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A184822
a(n) = n + floor(n*t) + floor(n*t^2), where t is the tribonacci constant.
3
5, 11, 18, 24, 30, 37, 42, 49, 55, 61, 68, 74, 79, 86, 92, 99, 105, 111, 117, 123, 130, 136, 142, 149, 154, 160, 167, 173, 180, 186, 192, 198, 204, 211, 217, 223, 230, 235, 241, 248, 254, 261, 267, 272, 279, 285, 291, 298, 304, 310, 316, 322, 329, 335, 342, 348, 353, 360, 366, 372, 379, 385, 391, 397, 403, 410, 416, 423, 428, 434, 441, 447, 453, 460, 465, 472, 478, 484, 491, 497, 503, 509, 515, 522, 528, 534, 541, 546, 553, 559, 565, 572, 578, 583, 590, 596, 603, 609, 615, 621, 627, 634, 640, 646, 653, 658, 664, 671, 677, 684, 690, 696, 702, 708, 715, 721, 727, 734, 739, 745, 752, 758, 765, 771
OFFSET
1,1
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^3 = 6.2222625231...
a(n) = n + floor(n*q/p) + floor(n*r/p), 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^3 = 1 + t + 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