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A228074
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A Fibonacci-Pascal triangle read by rows: T(n,0) = Fibonacci(n), T(n,n) = n and for n > 0: T(n,k) = T(n-1,k-1) + T(n-1,k), 0 < k < n.
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35
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0, 1, 1, 1, 2, 2, 2, 3, 4, 3, 3, 5, 7, 7, 4, 5, 8, 12, 14, 11, 5, 8, 13, 20, 26, 25, 16, 6, 13, 21, 33, 46, 51, 41, 22, 7, 21, 34, 54, 79, 97, 92, 63, 29, 8, 34, 55, 88, 133, 176, 189, 155, 92, 37, 9, 55, 89, 143, 221, 309, 365, 344, 247, 129, 46, 10
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,5
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COMMENTS
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Sum of n-th row is 2^(n+1) - F(n+1) - 1 = A228078(n+1). - Greg Dresden and Sadek Mohammed, Aug 30 2022
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LINKS
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Reinhard Zumkeller, Rows n = 0..120 of table, flattened
Index entries for triangles and arrays related to Pascal's triangle
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EXAMPLE
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. 0: 0
. 1: 1 1
. 2: 1 2 2
. 3: 2 3 4 3
. 4: 3 5 7 7 4
. 5: 5 8 12 14 11 5
. 6: 8 13 20 26 25 16 6
. 7: 13 21 33 46 51 41 22 7
. 8: 21 34 54 79 97 92 63 29 8
. 9: 34 55 88 133 176 189 155 92 37 9
. 10: 55 89 143 221 309 365 344 247 129 46 10
. 11: 89 144 232 364 530 674 709 591 376 175 56 11
. 12: 144 233 376 596 894 1204 1383 1300 967 551 231 67 12 .
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MAPLE
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with(combinat);
T:= proc (n, k) option remember;
if k = 0 then fibonacci(n)
elif k = n then n
else T(n-1, k-1) + T(n-1, k)
end if
end proc;
seq(seq(T(n, k), k = 0..n), n = 0..12); # G. C. Greubel, Sep 05 2019
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MATHEMATICA
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T[n_, k_]:= T[n, k]= If[k==0, Fibonacci[n], If[k==n, n, T[n-1, k-1] + T[n -1, k]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] (* G. C. Greubel, Sep 05 2019 *)
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PROG
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(Haskell)
a228074 n k = a228074_tabl !! n !! k
a228074_row n = a228074_tabl !! n
a228074_tabl = map fst $ iterate
(\(u:_, vs) -> (vs, zipWith (+) ([u] ++ vs) (vs ++ [1]))) ([0], [1, 1])
(PARI) T(n, k) = if(k==0, fibonacci(n), if(k==n, n, T(n-1, k-1) + T(n-1, k)));
for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Sep 05 2019
(Sage)
def T(n, k):
if (k==0): return fibonacci(n)
elif (k==n): return n
else: return T(n-1, k) + T(n-1, k-1)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Sep 05 2019
(GAP)
T:= function(n, k)
if k=0 then return Fibonacci(n);
elif k=n then return n;
else return T(n-1, k-1) + T(n-1, k);
fi;
end;
Flat(List([0..12], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Sep 05 2019
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CROSSREFS
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Cf. A000045 (left edge), A001477 (right edge), A228078 (row sums), A027988 (maxima per row);
diagonals T(*,k): A000045, A000071, A001924, A014162, A014166, A053739, A053295, A053296, A053308, A053309;
diagonals T(k,*): A001477, A001245, A004006, A027927, A027928, A027929, A027930, A027931, A027932, A027933;
some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A105809, A109906, A111006, A114197, A162741.
Sequence in context: A057748 A057747 A281965 * A152803 A347102 A187181
Adjacent sequences: A228071 A228072 A228073 * A228075 A228076 A228077
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KEYWORD
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nonn,tabl,look
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AUTHOR
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Reinhard Zumkeller, Aug 15 2013
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STATUS
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approved
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