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 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

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Last modified February 22 14:25 EST 2020. Contains 332136 sequences. (Running on oeis4.)