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 1-9 of 9 results.

A110316 a(n) is the number of different shapes of balanced binary trees with n nodes. The tree is balanced if the total number of nodes in the left and right branch of every node differ by at most one.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 4, 1, 8, 16, 32, 16, 32, 16, 8, 1, 16, 64, 256, 256, 1024, 1024, 1024, 256, 1024, 1024, 1024, 256, 256, 64, 16, 1, 32, 256, 2048, 4096, 32768, 65536, 131072, 65536, 524288, 1048576, 2097152, 1048576, 2097152, 1048576, 524288, 65536, 524288
Offset: 0

Views

Author

Jeffrey Barnett, Jun 23 2007

Keywords

Comments

The value of a(n) is always a power of 2.

References

  • Hsien-Kuei Hwang, S Janson, TH Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint, 2016; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585

Links

Crossrefs

Column k=2 of A221857. - Alois P. Heinz, Apr 17 2013
Cf. A000225.

Programs

  • Maple
    a:= proc(n) option remember; local r; `if`(n<2, 1,
      `if`(irem(n, 2, 'r')=0, 2*a(r)*a(r-1), a(r)^2))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 10 2013
  • Mathematica
    a[0] = a[1] = 1; a[n_] := a[n] = If[EvenQ[n], 2 a[n/2] a[n/2-1], a[(n-1)/2 ]^2]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 31 2016 *)
  • Python
    def A110316(n): return 1<<(k:=n+1)-(sum(i.bit_count() for i in range(1,k))<<1)+k*(m:=k.bit_length())-(1<Chai Wah Wu, Mar 02 2023

Formula

a(0) = a(1) = 1; a(2*n) = 2*a(n)*a(n-1); a(2*n+1) = a(n)*a(n).
a(n) = 1 <=> n in { A000225 }. - Orson R. L. Peters, Mar 12 2024

A131889 a(n) is the number of shapes of balanced trees with constant branching factor 3 and n nodes. The node is balanced if the size, measured in nodes, of each pair of its children differ by at most one node.

Original entry on oeis.org

1, 1, 3, 3, 1, 9, 27, 27, 81, 81, 27, 27, 9, 1, 27, 243, 729, 6561, 19683, 19683, 59049, 59049, 19683, 177147, 531441, 531441, 1594323, 1594323, 531441, 531441, 177147, 19683, 59049, 59049, 19683, 19683, 6561, 729, 243, 27, 1, 81, 2187, 19683, 531441, 4782969
Offset: 0

Views

Author

Jeffrey Barnett, Jul 24 2007

Keywords

Comments

a(n) is always an integer power of 3.

Links

Crossrefs

Cf. A110316, A131890, A131891, A131892, A131893.
Column k=3 of A221857. - Alois P. Heinz, Apr 17 2013

Programs

  • Maple
    a:= proc(n) option remember; local m, r; if n<2 then 1 else
      r:= iquo(n-1, 3, 'm'); binomial(3, m) *a(r+1)^m *a(r)^(3-m) fi
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 10 2013
  • Mathematica
    a[n_, k_] := a[n, k] = Module[{m, r}, If[n < 2 || k == 1, 1, If[k == 0, 0, {r, m} = QuotientRemainder[n - 1, k]; Binomial[k, m]*a[r + 1, k]^m*a[r, k]^(k - m)]]];
a[n_] := a[n, 3];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

Formula

a(0) = a(1) = 1; a(3n+1+m) = (3 choose m) * a(n+1)^m * a(n)^(3-m), where n >= 0 and 0 <= m <= 3.

A131890 a(n) is the number of shapes of balanced trees with constant branching factor 4 and n nodes. The node is balanced if the size, measured in nodes, of each pair of its children differ by at most one node.

Original entry on oeis.org

1, 1, 4, 6, 4, 1, 16, 96, 256, 256, 1536, 3456, 3456, 1296, 3456, 3456, 1536, 256, 256, 96, 16, 1, 64, 1536, 16384, 65536, 1572864, 14155776, 56623104, 84934656, 905969664, 3623878656, 6442450944, 4294967296, 17179869184, 25769803776, 17179869184, 4294967296
Offset: 0

Views

Author

Jeffrey Barnett, Jul 24 2007

Keywords

Links

Crossrefs

Cf. A110316, A131889, A131891, A131892, A131893.
Column k=4 of A221857. - Alois P. Heinz, Apr 17 2013

Programs

  • Maple
    a:= proc(n) option remember; local m, r; if n<2 then 1 else
      r:= iquo(n-1, 4, 'm'); binomial(4, m) *a(r+1)^m *a(r)^(4-m) fi
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 10 2013
  • Mathematica
    a[n_, k_] := a[n, k] = Module[{m, r}, If[n < 2 || k == 1, 1, If[k == 0, 0, {r, m} = QuotientRemainder[n - 1, k]; Binomial[k, m]*a[r + 1, k]^m*a[r, k]^(k - m)]]];
a[n_] := a[n, 4];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

Formula

a(0) = a(1) = 1; a(4n+1+m) = (4 choose m) * a(n+1)^m * a(n)^(4-m), where n >= 0 and 0 <= m <= 4.

A131891 a(n) is the number of shapes of balanced trees with constant branching factor 5 and n nodes. The node is balanced if the size, measured in nodes, of each pair of its children differ by at most one node.

Original entry on oeis.org

1, 1, 5, 10, 10, 5, 1, 25, 250, 1250, 3125, 3125, 31250, 125000, 250000, 250000, 100000, 500000, 1000000, 1000000, 500000, 100000, 250000, 250000, 125000, 31250, 3125, 3125, 1250, 250, 25, 1, 125, 6250, 156250, 1953125, 9765625, 488281250, 9765625000
Offset: 0

Views

Author

Jeffrey Barnett, Jul 24 2007

Keywords

Links

Crossrefs

Cf. A110316, A131889, A131890, A131892, A131893.
Column k=5 of A221857. - Alois P. Heinz, Apr 17 2013

Programs

  • Maple
    a:= proc(n) option remember; local m, r; if n<2 then 1 else
      r:= iquo(n-1, 5, 'm'); binomial(5, m) *a(r+1)^m *a(r)^(5-m) fi
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 10 2013
  • Mathematica
    a[n_, k_] := a[n, k] = Module[{m, r}, If[n < 2 || k == 1, 1, If[k == 0, 0, {r, m} = QuotientRemainder[n - 1, k]; Binomial[k, m]*a[r + 1, k]^m*a[r, k]^(k - m)]]];
a[n_] := a[n, 5];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

Formula

a(0) = a(1) = 1; a(5n+1+m) = (5 choose m) * a(n+1)^m * a(n)^(5-m), where n >= 0 and 0 <= m <= 5.

A131892 a(n) is the number of shapes of balanced trees with constant branching factor 6 and n nodes. The node is balanced if the size, measured in nodes, of each pair of its children differ by at most one node.

Original entry on oeis.org

1, 1, 6, 15, 20, 15, 6, 1, 36, 540, 4320, 19440, 46656, 46656, 699840, 4374000, 14580000, 27337500, 27337500, 11390625, 91125000, 303750000, 540000000, 540000000, 288000000, 64000000, 288000000, 540000000, 540000000, 303750000, 91125000, 11390625, 27337500
Offset: 0

Views

Author

Jeffrey Barnett, Jul 24 2007

Keywords

Links

Crossrefs

Cf. A110316, A131889, A131890, A131891, A131893.
Column k=6 of A221857. - Alois P. Heinz, Apr 17 2013

Programs

  • Maple
    a:= proc(n) option remember; local m, r; if n<2 then 1 else
      r:= iquo(n-1, 6, 'm'); binomial(6, m) *a(r+1)^m *a(r)^(6-m) fi
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 10 2013
  • Mathematica
    a[n_, k_] := a[n, k] = Module[{m, r}, If[n < 2 || k == 1, 1, If[k == 0, 0, {r, m} = QuotientRemainder[n - 1, k]; Binomial[k, m]*a[r + 1, k]^m*a[r, k]^(k - m)]]];
a[n_] := a[n, 6];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

Formula

a(0) = a(1) = 1; a(6n+1+m) = (6 choose m) * a(n+1)^m * a(n)^(6-m), where n >= 0 and 0 <= m <= 6.

A131893 a(n) is the number of shapes of balanced trees with constant branching factor 7 and n nodes. The node is balanced if the size, measured in nodes, of each pair of its children differ by at most one node.

Original entry on oeis.org

1, 1, 7, 21, 35, 35, 21, 7, 1, 49, 1029, 12005, 84035, 352947, 823543, 823543, 17294403, 155649627, 778248135, 2334744405, 4202539929, 4202539929, 1801088541, 21012699645, 105063498225, 291843050625, 486405084375, 486405084375, 270225046875, 64339296875
Offset: 0

Views

Author

Jeffrey Barnett, Jul 24 2007

Keywords

Links

Crossrefs

Cf. A110316, A131889, A131890, A131891, A131892.
Column k=7 of A221857. - Alois P. Heinz, Apr 17 2013

Programs

  • Maple
    a:= proc(n) option remember; local m, r; if n<2 then 1 else
      r:= iquo(n-1, 7, 'm'); binomial(7, m) *a(r+1)^m *a(r)^(7-m) fi
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 10 2013
  • Mathematica
    a[n_, k_] := a[n, k] = Module[{m, r}, If[n < 2 || k == 1, 1, If[k == 0, 0, {r, m} = QuotientRemainder[n - 1, k]; Binomial[k, m]*a[r + 1, k]^m*a[r, k]^(k - m)]]];
a[n_] := a[n, 7];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

Formula

a(0) = a(1) = 1; a(7n+1+m) = (7 choose m) * a(n+1)^m * a(n)^(7-m), where n >= 0 and 0 <= m <= 7.

A229393 Number of shapes of balanced 8-ary trees with n nodes, where a tree is balanced if the total number of nodes in subtrees corresponding to the branches of any node differ by at most one.

Original entry on oeis.org

1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 64, 1792, 28672, 286720, 1835008, 7340032, 16777216, 16777216, 469762048, 5754585088, 40282095616, 176234168320, 493455671296, 863547424768, 863547424768, 377801998336, 6044831973376, 42313823813632, 169255295254528
Offset: 0

Views

Author

Alois P. Heinz, Sep 21 2013

Keywords

Comments

a(n) = 1 for n in { A023001 }.

Links

Crossrefs

Column k=8 of A221857.

Programs

  • Maple
    a:= proc(n) option remember; local m, r; if n<2 then 1 else
      r:= iquo(n-1, 8, 'm'); binomial(8, m) *a(r+1)^m *a(r)^(8-m) fi
    end:
seq(a(n), n=0..73);

A229394 Number of shapes of balanced 9-ary trees with n nodes, where a tree is balanced if the total number of nodes in subtrees corresponding to the branches of any node differ by at most one.

Original entry on oeis.org

1, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 81, 2916, 61236, 826686, 7440174, 44641044, 172186884, 387420489, 387420489, 13947137604, 223154201664, 2082772548864, 12496635293184, 49986541172736, 133297443127296, 228509902503936, 228509902503936, 101559956668416
Offset: 0

Views

Author

Alois P. Heinz, Sep 21 2013

Keywords

Comments

a(n) = 1 for n in { A002452 }.

Links

Crossrefs

Column k=9 of A221857.
Cf. A175673.

Programs

  • Maple
    a:= proc(n) option remember; local m, r; if n<2 then 1 else
      r:= iquo(n-1, 9, 'm'); binomial(9, m) *a(r+1)^m *a(r)^(9-m) fi
    end:
seq(a(n), n=0..91);

A229395 Number of shapes of balanced 10-ary trees with n nodes, where a tree is balanced if the total number of nodes in subtrees corresponding to the branches of any node differ by at most one.

Original entry on oeis.org

1, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 100, 4500, 120000, 2100000, 25200000, 210000000, 1200000000, 4500000000, 10000000000, 10000000000, 450000000000, 9112500000000, 109350000000000, 861131250000000, 4650108750000000, 17437907812500000
Offset: 0

Views

Author

Alois P. Heinz, Sep 21 2013

Keywords

Comments

a(n) = 1 for n in { A002275 }.

Links

Crossrefs

Column k=10 of A221857.

Programs

  • Maple
    a:= proc(n) option remember; local m, r; if n<2 then 1 else
      r:= iquo(n-1, 10, 'm'); binomial(10, m) *a(r+1)^m *a(r)^(10-m) fi
    end:
seq(a(n), n=0..111);
Showing 1-9 of 9 results.

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