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A123596
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Squares alternating with triangular numbers.
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8
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0, 0, 1, 1, 4, 3, 9, 6, 16, 10, 25, 15, 36, 21, 49, 28, 64, 36, 81, 45, 100, 55, 121, 66, 144, 78, 169, 91, 196, 105, 225, 120, 256, 136, 289, 153, 324, 171, 361, 190, 400, 210, 441, 231, 484, 253, 529, 276, 576, 300, 625, 325, 676, 351, 729, 378, 784, 406, 841, 435
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OFFSET
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0,5
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COMMENTS
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For n >= 2, a(n) is the number of distinct positive slopes of least squares regression lines fitted to n points (j,y_j), 1 <= j <= n, where all y_j are 0 or 1. The total number of distinct slopes is 2*a(n)+1 (a(n) positive, a(n) negative, and the zero slope). - Pontus von Brömssen, Mar 10 2024
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LINKS
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FORMULA
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a(2*n) = n^2, a(2*n+1) = (n^2+n)/2.
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).
G.f.: x^2*(1+x+x^2)/((1-x)^3*(1+x)^3). (End)
a(n) = (3*n^2-1+(n^2+1)*(-1)^n)/16. - Luce ETIENNE, May 30 2015
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MATHEMATICA
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CoefficientList[Series[x^2*(1+x+x^2)/((1-x)^3*(1+x)^3), {x, 0, 50}], x] (* or *) Table[(3*n^2-1+(n^2+1)*(-1)^n)/16, {n, 0, 50}] (* G. C. Greubel, Oct 26 2017 *)
With[{nn=30}, Riffle[Range[0, nn]^2, Accumulate[Range[0, nn]]]] (* or *) LinearRecurrence[{0, 3, 0, -3, 0, 1}, {0, 0, 1, 1, 4, 3}, 60] (* Harvey P. Dale, Feb 11 2020 *)
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PROG
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(PARI) {a(n) = if(n%2, (n^2-1)/8, n^2/4)} \\ Michael Somos, Nov 18 2006
(Magma) [(3*n^2-1+(n^2+1)*(-1)^n)/16: n in [0..10]]; // G. C. Greubel, Oct 26 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Peter Hansen (babyskbaby(AT)web.de), Nov 14 2006
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EXTENSIONS
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Edited by Michael Somos, and several other correspondents, Nov 14 2005
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STATUS
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approved
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