login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A228572 Triangle read by rows, formed from antidiagonals of triangle A228570: T(n,k) = A034851(n-3*k, k) for n >= 0 and 0 <= k <= floor(n/4). 26
1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 3, 2, 1, 4, 4, 1, 4, 6, 1, 5, 9, 1, 1, 5, 12, 2, 1, 6, 16, 6, 1, 6, 20, 10, 1, 7, 25, 19, 1, 1, 7, 30, 28, 3, 1, 8, 36, 44, 9, 1, 8, 42, 60, 19, 1, 9, 49, 85, 38, 1, 1, 9, 56, 110, 66, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,10
COMMENTS
The row sums of this triangle are A192928.
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. Observe A102541 and A228570 for the same phenomenom. The number of zeros in the columns for these three triangles are multiples of 2, 3 and 4 respectively.
Also the number of equivalence classes of ways of placing k 4 X 4 tiles in an n X 4 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Apr 24 2015
LINKS
FORMULA
T(n, k) = A034851(n-3*k, k) for n >= 0 and 0 <= k <= floor(n/4).
T(n, k) = T(n-1, k) + T(n-4, k-1) - C((n+3)/2 - 3*(k+1)/2-1, (k+1)/2-1), where the last term is present only if n odd and k odd; T(0, 0) = 1, T(1, 0) = 1, T(2, 0) = 1, T(3, 0) = 1, T(n, k) = 0 for n < 0 and T(n, k) = 0 for k < 0 and k > floor(n/4).
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
4: 1, 1
5: 1, 1
6: 1, 2
7: 1, 2
8: 1, 3, 1
9: 1, 3, 2
10: 1, 4, 4
11: 1, 4, 6
12: 1, 5, 9, 1
MAPLE
T := proc(n, k) option remember: if n <0 then return(0) fi: if k < 0 or k > floor(n/4) then return(0) fi: A034851(n-3*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/4)), n=0..21); # 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 or n=3 and k=0 then return(1) fi: if k <0 or k > floor(n/4) then return(0) fi: if type(n, odd) and type(k, odd) then procname(n-1, k) + procname(n-4, k-1) - binomial((n+3)/2 - 3*(k+1)/2 - 1, (k+1)/2-1) else procname(n-1, k) + procname(n-4, k-1) fi: end: seq(seq(T(n, k), k=0..floor(n/4)), n=0..21); # End second program
MATHEMATICA
T[n_, k_] := (Binomial[n - 3k, k] + Boole[EvenQ[k] || EvenQ[n]]* Binomial[(n - 3k - Mod[k, 2] - Mod[n, 2])/2, Quotient[k, 2]])/2; Table[T[n, k], {n, 0, 20}, {k, 0, Quotient[n, 4]}] // Flatten (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)
PROG
(PARI)
T(n, k)={(binomial(n-3*k, k) + (k%2==0||n%2==0)*binomial((n-3*k-k%2-n%2)/2, k\2))/2}
for(n=1, 20, for(k=0, (n\4), print1(T(n, k), ", ")); print) \\ Andrew Howroyd, May 29 2017
CROSSREFS
Sequence in context: A182593 A201167 A202853 * A334675 A078380 A062356
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 | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 10:44 EDT 2024. Contains 371268 sequences. (Running on oeis4.)