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A119457
Triangle read by rows: T(n, 1) = n, T(n, 2) = 2*(n-1) for n>1 and T(n, k) = T(n-1, k-1) + T(n-2, k-2) for 2 < k <= n.
9
1, 2, 2, 3, 4, 3, 4, 6, 6, 5, 5, 8, 9, 10, 8, 6, 10, 12, 15, 16, 13, 7, 12, 15, 20, 24, 26, 21, 8, 14, 18, 25, 32, 39, 42, 34, 9, 16, 21, 30, 40, 52, 63, 68, 55, 10, 18, 24, 35, 48, 65, 84, 102, 110, 89, 11, 20, 27, 40, 56, 78, 105, 136, 165, 178, 144, 12, 22, 30, 45, 64, 91, 126, 170, 220, 267, 288, 233
OFFSET
1,2
LINKS
Eric Weisstein's World of Mathematics, Fibonacci Number
FORMULA
T(n, k) = (n-k+1)*T(k,k) for 1 <= k < n, with T(n, n) = A000045(n+1).
From G. C. Greubel, Apr 15 2025: (Start)
T(n, k) = (n-k+1)*Fibonacci(k+1).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1-(-1)^n)*A023652(floor((n+1)/2)) + (1+(-1)^n)*A001891(floor(n/2)).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1-(-1)^n)*A112469(floor((n-1)/2)) + (1+(-1)^n)*A355020(floor((n-2)/2)). (End)
EXAMPLE
Triangle begins as:
1;
2, 2;
3, 4, 3;
4, 6, 6, 5;
5, 8, 9, 10, 8;
6, 10, 12, 15, 16, 13;
7, 12, 15, 20, 24, 26, 21;
8, 14, 18, 25, 32, 39, 42, 34;
9, 16, 21, 30, 40, 52, 63, 68, 55;
10, 18, 24, 35, 48, 65, 84, 102, 110, 89;
11, 20, 27, 40, 56, 78, 105, 136, 165, 178, 144;
12, 22, 30, 45, 64, 91, 126, 170, 220, 267, 288, 233;
MATHEMATICA
(* First program *)
T[n_, 1] := n;
T[n_ /; n > 1, 2] := 2 n - 2;
T[n_, k_] /; 2 < k <= n := T[n, k] = T[n - 1, k - 1] + T[n - 2, k - 2];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 01 2021 *)
(* Alternative: *)
A119457[n_, k_]:= (n-k+1)*Fibonacci[k+1];
Table[A119457[n, k], {n, 13}, {k, n}]//Flatten (* G. C. Greubel, Apr 16 2025 *)
PROG
(Magma)
A119457:= func< n, k | (n-k+1)*Fibonacci(k+1) >;
[A119457(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 16 2025
(SageMath)
def A119457(n, k): return (n-k+1)*fibonacci(k+1)
print(flatten([[A119457(n, k) for k in range(1, n+1)] for n in range(1, 13)])) # G. C. Greubel, Apr 16 2025
CROSSREFS
Main diagonal: A023607(n).
Sums: A001891 (row), A355020 (signed row).
Columns: A000027(n) (k=1), A005843(n-1) (k=2), A008585(n-2) (k=3), A008587(n-3) (k=4), A008590(n-4) (k=5), A008595(n-5) (k=6), A008603(n-6) (k=7).
Diagonals: A000045(n+1) (k=n), A006355(n+1) (k=n-1), A022086(n-1) (k=n-2), A022087(n-2) (k=n-3), A022088(n-3) (k=n-4), A022089(n-4) (k=n-5), A022090(n-5) (k=n-6).
Sequence in context: A368310 A003991 A131923 * A241356 A065157 A390934
KEYWORD
nonn,tabl
AUTHOR
Reinhard Zumkeller, May 20 2006
STATUS
approved