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A184820
a(n) = n + floor(n/t) + floor(n/t^2), where t is the tribonacci constant.
6
1, 3, 4, 7, 8, 10, 12, 14, 15, 17, 19, 21, 23, 25, 27, 28, 31, 32, 34, 35, 38, 39, 41, 44, 45, 47, 48, 51, 52, 54, 56, 58, 59, 62, 64, 65, 67, 69, 71, 72, 75, 76, 78, 80, 82, 84, 85, 88, 89, 91, 93, 95, 96, 98, 100, 102, 103, 106, 108, 109, 112, 113, 115, 116, 119, 120, 122, 124, 126, 128, 129, 132, 133, 135, 137, 139, 140, 143, 144, 146, 148, 150, 152, 153, 156, 157, 159, 161, 163, 164, 166, 169, 170, 172, 174, 176, 177, 179, 181, 183, 184, 187, 188, 190, 193, 194, 196, 197, 200, 201, 203, 205, 207, 208, 210, 213, 214, 216, 218, 220, 221, 224, 225, 227, 228, 231, 233, 234, 237, 238, 240, 242, 244, 245, 247
OFFSET
1,2
COMMENTS
This is one of three sequences that partition the positive integers.
Given t is the tribonacci constant, then the following sequences are disjoint:
. A184820(n) = n + [n/t] + [n/t^2],
. A184821(n) = n + [n*t] + [n/t],
. A184822(n) = n + [n*t] + [n*t^2], where []=floor.
This is a special case of Clark Kimberling's results given in A184812.
FORMULA
Limit a(n)/n = t = 1.8392867552141611325518525646532866...
a(n) = n + floor(n*p/r) + floor(n*q/r), where p=t, q=t^2, r=t^3, and t is the tribonacci constant (see Clark Kimberling's formula in A184812).
EXAMPLE
Let t be the tribonacci constant, then t = 1 + 1/t + 1/t^2 where:
t = 1.8392867552..., t^2 = 3.3829757679..., t^3 = 6.2222625231...
PROG
(PARI) {a(n)=local(t=real(polroots(1+x+x^2-x^3)[1])); n+floor(n/t)+floor(n/t^2)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 22 2011
STATUS
approved