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A083392
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Alternating partial sums of A000217.
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11
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0, -1, 2, -4, 6, -9, 12, -16, 20, -25, 30, -36, 42, -49, 56, -64, 72, -81, 90, -100, 110, -121, 132, -144, 156, -169, 182, -196, 210, -225, 240, -256, 272, -289, 306, -324, 342, -361, 380, -400, 420, -441, 462, -484, 506, -529, 552, -576, 600, -625, 650, -676, 702
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OFFSET
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0,3
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COMMENTS
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Conjecture: for n > 0, a(n-1) is equal to the determinant of an n X n symmetric Toeplitz matrix M(n) whose first row consists of a single zero followed by successive positive integers repeated (A004526). - Stefano Spezia, Jan 10 2020
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LINKS
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FORMULA
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a(n) = Sum_{i=0..n} (-1)^i*t(i) where t(i) = i*(i+1)/2.
a(n) = -2*a(n-1) + 2*a(n-3) + a(n-4) for n > 3.
G.f.: x/((x-1)*(1+x)^3). (End)
E.g.f.: (1/4)*((x - 3)*x*cosh(x) - (x^2 - 3*x + 1)*sinh(x)). - Stefano Spezia, Jan 11 2020
Negative of the Euler transform of length 2 sequence [-2, 3]. - Michael Somos, Apr 27 2020
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EXAMPLE
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a(4) = t(0) - t(1) + t(2) - t(3) + t(4) = 0 - 1 + 3 - 6 + 10 = 6.
G.f. = - x + 2*x^2 - 4*x^3 + 6*x^4 - 9*x^5 + 12*x^6 - 16*x^7 + ... - Michael Somos, Apr 27 2020
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MATHEMATICA
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LinearRecurrence[{-2, 0, 2, 1}, {0, -1, 2, -4}, 60] (* Harvey P. Dale, Mar 16 2016 *)
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PROG
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(PARI) t(n)=n*(n+1)/2;
for (n=0, 30, print1(sum(i=0, n, (-1)^i*t(i)), ", "))
(Magma) [(-1)^n*((n^2+n)/2 - Floor(n^2/4)): n in [0..50]]; // G. C. Greubel, Oct 29 2017
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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