A173964 Sequence derived from a memorization technique.

1, 2, 1, 2, 3, 2, 3, 1, 2, 3, 4, 3, 4, 2, 3, 4, 1, 2, 3, 4, 5, 4, 5, 3, 4, 5, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 5, 6, 4, 5, 6, 3, 4, 5, 6, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 6, 7, 5, 6, 7, 4, 5, 6, 7, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8

Offset: 1

Author

Ibrahima Faye (ifaye2001(AT)yahoo.fr), Feb 22 2010

Keywords

• nonn

Comments

To memorize a poem composed of several verses, the following technique can be used. -Memorize verse 1. -Memorize verse 2. -Memorize (revise) verse 1, and then verse 2. -Memorize verse 3. -Memorize (revise) verse 2, and then verse 3. -Memorize (revise) verse 1, then verse 2, and then verse 3. -Memorize verse 4. -Memorize (revise) verse 3, and then verse 4. -Memorize (revise) verse 2, then verse 3, and then verse 4. -Memorize (revise) verse 1, then verse 2, then verse 3, and then verse 4. Etc. Listing down the verses in the order they are memorized or revised gives the sequence.

The sequence is self-similar: striking out the n first occurrences of each integer n gives the initial sequence.

The sequence can be created by writing down the strings 12, 123, 1234, 12345, etc, on separate rows, and then following each string with its last substring of length 2, then last substring of length 3, last substring of length 4, etc, stopping before the whole string is repeated. - Andrew Woods, Aug 03 2011

The first appearance of n >= 1 is at a(1+(n-1)*n*(n+1)/6). - Andrew Woods, Aug 03 2011

For n > 1: a(A000292(n)+1) = n.

Links

• Andrew Woods, Table of n, a(n) for n = 1..10000

Programs

Haskell

a173964 n = a173964_list !! (n-1)
a173964_list = concat $ [1] : f [[1]] where
   f xss = yss ++ f yss where
     yss = [y] : map (++ [y]) xss
     y = head (head xss) + 1
-- Reinhard Zumkeller, Nov 16 2013

Maple

S := n -> (1/6)*n*(n+1)*(n+2): invS := n -> ceil((1/3)*(81*n+3*sqrt(-3+729*n^2))^(1/3)+1/(81*n+3*sqrt(-3+729*n^2))^(1/3)-1):
A := n -> (1/2)*n*(n+1): invA := n -> floor(-1/2+(1/2)*sqrt(1+8*n)):
A173964 := n -> invS(n)-invA(n-1-S(invS(n)-1))+n-1-S(invS(n)-1)-A(invA(n-1-S(invS(n)-1)));