A256915 -id:A256915 - cp's OEIS frontend

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

Clark Kimberling, Apr 14 2015

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.

			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

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

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

Cf. A257053 (primes).

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

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 *)

A256914 Trace of the enhanced squares representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 2, 3, 1, 9, 1, 2, 3, 4, 1, 2, 16, 1, 2, 3, 4, 1, 2, 3, 1, 25, 1, 2, 3, 4, 1, 2, 3, 1, 9, 1, 36, 1, 2, 3, 4, 1, 2, 3, 1, 9, 1, 2, 3, 49, 1, 2, 3, 4, 1, 2, 3, 1, 9, 1, 2, 3, 4, 1, 64, 1, 2, 3, 4, 1, 2, 3, 1, 9, 1, 2, 3, 4, 1, 2, 16, 81, 1, 2

Offset: 0

Clark Kimberling, Apr 14 2015

See A256913 for definitions.

a(A257046(n)) = 1; a(A257047(n)) != 1. - Reinhard Zumkeller, Apr 15 2015

			R(0) = 0, so trace = 0.
R(1) = 1, so trace = 1.
R(8) = 4 + 3 + 1, so trace = 1.
R(43) = 36 + 4 + 3, so trace = 3.

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

Cf. A000290, A256913, A256915 (number of terms).

Cf. A257046, A257047, A257070.

a256914 = last . a256913_row  -- Reinhard Zumkeller, Apr 15 2015

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 *)

A257071 Length of enhanced squares representation of n-th prime, cf. A256913.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 2, 4, 2, 4, 2, 2, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 4, 2, 3, 3, 4, 5, 2, 2, 4, 3, 2, 2, 4, 4, 2, 3, 2, 3, 3, 4, 4, 2, 3, 2, 3, 4, 5, 3, 3, 3, 4, 2, 3, 4, 3, 3, 3, 4, 3, 2, 2, 2, 3, 4, 4, 5, 3, 2

Offset: 1

Reinhard Zumkeller, Apr 15 2015

nonn

a(n) = A256915(A000040(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A257053, A256913, A000040, A256915.

Haskell
```
a257071 = length . a257053_row
```

cp's OEIS Frontend

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

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A256914 Trace of the enhanced squares representation of n.

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A257071 Length of enhanced squares representation of n-th prime, cf. A256913.

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