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 A095800 Triangle T(n,k) = abs( k *( (2*n+1)*(-1)^(n+k)+2*k-1) /4 ) read by rows, 1<=k<=n. 2
 1, 1, 4, 2, 2, 9, 2, 6, 3, 16, 3, 4, 12, 4, 25, 3, 8, 6, 20, 5, 36, 4, 6, 15, 8, 30, 6, 49, 4, 10, 9, 24, 10, 42, 7, 64, 5, 8, 18, 12, 35, 12, 56, 8, 81, 5, 12, 12, 28, 15, 48, 14, 72, 9, 100, 6, 10, 21, 16, 40, 18, 63, 16, 90, 10, 121, 6, 14, 15, 32, 20, 54, 21, 80, 18, 110, 11, 144, 7, 12, 24, 20, 45, 24, 70, 24, 99, 20, 132 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 1. Triangles of increasing sizes are subdivided using a triangular array. Then as shown on p. 83 of Conway and Guy, the series A002717 (1, 5, 13, 27, 48, 78, 118...) denotes the total number of triangles in each figure. 2. As a conjecture, each row of A095800 could be a distribution governing distinct subsets of types of triangles having the sum in the "How Many Triangles" series A002717. Thus 1 = 1; 5 = (1 + 4), 13 = (2 + 2 + 9)...etc. 3. Powers of the matrices have alternating signs such that odd rows begin with (+) and even rows begin with (-), as: 1; -1, 4; 2, -2, 9; -2, 6, -3, 16; 3, -4, 12, -4, 25;... Signed row sums = A049778: 1, 3, 9, 17, 32, 48... REFERENCES J. H. Conway and R. K. Guy, The Book of Numbers, Springer-Verlag New York, 1996, p. 83. LINKS Indranil Ghosh, Rows 1..125, flattened FORMULA Let M(n,k) = (-1)^(k+1)*k, 1<=k<=n be the infinite lower triangular matrix with 1, -2, 3,.. up to the diagonal, and the upper triangular part all zeros. The 3x3 submatrix would be [1 0 0 / 1 -2 0 / 1 -2 3]. The current triangle contains the absolute values of the matrix square M^2. EXAMPLE 1. [1 0 0 / 1 -2 0 / 1 -2 3]^2 = [1 0 0 / 1 -4 0 / 2 -2 9]. Then change the (-) signs to (+) getting the first 3 rows of the triangle: 1; 1, 4; 2, 2, 9; 2, 6, 3, 16; MAPLE A095800 := proc(n, k) k/4*( (2*n+1)*(-1)^(n+k)+2*k-1) ; abs(%) ; end proc: seq(seq(A095800(n, k), k=1..n), n=1..16) ; # R. J. Mathar, Apr 17 2011 PROG (PARI) T(n, k) = abs( k *( (2*n+1)*(-1)^(n+k)+2*k-1) /4 ); for(n=1, 20, for(m=1, n, print1(T(n, m), ", "))); \\ Joerg Arndt, Mar 05 2014 (Python) #Generates the b-file i=1 for n in range(1, 126): ....for k in range(1, n+1): ........print str(i)+" "+str(abs(k*((2*n+1)*(-1)**(n+k)+2*k-1)/4)) ........i+=1 # Indranil Ghosh, Feb 17 2017 CROSSREFS Cf. A002717 (row sums), A049778. Sequence in context: A098134 A079191 A079184 * A055630 A182700 A136202 Adjacent sequences:  A095797 A095798 A095799 * A095801 A095802 A095803 KEYWORD nonn,tabl,changed AUTHOR Gary W. Adamson, Jun 07 2004 EXTENSIONS Replaced NAME by closed form and inserted a missing row. - R. J. Mathar, Apr 17 2011 STATUS approved

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Last modified August 24 02:32 EDT 2019. Contains 326260 sequences. (Running on oeis4.)