A083392 - OEIS

%I #46 Sep 08 2022 08:45:10

%S 0,-1,2,-4,6,-9,12,-16,20,-25,30,-36,42,-49,56,-64,72,-81,90,-100,110,

%T -121,132,-144,156,-169,182,-196,210,-225,240,-256,272,-289,306,-324,

%U 342,-361,380,-400,420,-441,462,-484,506,-529,552,-576,600,-625,650,-676,702

%N Alternating partial sums of A000217.

%C 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

%H William A. Tedeschi, <a href="/A083392/b083392.txt">Table of n, a(n) for n = 0..10000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-2,0,2,1).

%F a(n) = Sum_{i=0..n} (-1)^i*t(i) where t(i) = i*(i+1)/2.

%F From _R. J. Mathar_, Feb 09 2010: (Start)

%F a(n) = -2*a(n-1) + 2*a(n-3) + a(n-4) for n > 3.

%F G.f.: x/((x-1)*(1+x)^3). (End)

%F a(n) = (-1)^n * ((n^2+n)/2 - floor(n^2/4)). - _William A. Tedeschi_, Aug 24 2010

%F E.g.f.: (1/4)*((x - 3)*x*cosh(x) - (x^2 - 3*x + 1)*sinh(x)). - _Stefano Spezia_, Jan 11 2020

%F Negative of the Euler transform of length 2 sequence [-2, 3]. - _Michael Somos_, Apr 27 2020

%e a(4) = t(0) - t(1) + t(2) - t(3) + t(4) = 0 - 1 + 3 - 6 + 10 = 6.

%e 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

%t LinearRecurrence[{-2,0,2,1},{0,-1,2,-4},60] (* _Harvey P. Dale_, Mar 16 2016 *)

%o (PARI) t(n)=n*(n+1)/2;

%o for (n=0,30,print1(sum(i=0,n,(-1)^i*t(i)), ", "))

%o (Magma) [(-1)^n*((n^2+n)/2 - Floor(n^2/4)): n in [0..50]]; // _G. C. Greubel_, Oct 29 2017

%Y Cf. A000217, A002620, A004526.

%K sign,easy

%O 0,3

%A _Jon Perry_, Jun 11 2003

%E More terms from _David W. Wilson_, Jun 14 2003