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Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the pentagonal numbers and the octagonal numbers. Complement of A186345.
4

%I #4 Mar 30 2012 18:57:18

%S 2,3,5,7,8,10,12,13,15,17,19,20,22,24,25,27,29,30,32,34,36,37,39,41,

%T 42,44,46,48,49,51,53,54,56,58,59,61,63,65,66,68,70,71,73,75,77,78,80,

%U 82,83,85,87,88,90,92,94,95,97,99,100,102,104,106,107,109,111,112,114,116,118,119,121,123,124,126,128,129,131,133,135,136,138,140,141,143,145,147,148,150,152,153,155,157,158,160,162,164,165,167,169,170

%N Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the pentagonal numbers and the octagonal numbers. Complement of A186345.

%e First, write

%e 1..5...12....22..35..... (pentagonal)

%e 1....8....21........40.. (octagonal)

%e Then replace each number by its rank, where ties are settled by ranking the pentagonal number after the octagonal:

%e a=(2,3,5,7,8,10,12,13,15,....)=A186344

%e b=(1,4,6,9,11,14,16,19,21,...)=A186345.

%t (* adjusted joint ranking; general formula *)

%t d=-1/2; u=3/2; v=-1/2; w=0; x=3; y=-2; z=0;

%t h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);

%t a[n_]:=n+Floor[h[n]/(2x)];

%t k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);

%t b[n_]:=n+Floor[k[n]/(2u)];

%t Table[a[n], {n, 1, 100}] (* A186344 *)

%t Table[b[n], {n, 1, 100}] (* A186345 *)

%Y Cf. A186342, A186343, A186345.

%K nonn

%O 1,1

%A _Clark Kimberling_, Feb 18 2011