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%I #22 Jun 13 2015 00:52:16
%S 0,0,1,1,3,4,6,9,10,16,15,25,21,36,28,49,36,64,45,81,55,100,66,121,78,
%T 144,91,169,105,196,120,225,136,256,153,289,171,324,190,361,210,400,
%U 231,441,253,484,276,529,300,576,325,625,351,676,378,729,406,784,435,841
%N Triangular numbers alternating with squares.
%H Colin Barker, <a href="/A124093/b124093.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).
%F a(n) = n(n+2)/8 if n is even; a(n) = (n-1)^2/4 if n is odd (n>=0). - _Emeric Deutsch_, Nov 29 2006
%F a(n) = (3*n^2-2*n+2-(n^2-6*n+2)*(-1)^n)/16. - _Luce ETIENNE_, May 28 2015
%F a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>5. - _Colin Barker_, May 28 2015
%F G.f.: -x^2*(x^3+x+1) / ((x-1)^3*(x+1)^3). - _Colin Barker_, May 28 2015
%p a:=proc(n) if n mod 2 = 0 then n*(n+2)/8 else (n-1)^2/4 fi end: seq(a(n),n=0..70); # _Emeric Deutsch_, Nov 29 2006
%t tr=Table[{k(k+1)/2,k^2},{k,0,100}]//Flatten (Seidov)
%t With[{nn=30},Riffle[Accumulate[Range[0,nn]],Range[0,nn]^2]] (* _Harvey P. Dale_, Jul 13 2014 *)
%o (PARI) concat([0,0], Vec(-x^2*(x^3+x+1)/((x-1)^3*(x+1)^3) + O(x^100))) \\ _Colin Barker_, May 28 2015
%Y Cf. A123596. Rearrangement of A054686.
%K nonn,easy
%O 0,5
%A _N. J. A. Sloane_, based on a suggestion from _Robert G. Wilson v_, Nov 27 2006
%E More terms from _Zak Seidov_, Nov 28 2006
%E More terms from _Emeric Deutsch_, Nov 29 2006
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