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A026268 Triangle, T(n, k): T(n,k) = 1 for n < 3, T(3,1) = T(3,2) = T(3,3) = 2, T(n,0) = 1, T(n,1) = n-1, T(n,n) = T(n-1,n-2) + T(n-1,n-1), otherwise T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k), read by rows. 14
1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 3, 5, 6, 4, 1, 4, 9, 14, 15, 10, 1, 5, 14, 27, 38, 39, 25, 1, 6, 20, 46, 79, 104, 102, 64, 1, 7, 27, 72, 145, 229, 285, 270, 166, 1, 8, 35, 106, 244, 446, 659, 784, 721, 436, 1, 9, 44, 149, 385, 796, 1349, 1889, 2164, 1941, 1157, 1, 10, 54, 202, 578, 1330, 2530, 4034, 5402, 5994, 5262, 3098 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,8
COMMENTS
a(n) = number of strings s(0)..s(n) such that s(n) = n-k, where s(0) = 0, s(1) = 1, |s(i)-s(i-1)| <= 1 for i >= 2; |s(2)-s(1)| = 1, and |s(3)-s(2)| = 1 if s(2) = 1.
LINKS
Clark Kimberling, Rows 0..100, flattened
FORMULA
From G. C. Greubel, Sep 24 2022: (Start)
T(n, 1) = A000027(n-1), n >= 1.
T(n, 2) = A212342(n-1), n >= 2.
T(n, n-1) = A026270(n), n >= 2.
T(n, n-2) = A026288(n), n >= 2.
T(n, n-3) = A026289(n), n >= 3.
T(n, n-4) = A026290(n), n >= 4.
T(n, n) = A026269(n), n >= 2.
T(n, floor(n/2)) = A026297(n), n >= 0.
T(2*n, n) = A026292(n).
T(2*n, n-1) = A026295(n), n >= 1.
T(2*n, n+1) = A026296(n), n >= 1.
T(2*n-1, n-1) = A026291(n), n >= 2.
T(3*n, n) = A026293(n), n >= 0.
T(4*n, n) = A026294(n), n >= 0.
Sum_{k=0..n} T(n, k) = A026299(n-1), n >= 3.(End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 1, 1;
1, 2, 2, 2;
1, 3, 5, 6, 4;
1, 4, 9, 14, 15, 10;
1, 5, 14, 27, 38, 39, 25;
1, 6, 20, 46, 79, 104, 102, 64;
1, 7, 27, 72, 145, 229, 285, 270, 166;
1, 8, 35, 106, 244, 446, 659, 784, 721, 436;
1, 9, 44, 149, 385, 796, 1349, 1889, 2164, 1941, 1157;
MATHEMATICA
T[n_, k_]:= T[n, k]= If[n<3 || k==0, 1, If[k==1, n-1, If[k==2, (n^2-n-2)/2 + Boole[n==2], If[k==n, T[n-1, n-2] +T[n-1, n-1], T[n-1, k-2] + T[n-1, k-1] + T[n -1, k] ]]]];
Table[T[n, k], {n, 0, 14}, {k, 0, n}]//Flatten (* corrected by G. C. Greubel, Sep 24 2022 *)
PROG
(Magma)
f:= func< n | n eq 2 select 1 else (n^2 -n -2)/2 >;
function T(n, k) // T = A026268
if k eq 0 or n lt 3 then return 1;
elif k eq 1 then return n-1;
elif k eq 2 then return f(n);
elif k eq n then return T(n-1, n-2) + T(n-1, n-1);
else return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k);
end if; return T;
end function;
[T(n, k): k in [0..n], n in [0..14]]; // G. C. Greubel, Sep 24 2022
(SageMath)
def T(n, k): # T = A026268
if n<3 or k==0: return 1
elif k==1: return n-1
elif k==2: return (n^2 -n -2)//2 + int(n==2)
elif k==n: return T(n-1, n-2) + T(n-1, n-1)
else: return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
flatten([[T(n, k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Sep 24 2022
CROSSREFS
Sequence in context: A055253 A103626 A238224 * A089258 A004065 A127496
KEYWORD
nonn,tabl
AUTHOR
EXTENSIONS
Updated by Clark Kimberling, Aug 29 2014
Indices of b-file corrected by Sidney Cadot, Jan 06 2023.
STATUS
approved

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Last modified July 16 10:51 EDT 2024. Contains 374345 sequences. (Running on oeis4.)