A005041 A self-generating sequence. (Formerly M0258)

1, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18

Offset

0,3

Comments

See A008620 for run lengths: each k occurs A008620(k+2) times. - Reinhard Zumkeller, Mar 16 2012

References

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

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

James Propp, Problem 1047, Math. Mag., 52 (1979), 265.

Jeffrey Shallit, Letter to N. J. A. Sloane, Nov 10 1979. Attached: James Propp, Problem 1047, Math. Mag., 52 (1979), 265. [Annotated scanned copy]

Aaron Snook, Augmented Integer Linear Recurrences, Thesis, 2012. - From N. J. A. Sloane, Dec 19 2012

Formula

For any k in {0, 1, 2, ...} and r in {0, 1, 2), we have: if n = 6*k + (3/2)*k*(k-1) + r*(k+2), then a(n) = 3*k + r + 1. E.g., for k=3 and r=1, we have n = 6*3 + (3/2)*3*(3-1) + 1*(3+2) = 32 and so a(32) = 3*3 + 1 + 1 = 11. - Francois Jooste (phukraut(AT)hotmail.com), Mar 12 2002

Mathematica

Table[n+1, {n, 0, 20}, {Ceiling[(n+1)/3]+1}] // Flatten (* Jean-François Alcover, Dec 10 2014 *)

Prog

(Haskell)
a005041 n = a005041_list !! n
a005041_list = 1 : f 1 1 (tail ts) where
   f y i gs'@((j, a):gs) | i < j  = y : f y (i+1) gs'
                        | i == j = a : f a (i+1) gs
   ts = [(6*k + 3*k*(k-1) `div` 2 + r*(k+2), 3*k+r+1) |
         k <- [0..], r <- [0, 1, 2]]
-- Reinhard Zumkeller, Mar 16 2012

Crossrefs

Cf. A005038, A005039, A005040, A005043, A005044, A055086, A001462, A082462, A024417, A084500.

Sequence in context: A357021 A082462 A276581 * A030530 A084500 A084557

Adjacent sequences: A005038 A005039 A005040 * A005042 A005043 A005044

Keyword

nonn,nice,easy

Author

N. J. A. Sloane, Jeffrey Shallit

Extensions

More terms from Samuel Hilliard (sam_spade1977(AT)hotmail.com), Apr 11 2004

Status

approved