A280054 Index of first occurrence of n in A280053, the nachos numbers based on squares.

1, 2, 3, 4, 9, 23, 53, 193, 1012, 11428, 414069, 89236803, 281079668014, 49673575524946259, 3690344289594918623401179, 2363083530686659576336864121757607550, 1210869542685904980187672572977511794639836071291151196

Offset: 1

Author

N. J. A. Sloane, Jan 07 2017

Keywords

• nonn

Comments

Analysis from Lars Blomberg, Jan 08 2017 (Start)

Consider the sequence of sums of squares, q(n), n=1,2,3,... (A000330):

1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, ...

which has formula q(n) = n*(n+1)*(2*n+1)/6.

The term A280053(x) can be computed by repeatedly subtracting the largest q(n)<=x from x until 0 is reached. For example, 8 = 5+1+1+1, so A280053(8)=4

Note that A280054 is strictly increasing. Let r be the last term so far in A280054, and s the next term. We must find the smallest term in q such that s-q(n-1) = r, or s=q(n-1)+r. Therefore s will have one more phase than r, and it will be the smallest possible s.

We also require that s

Calculate n=floor(sqrt(r))+1 and from this we get s=q(n-1)+r.

Note that the q sequence need not be explicitly calculated and stored.

Examples:

r.........n....q(n-1).......q(n)........s..phases

4.........3.........5........14.........9.......5

9.........4........14........30........23.......6

23........5........30........55........53.......7

53........8.......140.......204.......193.......8

193......14.......819......1015......1012.......9

1012.....32.....10416.....11440.....11428......10

11428...107....402641....414090....414069......11

414069..644..88822734..89237470..89236803......12

...

The above values were confirmed by direct calculation.

(End)

Links

• Lars Blomberg, Table of n, a(n) for n = 1..24

Crossrefs

Cf. A280053.

Extensions

More terms from Lars Blomberg, Jan 08 2017

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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