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%I #34 Apr 06 2024 20:32:08
%S 1,2,4,5,8,10,13,15,19,22,26,29,34,38,43,47,53,58,64,69,76,82,89,95,
%T 103,110,118,125,134,142,151,159,169,178,188,197,208,218,229,239,251,
%U 262,274,285,298,310,323,335,349,362,376,389,404,418,433,447,463,478,494,509,526,542,559,575,593,610,628,645,664,682,701,719,739,758,778,797,818,838,859,879,901,922,944,965,988,1010
%N Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=8i and g(j)=j^2. Complement of A186348.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,-2,1).
%F a(n) = n + floor((n^2 - 1)/8).
%F a(n) = n + ceiling(n^2/8) - 1. - _Wesley Ivan Hurt_, Jun 28 2013
%F From _Bruno Berselli_, Jul 05 2013: (Start)
%F G.f.: x*(1 + x^2 - x^3 + x^4 - x^5)/((1+x)*(1+x^2)*(1-x)^3).
%F a(n) = (2*n*(n+8) - (1+(-1)^n)*(5+2*i^(n*(n+1))) - 2)/16 where i=sqrt(-1). (End)
%F E.g.f.: (8 - 2*cos(x) + (x^2 + 9*x - 6)*cosh(x) + (x^2 + 9*x - 1)*sinh(x))/8. - _Stefano Spezia_, Apr 06 2024
%e First, write
%e .....8...16..24..32..40..48..56..64..72..80.. (8i)
%e 1..4..9..16...25...36.....49.....64.......81. (squares)
%e Then replace each number by its rank, where ties are settled by ranking 8i after the square:
%e p = (3,6,7,9,11,12,14,16,17,...) = A186348 = n + floor(sqrt(8n+1/2)).
%e q = (1,2,4,5,8,10,13,15,19,...) = a(n).
%p seq(k+ceil(k^2/8)-1,k=1..100); # _Wesley Ivan Hurt_, Jun 28 2013
%t (* adjusted joint rank sequences p and q (=a(n)), using general formula for ranking 1st degree u*n+v and 2nd degree x*n^2 + y*n + z *)
%t d=-1/2; u=8; v=0; x=1; y=0;
%t k[n_]:=(x*n^2+y*n-v+d)/u;
%t a[n_]:=n+Floor[k[n]];
%t Table[a[n], {n, 1, 100}]
%o (Magma) m:=90; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x^2-x^3+x^4-x^5)/((1+x)*(1+x^2)*(1-x)^3))); // _Bruno Berselli_, Jul 05 2013
%o (PARI) a(n)=(n^2-1)\8+n \\ _Charles R Greathouse IV_, Jul 05 2013
%o (Maxima) makelist((2*n*(n+8)-(1+(-1)^n)*(5+2*%i^(n*(n+1)))-2)/16, n, 1, 90); /* _Bruno Berselli_, Jul 05 2013 */
%Y Cf. A186346, A186347, A186348.
%K nonn,easy,changed
%O 1,2
%A _Clark Kimberling_, Feb 20 2011
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