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.

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A132852 Number of sequences {c(i), i=0..n} that form the initial terms of a self-convolution square of an integer sequence such that 0 < c(n) <= 2*c(n-1) for n>0 with c(0)=1.

Original entry on oeis.org

1, 1, 2, 4, 14, 62, 462, 5380, 105626, 3440686, 196429906, 19603795552, 3496015313038, 1120368106124268, 653253602487886098, 697073727912597623594, 1371575342274982257650434
Offset: 0

Views

Author

Paul D. Hanna, Sep 19 2007, Oct 06 2007

Keywords

Comments

Equals the number of nodes at generation n in the 2-convoluted tree. The minimal path in the 2-convoluted tree is A083952 and the maximal path is A132831. The 2-convoluted tree is defined as follows: tree of all finite sequences {c(k), k=0..n} that form the initial terms of a self-convolution square of some integer sequence such that 0 < c(n) <= 2*c(n-1) for n>0 with a(0)=1.

Examples

			a(n) counts the nodes in generation n of the following tree.
Generations 0..5 of the 2-convoluted tree are as follows;
The path from the root is shown, with child nodes enclosed in [].
GEN.0: [1];
GEN.1: 1->[2];
GEN.2: 1-2->[1,3];
GEN.3:
1-2-1->[2]
1-2-3->[2,4,6];
GEN.4:
1-2-1-2->[2,4]
1-2-3-2->[1,3]
1-2-3-4->[1,3,5,7]
1-2-3-6->[1,3,5,7,9,11];
GEN.5:
1-2-1-2-2->[2,4]
1-2-1-2-4->[2,4,6,8]
1-2-3-2-1->[2]
1-2-3-2-3->[2,4,6]
1-2-3-4-1->[2]
1-2-3-4-3->[2,4,6]
1-2-3-4-5->[2,4,6,8,10]
1-2-3-4-7->[2,4,6,8,10,12,14]
1-2-3-6-1->[2]
1-2-3-6-3->[2,4,6]
1-2-3-6-5->[2,4,6,8,10]
1-2-3-6-7->[2,4,6,8,10,12,14]
1-2-3-6-9->[2,4,6,8,10,12,14,16,18]
1-2-3-6-11->[2,4,6,8,10,12,14,16,18,20,22].
Each path in the tree from the root node forms the initial terms of a self-convolution square of a sequence with integer terms.

Links

Crossrefs

Cf. A132853, A132854, A132855, A132856.
Cf. A083952, A132831.

Extensions

Extended by Martin Fuller, Sep 24 2007.

A111603 Consider the array T(n, m) where the n-th row is the sequence of integer coefficients of A(x), where 1<=a(n)<=n, such that A(x)^(1/n) consists entirely of integer coefficients and where m is the (m+1)-th coefficient. This is the antidiagonal read from upper right to lower left.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 3, 4, 1, 1, 2, 1, 2, 5, 1, 1, 2, 3, 4, 5, 6, 1, 1, 1, 3, 3, 5, 3, 7, 1, 1, 2, 3, 4, 5, 2, 7, 8, 1, 1, 2, 3, 4, 1, 3, 7, 4, 9, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 1, 1, 3, 1, 5, 6, 7, 2, 3, 5, 11, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 1, 1, 3, 4, 5, 3, 1, 4, 9, 10
Offset: 1

Views

Author

Paul D. Hanna and Robert G. Wilson v, Aug 01 2005

Keywords

Examples

			Table begins
k= 0 1 2 3 4 5 6 7 8 9 10 11 12 13
n\
1| 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2| 1 2 1 2 2 2 1 2 2 2 1 2 1 2
3| 1 3 3 1 3 3 3 3 3 3 3 3 1 3
4| 1 4 2 4 3 4 4 4 1 4 4 4 3 4
5| 1 5 5 5 5 1 5 5 5 5 4 5 5 5
6| 1 6 3 2 3 6 6 6 3 4 6 6 6 6
7| 1 7 7 7 7 7 7 1 7 7 7 7 7 7
8| 1 8 4 8 2 8 4 8 7 8 8 8 4 8
9| 1 9 9 3 9 9 3 9 9 1 9 9 6 9
10| 1 10 5 10 10 2 5 10 10 10 3 10 5 10
11| 1 11 11 11 11 11 11 11 11 11 11 1 11 11
12| 1 12 6 4 9 12 4 12 12 8 6 12 6 12
13| 1 13 13 13 13 13 13 13 13 13 13 13 13 1
14| 1 14 7 14 7 14 14 2 7 14 14 14 14 14
15| 1 15 15 5 15 3 10 15 15 10 15 15 5 15
16| 1 16 8 16 4 16 8 16 10 16 8 16 12 16

Crossrefs

Cf. A111613, A083952, A083953, A083954, A083945, A083946, A083947, A083948, A083949, A083950, A084066, A084067.
Cf. A109626, A111604.

Programs

  • Mathematica
    f[n_] := f[n] = Block[{a}, a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[ a[j], {j, 0, 32}]]; g[n_, m_] := f[n][[m]]; Flatten[ Table[ f[i, n - i], {n, 15}, {i, n - 1, 1, -1}]]

A111363 a(n) = A108783(n) + 1.

Original entry on oeis.org

1, 3, 7, 11, 13, 27, 31, 33, 37, 51, 53, 57, 61, 63, 127, 131, 133, 137, 151, 153, 161, 165, 167, 171, 173, 175, 177, 181, 185, 193, 195, 199, 201, 203, 215, 217, 221, 227, 229, 231, 235, 237, 241, 243, 245, 261, 263, 265, 273, 275, 279, 283, 287, 289, 291, 295
Offset: 1

Views

Author

Paul D. Hanna and Robert G. Wilson v, Nov 08 2005

Keywords

Comments

Positions of the 1's in A083952 but indexing that sequence starting at 1. A108783 is the preferred version.

Crossrefs

Cf. A083952. See A108783 for another version.

Programs

  • Mathematica
    a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/2), {x, 0, l}], x]]] != True, k++ ]; k]; Position[ Table[a[n], {n, 0, 300}], 1] // Flatten

Formula

It appears that a(n)~4.1*n.

A111604 Consider the array T(n, m) where the n-th row is the sequence of integer coefficients of A(x), where 1<=a(n)<=n, such that A(x)^(1/n) consists entirely of integer coefficients and where m is the (m+1)-th coefficient. This is the antidiagonal read zig-zag.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 4, 3, 2, 1, 1, 2, 1, 2, 5, 1, 1, 6, 5, 4, 3, 2, 1, 1, 1, 3, 3, 5, 3, 7, 1, 1, 8, 7, 2, 5, 4, 3, 2, 1, 1, 2, 3, 4, 1, 3, 7, 4, 9, 1, 1, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 1, 3, 1, 5, 6, 7, 2, 3, 5, 11, 1, 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 1, 3, 4, 5, 3, 1, 4, 9, 10, 11
Offset: 1

Views

Author

Paul D. Hanna and Robert G. Wilson v, Aug 01 2005

Keywords

Comments

T(n,n)=T(n,n+2)=A111627.

Examples

			Table begins
\k...0...1....2....3....4....5....6....7....8....9...10...11...12...13
n\
1| 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2| 1 2 1 2 2 2 1 2 2 2 1 2 1 2
3| 1 3 3 1 3 3 3 3 3 3 3 3 1 3
4| 1 4 2 4 3 4 4 4 1 4 4 4 3 4
5| 1 5 5 5 5 1 5 5 5 5 4 5 5 5
6| 1 6 3 2 3 6 6 6 3 4 6 6 6 6
7| 1 7 7 7 7 7 7 1 7 7 7 7 7 7
8| 1 8 4 8 2 8 4 8 7 8 8 8 4 8
9| 1 9 9 3 9 9 3 9 9 1 9 9 6 9
10| 1 10 5 10 10 2 5 10 10 10 3 10 5 10
11| 1 11 11 11 11 11 11 11 11 11 11 1 11 11
12| 1 12 6 4 9 12 4 12 12 8 6 12 6 12
13| 1 13 13 13 13 13 13 13 13 13 13 13 13 1
14| 1 14 7 14 7 14 14 2 7 14 14 14 14 14
15| 1 15 15 5 15 3 10 15 15 10 15 15 5 15
16| 1 16 8 16 4 16 8 16 10 16 8 16 12 16

Crossrefs

Cf. A111613, A083952, A083953, A083954, A083945, A083946, A083947, A083948, A083949, A083950, A084066, A084067.
Cf. A109626, A111603.

Programs

  • Mathematica
    f[n_] := f[n] = Block[{a}, a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[a[j], {j, 0, 32}]]; g[n_, m_] := f[n][[m]];

A108335 A084202 read mod 4.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Jul 02 2005

Keywords

Crossrefs

Cf. A083952, A084202, A108336.

A134779 Integer coefficients of A(x), where a(n) is the least integer greater than a(n-1)+2 such that A(x)^(1/2) has integer coefficients.

Original entry on oeis.org

1, 4, 6, 8, 11, 14, 16, 18, 21, 24, 27, 30, 33, 36, 38, 40, 43, 46, 49, 52, 55, 58, 61, 64, 66, 68, 71, 74, 76, 78, 80, 82, 85, 88, 90, 92, 95, 98, 101, 104, 106, 108, 111, 114, 117, 120, 122, 124, 126, 128, 131, 134, 137, 140, 143, 146, 148, 150, 153, 156, 159, 162, 164
Offset: 0

Views

Author

Paul D. Hanna & Robert G. Wilson v, Nov 04 2007; definition corrected Sep 19 2008

Keywords

Comments

If the condition for a(n) is simply to be greater than a(n-1) then the sequence is A000027: the natural numbers.
a(n)~5/2*n.

Links

Crossrefs

Cf. A083952, A134780.

Programs

  • Mathematica
    a[n_] := a[n] = Block[{k = a[n - 1] + 2, s = Sum[ a[i]*x^i, {i, 0, n - 1}]}, While[ !IntegerQ@ Last@ CoefficientList[ Series[ Sqrt[s + k*x^n], {x, 0, n}], x], k++ ]; k]; a[0] = 1; Table[ a[n], {n, 0, 63}]

A108115 Let B(x) = Sum_{i >= 0} A108336(i)*x^i; sequence gives coefficients of B(x)^2.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 5, 2, 2, 2, 1, 2, 1, 2, 2, 2, 6, 2, 6, 6, 2, 6, 2, 2, 2, 2, 5, 2, 6, 6, 5, 14, 5, 6, 6, 2, 5, 2, 2, 2, 2, 6, 2, 6, 6, 2, 6, 2, 2, 2, 1, 2, 1, 2, 2, 2, 5, 2, 2, 2, 1, 2, 1, 2, 2, 2, 6, 2, 6, 6, 2, 6, 2, 2, 2, 2, 6, 2, 6, 6, 6, 14, 6, 10, 6, 6, 10, 6, 14, 10, 10, 18, 10, 18, 10, 10, 14
Offset: 0

Views

Author

N. J. A. Sloane, Jul 03 2005

Keywords

Links

Crossrefs

Cf. A108336.
a(n) mod 4 = A083952(n).

Programs

  • Maple
    S:= 0: SS:= 0:
for i from 0 to 100 do
  s:= coeff(SS, x, i) mod 4;
  if s = 0 or s = 3 then
     SS:= SS + 2*expand(S*x^i)+x^(2*i); S:= S + x^i;
  fi
od:
seq(coeff(SS, x, i), i=0..100); # Robert Israel, May 14 2019

A111581 G.f.: A(x) = ( G(x)^2 - G(x^2) - 2*x/(1-x^2) )/(4*x^2) where G(x) is the g.f. of A110627.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 14, 14, 15, 16, 16, 17, 17, 18, 19, 19, 21, 20, 21, 22, 22, 24, 24, 25, 26, 27, 29, 29, 31, 32, 32, 34, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 44, 45, 46, 47, 48, 49, 49, 50, 50, 51, 51, 52, 53, 53, 55, 55, 56, 57, 57
Offset: 0

Views

Author

Paul D. Hanna, Aug 28 2005

Keywords

Comments

A110627 is the bisection of A083952; surprisingly, the self-convolution of A110627 is congruent modulo 4 to A083952, which consists entirely of 1's and 2's.

Crossrefs

Cf. A110627, A083952.
Previous Showing 21-28 of 28 results.

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

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