A223544 -id:A223544 - cp's OEIS frontend

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

Offset: 1

Amarnath Murthy, Sep 20 2002

(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
Conjecture that antidiagonal sums are A023855. - 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

			Triangle begins:
  1;
  2,  4;
  3,  6,  9;
  4,  8, 12, 16;
  5, 10, 15, 20, 25;
  6, 12, 18, 24, 30, 36;

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
L. E. Jeffery, Unit-primitive matrices

A002411 gives the sum of the n-th row. A141419 is similarly derived.
Cf. A066680, A223544.
Cf. A003991 (square multiplication table).
Main diagonal gives A000290.

a075362 n k = a075362_tabl !! (n-1) !! (k-1)
a075362_row n = a075362_tabl !! (n-1)
a075362_tabl = zipWith (zipWith (*)) a002260_tabl a002024_tabl
-- Reinhard Zumkeller, Nov 11 2012, Oct 04 2012

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

Table[NestList[n+#&,n,n-1],{n,15}]//Flatten (* Harvey P. Dale, Jun 14 2022 *)

T(n,k) = n*k, 1 <= k <= n. - Reinhard Zumkeller, Mar 07 2010
T(n,k) = A050873(n,k)*A051173(n,k), 1 <= k <= n. - Reinhard Zumkeller, Apr 25 2011
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
T(n,k) = A002260(n,k)*A002024(n,k) = (A215630(n,k)-A215631(n,k))/2, 1 <= k <= n. - Reinhard Zumkeller, Nov 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

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003

cp's OEIS Frontend

A075362 Triangle read by rows with the n-th row containing the first n multiples of n.

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