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A214297 a(0)=-1, a(1)=0, a(2)=-3; thereafter a(n+2) - 2*a(n+1) + a(n) has period 4: repeat -4, 8, -4, 2.

%I A214297 #54 Sep 08 2022 08:46:02
%S A214297 -1,0,-3,2,3,6,5,12,15,20,21,30,35,42,45,56,63,72,77,90,99,110,117,
%T A214297 132,143,156,165,182,195,210,221,240,255,272,285,306,323,342,357,380,
%U A214297 399,420,437,462,483,506,525,552,575,600,621,650,675,702,725,756,783,812,837,870,899,930,957,992,1023,1056,1085,1122,1155,1190
%N A214297 a(0)=-1, a(1)=0, a(2)=-3; thereafter a(n+2) - 2*a(n+1) + a(n) has period 4: repeat -4, 8, -4, 2.
%C A214297 Let a(n)/A000290(n) = [-1/0, 0/1, -3/4, 2/9, 3/16, 6/25, 5/36, 12/49, 15/64, 20/81, 21/100, 30/121, ...] = a(n)/b(n) (say).
%C A214297 Then b(n)-4*a(n)=4, 1, 16, 1 (period of length 4).
%C A214297 Permutation from a(n) to A061037(n): 1, 3, 2, 7, 5, 11, 4, 15, 9, 19, 6, ... = shifted A145979 + 1.
%C A214297 A061037(n) - a(n) = 0, 3, -3, -3, 0, -15, 3, -33, 0 -57, 15, -87, 0, -123, ...
%C A214297 First 3 rows:
%C A214297 -1 0 -3 2  3  6  5 12 15 20 21 30 35
%C A214297 1 -3  5 1  3 -1  7  3  5  1  9  5  7
%C A214297 -4 8 -4 2 -4  8 -4  2 -4  8 -4  2 -4.
%C A214297 Note that the terms of a(n) increase from 12. Compare to increasing terms permutation of A061037(n): -3,-1,0,2,3,5,6,12,15, .... and A129647.
%C A214297 c(n) =  0, -1, 0, -1, 2, 1, 2, 1, 4, 3, 4, 3, 6, 5, 6, 5, ... (cf. A134967)
%C A214297 d(n) = -1,  1, 1,  3, 1, 3, 3, 5, 3, 5, 5, 7, 5, 7, 7, 9, ..., hence:
%C A214297 a(n) = c(n+1) * d(n+1).
%H A214297 G. C. Greubel, <a href="/A214297/b214297.txt">Table of n, a(n) for n = 0..10000</a>
%H A214297 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,-2,1).
%F A214297 a(k+4)  - a(k)    =  2*k + 4.
%F A214297 a(k+2)  - a(k-2)  =  2*k.
%F A214297 a(k+6)  - a(k-6)  =  6*k.
%F A214297 a(k+10) - a(k-10) = 10*k.
%F A214297 a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
%F A214297 a(2*k)   = -1, -3, followed by 3, 5, 15, 21, 35, 45, ... (A142717);
%F A214297 a(2*k+1) = k*(k+1) (see A002378).
%F A214297 A198442(n) = -1,0,0,2,3,6,8,12,  minus 3 at A198442(4*n+2).
%F A214297 G.f. -( 1-2*x+4*x^2-8*x^3+3*x^4 )/( (1-x)^2*(1-x^4) ). - _R. J. Mathar_, Jul 17 2012; edited by _N. J. A. Sloane_, Jul 22 2012
%F A214297 From _R. J. Mathar_, Jun 28 2013: (Start)
%F A214297 a(4*k)   = A000466(k);
%F A214297 a(4*k+1) = A002943(k);
%F A214297 a(4*k+2) = A078371(k-1) for k>0;
%F A214297 a(4*k+3) = A002939(k+1). (End)
%F A214297 a(n) = (2*n^2-11-9*(-1)^n+6*((-1)^((2*n+1-(-1)^n)/4)+(-1)^((2*n-1+(-1)^n)/4)))/8. - _Luce ETIENNE_, Oct 27 2016
%p A214297 A214297 := proc(n)
%p A214297     option remember;
%p A214297     if n <=5 then
%p A214297         op(n+1,[-1,0,-3,2,3,6]) ;
%p A214297     else
%p A214297         2*procname(n-1)-procname(n-2)+procname(n-4)-2*procname(n-5)+procname(n-6) ;
%p A214297     end if;
%p A214297 end proc: # _R. J. Mathar_, Jun 28 2013
%t A214297 Table[(2 n^2 - 11 - 9 (-1)^n + 6 ((-1)^((2 n + 1 - (-1)^n)/4) + (-1)^((2 n - 1 + (-1)^n)/4)))/8, {n, 0, 69}] (* or *)
%t A214297 CoefficientList[Series[-(1 - 2 x + 4 x^2 - 8 x^3 + 3 x^4)/((1 - x)^2*(1 - x^4)), {x, 0, 69}], x] (* _Michael De Vlieger_, Mar 24 2017 *)
%o A214297 (PARI) vector(100, n, n--; (2*n^2-11-9*(-1)^n+6*((-1)^((2*n+1-(-1)^n)/4)+(-1)^((2*n-1+(-1)^n)/4)))/8) \\ _G. C. Greubel_, Sep 19 2018
%o A214297 (Magma) [(2*n^2-11-9*(-1)^n+6*((-1)^((2*n+1-(-1)^n)/4)+(-1)^((2*n-1+(-1)^n)/4)))/8: n in [0..100]]; // _G. C. Greubel_, Sep 19 2018
%K A214297 sign,easy
%O A214297 0,3
%A A214297 _Paul Curtz_, Jul 11 2012
%E A214297 Edited by _N. J. A. Sloane_, Jul 22 2012