A005041 A self-generating sequence.

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

Author

N. J. A. Sloane, Jeffrey Shallit

Keywords

• nonn
• nice
• easy

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

Crossrefs

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

Programs

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

Mathematica

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

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

Extensions

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