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A184823
a(n) = n + floor(n/t) + floor(n/t^2) + floor(n/t^3), where t is the tetranacci constant.
6
1, 3, 4, 7, 8, 10, 11, 15, 16, 18, 19, 22, 23, 25, 28, 30, 31, 33, 35, 37, 38, 41, 43, 45, 46, 48, 51, 52, 55, 57, 59, 60, 62, 64, 66, 68, 70, 72, 74, 75, 78, 79, 82, 83, 86, 87, 89, 90, 93, 94, 97, 98, 101, 103, 104, 107, 108, 111, 112, 115, 116, 118, 119, 122, 124, 126, 128, 130, 131, 133, 135, 138, 139, 141, 143, 145, 146, 148, 151, 153, 155, 157, 159, 160, 162, 165, 167, 168, 170, 172, 174, 175, 178, 180, 182, 183, 186, 187, 189, 190, 194, 195, 197, 198, 201, 202, 204, 208, 209, 211, 212, 215, 216, 218, 220, 223, 224
OFFSET
1,2
COMMENTS
This is one of four sequences that partition the positive integers.
Given t is the tetranacci constant, then the following sequences are disjoint:
. A184823(n) = n + [n/t] + [n/t^2] + [n/t^3],
. A184824(n) = n + [n*t] + [n/t] + [n/t^2],
. A184825(n) = n + [n*t] + [n*t^2] + [n/t],
. A184826(n) = n + [n*t] + [n*t^2] + [n*t^3], where []=floor.
This is a special case of Clark Kimberling's results given in A184812.
FORMULA
Limit a(n)/n = t = 1.9275619754829253042619058...
a(n) = n + floor(n*p/s) + floor(n*q/s) + floor(n*r/s), where p=t, q=t^2, r=t^3, s=t^4, and t is the tetranacci constant.
EXAMPLE
Let t be the tetranacci constant, then t = 1 + 1/t + 1/t^2 + 1/t^3 and:
t = 1.92756197548..., t^2 = 3.71549516932..., t^3 = 7.16184720848..., t^4 = 13.8049043532...
PROG
(PARI) {a(n)=local(t=real(polroots(1+x+x^2+x^3-x^4)[2])); n+floor(n/t)+floor(n/t^2)+floor(n/t^3)}
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 23 2011
STATUS
approved