A003663 a(n) is smallest number != a(j) + a(k), j < k and a(1) = 1, a(2) = 6. (Formerly M4066)

1, 6, 8, 10, 12, 15, 17, 19, 24, 26, 28, 33, 35, 37, 42, 44, 46, 51, 53, 55, 60, 62, 64, 69, 71, 73, 78, 80, 82, 87, 89, 91, 96, 98, 100, 105, 107, 109, 114, 116, 118, 123, 125, 127, 132, 134, 136, 141, 143, 145, 150, 152, 154, 159, 161, 163, 168, 170, 172, 177, 179

Offset

1,2

Comments

Numbers congruent to {1, 6, 8} mod 9 plus the number 12.

References

R. K. Guy, "s-Additive sequences", preprint, 1994.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

S. R. Finch, Are 0-additive sequences always regular?, Amer. Math. Monthly, 99 (1992), 671-673.

R. K. Guy, s-Additive sequences, Preprint, 1994. (Annotated scanned copy)

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Formula

From Chai Wah Wu, Feb 21 2018: (Start)

a(n) = a(n-1) + a(n-3) - a(n-4) for n > 9.

G.f.: x*(2*x^8 + x^5 - 3*x^4 + x^3 + 2*x^2 + 5*x + 1)/(x^4 - x^3 - x + 1). (End)

Mathematica

f[s_List, j_Integer] := Block[{k = s[[-1]] + 1, ss = Union[Plus @@@ Subsets[s, {j}]]}, While[ MemberQ[ss, k], k++]; Append[s, k]]; Nest[ f[#, 2] &, {1, 6}, 65] (* Robert G. Wilson v, Jul 05 2014 *)
LinearRecurrence[{1, 0, 1, -1}, {1, 6, 8, 10, 12, 15, 17, 19, 24}, 70] (* Harvey P. Dale, Jul 25 2018 *)

Prog

(Magma) I:=[1, 6, 8, 10, 12, 15, 17, 19, 24]; [n le 9 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..70]]; // Vincenzo Librandi, Feb 22 2018

Crossrefs

Cf. A199162, A060469, A060470, A060471, A060472.

Cf. A003662, A005408, A026471, A026474, A033627, A051039, A051040, A244151.

Sequence in context: A026286 A187085 A303580 * A075396 A269135 A369666

Adjacent sequences: A003660 A003661 A003662 * A003664 A003665 A003666

Keyword

nonn

Author

N. J. A. Sloane, Mira Bernstein

Extensions

Name clarified by David A. Corneth, Mar 13 2023

Status

approved