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Search: id:A131805
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| A131805 |
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Row sums of triangular array T: T(j,k) = -(k+1)/2 for odd k, T(j,k) = 0 for k = 0, T(j,k) = j+1-k/2 for even k > 0; 0 <= k <= j. |
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+0
2
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| 0, -1, 1, 0, 4, 3, 9, 8, 16, 15, 25, 24, 36, 35, 49, 48, 64, 63, 81, 80, 100, 99, 121, 120, 144, 143, 169, 168, 196, 195, 225, 224, 256, 255, 289, 288, 324, 323, 361, 360, 400, 399, 441, 440, 484, 483, 529, 528, 576, 575, 625, 624, 676, 675, 729, 728, 784, 783, 841
(list; graph; listen)
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OFFSET
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0,5
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COMMENT
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Interleaving of A000290 and A067998 (starting at second term).
First differences are -1, 2, -1, 4, -1, 6, -1, 8, -1, 10, ...: a(n+1) - a(n) = (-1)^(n+1)*A124625(n+2).
Main diagonal of T is in A001057, antidiagonal sums are in A131804.
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FORMULA
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a(0) = 0; a(n) = a(n-1) - (n mod 2) + n*(1 - (n mod 2)) for n > 0.
G.f.: x*(-1+2*x+x^2)/((1-x)^3*(1+x)^2).
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EXAMPLE
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First seven rows of T are
[ 0 ],
[ 0, -1 ],
[ 0, -1, 2 ],
[ 0, -1, 3, -2 ],
[ 0, -1, 4, -2, 3 ],
[ 0, -1, 5, -2, 4, -3 ],
[ 0, -1, 6, -2, 5, -3, 4 ]
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PROGRAM
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(MAGMA) m:=59; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do for k:=2 to j do if k mod 2 eq 0 then M[j, k]:= -k div 2; else M[j, k]:=j-(k div 2); end if; end for; end for; [ &+[ M[j, k]: k in [1..j] ]: j in [1..m] ];
(MAGMA) m:=29; &cat[ [ n^2, n^2-1 ]: n in [0..m] ];
(PARI) {m=58; for(n=0, m, r=n%2; print1(((n-r)/2)^2-r, ", "))}
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CROSSREFS
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Cf. A000290 (n^2), A067998 (n^2-2*n), A124625, A001057, A131804.
Sequence in context: A123596 A094885 A094728 this_sequence A103218 A107381 A132192
Adjacent sequences: A131802 A131803 A131804 this_sequence A131806 A131807 A131808
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KEYWORD
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sign
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AUTHOR
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Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jul 18 2007
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