A256913 Enhanced squares representations for k = 0, 1, 2, ..., concatenated.

0, 1, 2, 3, 4, 4, 1, 4, 2, 4, 3, 4, 3, 1, 9, 9, 1, 9, 2, 9, 3, 9, 4, 9, 4, 1, 9, 4, 2, 16, 16, 1, 16, 2, 16, 3, 16, 4, 16, 4, 1, 16, 4, 2, 16, 4, 3, 16, 4, 3, 1, 25, 25, 1, 25, 2, 25, 3, 25, 4, 25, 4, 1, 25, 4, 2, 25, 4, 3, 25, 4, 3, 1, 25, 9, 25, 9, 1, 36

Offset: 0

Author

Clark Kimberling, Apr 14 2015

Keywords

• nonn
• easy
• tabf
• nice

Comments

Let B = {0,1,2,3,4,9,16,25,...}, so that B consists of the squares together with 2 and 3. We call B the enhanced basis of squares. Define R(0) = 0 and R(n) = g(n) + R(n - g(n)) for n > 0, where g(n) is the greatest number in B that is <= n. Thus, each n has an enhanced squares representation of the form R(n) = b(m(n)) + b(m(n-1)) + ... + b(m(k)), where b(n) > m(n-1) > ... > m(k) > 0, in which the last term, b(m(k)), is the trace.

The least n for which R(n) has 5 terms is given by R(168) = 144 + 16 + 4 + 3 + 1.

The least n for which R(n) has 6 terms is given by R(7224) = 7056 + 144 + 16 + 4 + 3 + 1.

Examples

			R(0) = 0
R(1) = 1
R(2) = 2
R(3) = 3
R(4) = 4
R(8) = 4 + 3 + 1
R(24) = 16 + 4 + 3 + 1
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000290, A256914 (trace), A256915 (number of terms), A256789 (minimal alternating squares representations).

Cf. A257053 (primes).

Programs

Haskell

a256913 n k = a256913_tabf !! n !! k
a256913_row n = a256913_tabf !! n
a256913_tabf = [0] : tail esr where
   esr = (map r [0..8]) ++
           f 9 (map fromInteger $ drop 3 a000290_list) where
     f x gs@(g:hs@(h:_))
       | x < h   = (g : genericIndex esr (x - g)) : f (x + 1) gs
       | otherwise = f x hs
     r 0 = []; r 8 = [4, 3, 1]
     r x = q : r (x - q) where q = [0,1,2,3,4,4,4,4,4] !! x
-- Reinhard Zumkeller, Apr 15 2015

Mathematica

b[n_] := n^2; bb = Insert[Table[b[n], {n, 0, 100}]  , 2, 3];
s[n_] := Table[b[n], {k, 1, 2 n + 1}];
h[1] = {0, 1, 2, 3}; h[n_] := Join[h[n - 1], s[n]];
g = h[100]; Take[g, 100]
r[0] = {0}; r[1] = {1}; r[2] = {2}; r[3] = {3}; r[8] = {4, 3, 1};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]];
t = Table[r[n], {n, 0, 120}] (* A256913, before concatenation *)
Flatten[t]  (* A256913 *)
Table[Last[r[n]], {n, 0, 120}]    (* A256914 *)
Table[Length[r[n]], {n, 0, 200}]  (* A256915 *)