  2

   8:   1,  3,  4

   9:   1,  4,  6,  1

  10:   1,  4,  9,  2

  11:   1,  5, 12,  6

MAPLE

T := proc(n, k) option remember: if n <0 then return(0) fi: if k < 0 or k > floor(n/3) then return(0) fi: A034851(n-2*k, k) end: A034851 := proc(n, k) option remember; local t; if k = 0 or k = n then return(1) fi; if n mod 2 = 0 and k mod 2 = 1 then t := binomial(n/2-1, (k-1)/2) else t := 0; fi; A034851(n-1, k-1) + A034851(n-1, k) - t; end: seq(seq(T(n, k), k=0..floor(n/3)), n=0..18);  # End first program

T := proc(n, k) option remember: if n=0 and k=0 or n=1 and k=0 or n=2 and k=0 then return(1) fi: if k <0 or k > floor(n/3) then return(0) fi: if type(n, even) and type(k, odd) then procname(n-1, k) + procname(n-3, k-1) - binomial((n-4)/2-2*(k-1)/2, (k-1)/2) else procname(n-1, k) + procname(n-3, k-1) fi: end: seq(seq(T(n, k), k=0..floor(n/3)), n=0..18); # End second program

MATHEMATICA

T[n_, k_] := (Binomial[n - 2k, k] + Boole[EvenQ[k] || OddQ[n]] Binomial[(n - 2k - Mod[n, 2])/2, Quotient[k, 2]])/2; Table[T[n, k], {n, 0, 20}, {k, 0, Quotient[n, 3]}] // Flatten (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)

PROG

(PARI)

T(n, k)={(binomial(n-2*k, k) + (k%2==0||n%2==1)*binomial((n-2*k-n%2)/2, k\2))/2}

for(n=1, 20, for(k=0, (n\3), print1(T(n, k), ", ")); print) \\ Andrew Howroyd, May 30 2017

CROSSREFS

Cf. A034851, A102541, A228572, A102543, A192928, A228571, A005691.

Sequence in context: A240060 A129264 A135840 * A173305 A233867 A319814

Adjacent sequences:  A228567 A228568 A228569 * A228571 A228572 A228573

KEYWORD

nonn,easy,tabf

AUTHOR

Johannes W. Meijer, Aug 26 2013

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 17 07:44 EST 2018. Contains 317275 sequences. (Running on oeis4.)