cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Apr 14 2015

Keywords

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

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 *)

A256915 Length of the enhanced squares representation of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 2, 3, 3, 1, 2, 2, 2, 2, 3, 3, 3, 4, 1, 2, 2, 2, 2, 3, 3, 3, 4, 2, 3, 1, 2, 2, 2, 2, 3, 3, 3, 4, 2, 3, 3, 3, 1, 2, 2, 2, 2, 3, 3, 3, 4, 2, 3, 3, 3, 3, 4, 1, 2, 2, 2, 2, 3, 3, 3, 4, 2, 3, 3, 3, 3, 4, 4, 2, 1, 2, 2, 2, 2
Offset: 0

Views

Author

Clark Kimberling, Apr 14 2015

Keywords

Comments

See A256913 for definitions.

Examples

			R(0) = 0, so length = 1.
R(1) = 1, so length = 1.
R(8) = 4 + 3 + 1, so length = 3.
R(7224) = 7056 + 144 + 16 + 4 + 3 + 1, so length = 6.

Links

Crossrefs

Cf. A000290, A256913, A256914 (trace).
Cf. A257071.

Programs

  • Haskell
    a256915 = length . a256913_row  -- 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 *)

A257047 Numbers not having trace 1 in their enhanced squares representation, see A256913.

Original entry on oeis.org

0, 2, 3, 4, 6, 7, 9, 11, 12, 13, 15, 16, 18, 19, 20, 22, 23, 25, 27, 28, 29, 31, 32, 34, 36, 38, 39, 40, 42, 43, 45, 47, 48, 49, 51, 52, 53, 55, 56, 58, 60, 61, 62, 64, 66, 67, 68, 70, 71, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 90, 92, 93, 94, 96
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 15 2015

Keywords

Comments

A256914(a(n)) != 1.

Links

Crossrefs

Cf. A256913, A256914, A257046 (complement).

Programs

  • Haskell
    a257047 n = a257047_list !! (n-1)
a257047_list = filter ((/= 1) . a256914) [0..]

A257046 Numbers having trace 1 in their enhanced squares representation, see A256913.

Original entry on oeis.org

1, 5, 8, 10, 14, 17, 21, 24, 26, 30, 33, 35, 37, 41, 44, 46, 50, 54, 57, 59, 63, 65, 69, 72, 74, 78, 82, 86, 89, 91, 95, 98, 101, 105, 108, 110, 114, 117, 122, 126, 129, 131, 135, 138, 142, 145, 149, 152, 154, 158, 161, 165, 168, 170, 174, 177, 179, 183, 186
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 15 2015

Keywords

Comments

A256914(a(n)) = 1.

Links

Crossrefs

Cf. A256913, A256914, A257047 (complement).

Programs

  • Haskell
    a257046 n = a257046_list !! (n-1)
a257046_list = filter ((== 1) . a256914) [0..]

A257070 Traces of primes in enhanced squares representation, cf. A256913.

Original entry on oeis.org

2, 3, 1, 3, 2, 4, 1, 3, 3, 4, 2, 1, 1, 3, 2, 4, 1, 3, 3, 3, 9, 2, 2, 1, 16, 1, 3, 3, 9, 4, 2, 1, 16, 2, 1, 3, 4, 3, 3, 4, 1, 3, 2, 1, 1, 3, 2, 2, 2, 4, 1, 1, 16, 1, 1, 3, 4, 2, 1, 25, 2, 4, 2, 2, 1, 3, 3, 4, 3, 25, 4, 1, 2, 3, 2, 2, 3, 36, 1, 9, 3, 1, 2, 1
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 15 2015

Keywords

Comments

a(n) = A256914(A000040(n)).

Links

Crossrefs

Cf. A257053, A256913, A000040, A256914.

Programs

  • Haskell
    a257070 = last . a257053_row
Showing 1-5 of 5 results.

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