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A075362
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Triangle read by rows with the n-th row containing the first n multiples of n.
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14
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1, 2, 4, 3, 6, 9, 4, 8, 12, 16, 5, 10, 15, 20, 25, 6, 12, 18, 24, 30, 36, 7, 14, 21, 28, 35, 42, 49, 8, 16, 24, 32, 40, 48, 56, 64, 9, 18, 27, 36, 45, 54, 63, 72, 81, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 12, 24, 36, 48, 60, 72, 84
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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(Conjecture) Let N=2*n and k=1,...,n. Let A_{N,0}, A_{N,1}, ..., A_{N,n-1} be the n X n unit-primitive matrices (see [Jeffery]) associated with N. Define the Chebyshev polynomials of the second kind by the recurrence U_0(x)=1, U_1(x)=2*x and U_r(x)=2*x*U_(r-1)(x)-U_(r-2)(x) (r>1). Define the column vectors V_(k-1)=(U_(k-1)(cos(Pi/N)), U_(k-1)(cos(3*Pi/N)), ..., U_(k-1)(cos((2*n-1)*Pi/N)))^T, where B^T denotes the transpose of matrix B. Let S_N=[V_0,V_1,...,V_(n-1)] be the n X n matrix formed by taking the components of vector V_(k-1) as the entries in column k-1 (V_(k-1) gives the ordered spectrum of A_{N,k-1}). Let X_N=[S_N]^T*S_N, and let [X_N]_(i,j) denote the entry in row i and column j of X_N, i,j in {0,...,n-1}. Then also T(n,k)=[X_N]_(k-1,k-1); that is, row n of the triangle is given by the main diagonal entries of X_N. Hence T(n,k) is the sum of squares T(n,k) = sum[m=1,...,n (U_(k-1)(cos((2*m-1)*Pi/N)))^2]=[V_(k-1)]^T*V_(k-1). - L. Edson Jeffery, Jan 20 2012
Viewed as a sequence of rows, consider the subsequences (of rows) that contain every positive integer. The lexicographically latest of these subsequences consists of the rows with row numbers in A066680 U {1}; this is the only one that contains its own row numbers only once. - Peter Munn, Dec 04 2019
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LINKS
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FORMULA
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T(n,k) = Sum_{i=1..k} i*binomial(k,i)*binomial(n+1-k,n-i), 1 <= k <= n. - Mircea Merca, Apr 11 2012
a(n) = A223544(n) - 1; a(n) = i*(t+1), where i = n - t*(t+1)/2, t = floor((-1 + sqrt(8*n-7))/2). - Boris Putievskiy, Jul 24 2013
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EXAMPLE
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Triangle begins:
1;
2, 4;
3, 6, 9;
4, 8, 12, 16;
5, 10, 15, 20, 25;
6, 12, 18, 24, 30, 36;
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MAPLE
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T(n, k):=piecewise(k<=n, sum(i*binomial(k, i)*binomial(n+1-k, n-i), i=1..k), k>n, 0) # Mircea Merca, Apr 11 2012
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MATHEMATICA
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Table[NestList[n+#&, n, n-1], {n, 15}]//Flatten (* Harvey P. Dale, Jun 14 2022 *)
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PROG
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(Haskell)
a075362 n k = a075362_tabl !! (n-1) !! (k-1)
a075362_row n = a075362_tabl !! (n-1)
a075362_tabl = zipWith (zipWith (*)) a002260_tabl a002024_tabl
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CROSSREFS
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Cf. A003991 (square multiplication table).
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KEYWORD
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AUTHOR
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EXTENSIONS
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More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003
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STATUS
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approved
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