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 A141387 Triangle read by rows: T(n,m) = n + 2*m*(n - m) (0 <= m <= n). 5
 0, 1, 1, 2, 4, 2, 3, 7, 7, 3, 4, 10, 12, 10, 4, 5, 13, 17, 17, 13, 5, 6, 16, 22, 24, 22, 16, 6, 7, 19, 27, 31, 31, 27, 19, 7, 8, 22, 32, 38, 40, 38, 32, 22, 8, 9, 25, 37, 45, 49, 49, 45, 37, 25, 9, 10, 28, 42, 52, 58, 60, 58, 52, 42, 28, 10 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Construct the infinite-dimensional matrix representation of angular momentum operators (J_1,J_2,J_3) in Jordan-Schwinger form (cf. Harter, Klee, Schwinger). The triangle terms T(n,k)=T(2j,j+m) satisfy:(1/4)T(2j,j+m) =   = . Matrices for J_1^2 and J_2^2 are sparse. These diagonal equalities and the off-diagonal equalities of A268759 determine the only nonzero entries. Comments on A268759 provide a conjecture for the clear interpretation of these numbers in the context of binomial coefficients and other geometrical sequences. - Bradley Klee, Feb 20 2016 This sequence appears in the probability of the coin tossing "Gambler's Ruin". Call the probability of winning a coin toss = p, and the probability of losing the toss is (1-p) = q,  and call z = q/p. A gambler starts with \$1, and tosses for \$1 stakes till he has \$0 (ruin) or has \$n (Wins). The average time Twinlose(n) of a game (win OR lose) is a well known function of z and n. The probabiltiy of the gambler winning Pwin(n) is also known, and is equal to (1-z)/(1-z^n). Twin(n) defined as the average time it takes the gambler to win such a game is not so well known (I have not found it in the literature). I calculated Twin(n) and found it to be Twin(n) = Pwin(n) * (the sum of T(n,m) * z^m  for m=0 to m=n) where T(n,m) is the given by the n-th row of this sequence. - Steve Newman, Oct 24 2016 REFERENCES R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139. LINKS Michael De Vlieger, Table of n, a(n) for n = 0..10010 (rows 0 <= n <= 70, flattened) Tomislav Došlić, On the Laplacian Szeged Spectrum of Paths, Iranian J. Math. Chem. (2020) Vol. 11, No. 1, 57-63. W. Harter, Principles of Symmetry, Dynamics, Spectroscopy, Wiley, 1993, Ch. 5, page 345-346. B. Klee, Quantum Angular Momentum Matrices, Wolfram Demonstrations Project, 2016. J. Schwinger, On Angular Momentum , Cambridge: Harvard University, Nuclear Development Associates, Inc., 1952. FORMULA T(n,m)=n+2*m*(n-m). EXAMPLE {0}, {1, 1}, {2, 4, 2}, {3, 7, 7, 3}, {4, 10, 12, 10, 4}, {5, 13, 17, 17, 13, 5}, {6, 16, 22, 24, 22, 16, 6}, {7, 19, 27, 31, 31, 27, 19, 7}, {8, 22, 32, 38, 40, 38, 32, 22, 8}, {9, 25, 37, 45, 49, 49, 45, 37, 25, 9}, {10, 28, 42, 52, 58, 60, 58, 52, 42, 28, 10} MATHEMATICA Clear[T, n, m, a]; T[n_, m_] = n + 2* m *(-m + n); a = Table[Table[T[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[a] Flatten[ Table[2 j + 2 j^2 - 2 m^2, {j, 0, 10, 1/2}, {m, -j, j}]] (* Bradley Klee, Feb 20 2016 *) PROG (PARI) {T(n, m) = if( m<0 || n

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Last modified September 25 20:01 EDT 2020. Contains 337344 sequences. (Running on oeis4.)