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A186223 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f and g are the triangular numbers and pentagonal numbers. Complement of A186224. 4
1, 3, 5, 6, 8, 9, 11, 13, 14, 16, 17, 19, 20, 22, 24, 25, 27, 28, 30, 31, 33, 35, 36, 38, 39, 41, 43, 44, 46, 47, 49, 50, 52, 54, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 73, 74, 76, 77, 79, 80, 82, 84, 85, 87, 88, 90, 91, 93, 95, 96, 98, 99, 101, 102, 104, 106, 107, 109, 110, 112, 114, 115, 117, 118, 120, 121, 123, 125, 126, 128, 129, 131, 132, 134, 136, 137, 139, 140, 142, 143, 145, 147, 148, 150, 151, 153, 155, 156, 158 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

See A186219 for a general description.

LINKS

Table of n, a(n) for n=1..100.

EXAMPLE

First, write

1..3...6..10....15...21.....28......36...45... (triangular)

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

Replace each number by its rank, where ties are settled by ranking the triangular number before the pentagonal:

a=(1,3,5,6,8,9,11,13,...)

b=(2,4,7,10,12,15,18,...).

MATHEMATICA

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

(* triangular & pentagonal *)

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

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

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

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

Table[a[n], {n, 1, 100}] (* A186223 *)

Table[b[n], {n, 1, 100}] (* A186224 *)

CROSSREFS

Cf. A186219, A186224, A186225, A186226,

A000217 (triangular), A000326 (pentagonal).

Sequence in context: A186324 A101358 A276210 * A047446 A058065 A218773

Adjacent sequences: A186220 A186221 A186222 * A186224 A186225 A186226

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 15 2011

STATUS

approved

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Last modified February 3 15:13 EST 2023. Contains 360035 sequences. (Running on oeis4.)