A186349 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.
1, 2, 4, 5, 8, 10, 13, 15, 19, 22, 26, 29, 34, 38, 43, 47, 53, 58, 64, 69, 76, 82, 89, 95, 103, 110, 118, 125, 134, 142, 151, 159, 169, 178, 188, 197, 208, 218, 229, 239, 251, 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
Offset: 1
Examples
First, write .....8...16..24..32..40..48..56..64..72..80.. (8i) 1..4..9..16...25...36.....49.....64.......81. (squares) Then replace each number by its rank, where ties are settled by ranking 8i after the square: p = (3,6,7,9,11,12,14,16,17,...) = A186348 = n + floor(sqrt(8n+1/2)). q = (1,2,4,5,8,10,13,15,19,...) = a(n).
Links
- Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).
Programs
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Magma
m:=90; R
:=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 -
Maple
seq(k+ceil(k^2/8)-1,k=1..100); # Wesley Ivan Hurt, Jun 28 2013
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Mathematica
(* 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 *) d=-1/2; u=8; v=0; x=1; y=0; k[n_]:=(x*n^2+y*n-v+d)/u; a[n_]:=n+Floor[k[n]]; Table[a[n], {n, 1, 100}] -
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 */
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PARI
a(n)=(n^2-1)\8+n \\ Charles R Greathouse IV, Jul 05 2013
Formula
a(n) = n + floor((n^2 - 1)/8).
a(n) = n + ceiling(n^2/8) - 1. - Wesley Ivan Hurt, Jun 28 2013
From Bruno Berselli, Jul 05 2013: (Start)
G.f.: x*(1 + x^2 - x^3 + x^4 - x^5)/((1+x)*(1+x^2)*(1-x)^3).
a(n) = (2*n*(n+8) - (1+(-1)^n)*(5+2*i^(n*(n+1))) - 2)/16 where i=sqrt(-1). (End)
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