A083952 -id:A083952 - cp's OEIS frontend

A084202 G.f. A(x) defined by: A(x)^2 consists entirely of integer coefficients between 1 and 2 (A083952); A(x) is the unique power series solution with A(0)=1.

1, 1, 0, 1, 0, 1, -1, 2, -2, 4, -6, 10, -16, 27, -44, 75, -127, 218, -375, 650, -1130, 1974, -3460, 6086, -10736, 18993, -33685, 59882, -106683, 190446, -340611, 610243, -1095102, 1968200, -3542468, 6384518, -11521308, 20815942, -37651528, 68176596, -123574852, 224204708, -407153894

Offset: 0

Paul D. Hanna, May 19 2003

sign

Limit a(n)/a(n+1) -> r = -0.530852489019085 where A(r)=0.

Let A_n(x) be the power series formed from the first n terms of this sequence. Then a(0) = 1, a(n) = floor(1 - [x^n] (A_(n-1)(x))^2/2). Replacing 2 with a larger integer k generates the related sequences A084203-A084212. - Charlie Neder, Jan 16 2019

Paul D. Hanna, Table of n, a(n) for n = 0..1024
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A083952, A084203-A084212.

a[n_] := a[n] = Block[{s = Sum[a[i]*x^i, {i, 0, n - 1}]}, If[ IntegerQ@ Last@ CoefficientList[ Series[ Sqrt[s + x^n], {x, 0, n}], x], 1, 2]]; Table[a[n], {n, 0, 42}]; CoefficientList[ Series[ Sqrt[ Sum[ a[i]*x^i, {i, 0, 42}]], {x, 0, 42}], x] (* Robert G. Wilson v, Nov 11 2007 *)

/* Using Charlie Neder's formula */
{a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); A[#A] = floor(1 - polcoeff( Ser(A)^2, #A-1)/2) ); A[n+1]}
for(n=0,50, print1(a(n),", ")) \\ Paul D. Hanna, Jan 17 2019

A108336 Unique sequence of 1's and 0's such that (Sum_{n >= 0} a(n)*x^n)^2 mod 4 has coefficients which are all 1's and 2's (A083952).

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0

Offset: 0

N. J. A. Sloane and Nadia Heninger, Jul 02 2005

Equals A084202 read mod 2.

Robert Israel, Table of n, a(n) for n = 0..10000
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A083952, A084202, A108335, A108337, A108340.

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

max = 98; (* a = A084202 *) a[n_] := a[n] = Block[{s = Sum[a[i]*x^i, {i, 0, n-1}]}, If[IntegerQ @ Last @ CoefficientList[Series[Sqrt[s + x^n], {x, 0, n}], x], 1, 2]]; Table[a[n], {n, 0, max}]; A108336 = CoefficientList[ Series[Sqrt[Sum[a[i]*x^i, {i, 0, max}]], {x, 0, max}], x] // Mod[#, 2]& (* Jean-François Alcover, Apr 01 2016, after Robert G. Wilson v *)

A108783 Positions of 1's in A083952, where A083952 gives the integer coefficients a(n) of A(x), where a(n) = 1 or 2 for all n, such that A(x)^(1/2) has only integer coefficients.

Original entry on oeis.org

0, 2, 6, 10, 12, 26, 30, 32, 36, 50, 52, 56, 60, 62, 126, 130, 132, 136, 150, 152, 160, 164, 166, 170, 172, 174, 176, 180, 184, 192, 194, 198, 200, 202, 214, 216, 220, 226, 228, 230, 234, 236, 240, 242, 244, 260, 262, 264, 272, 274, 278, 282, 286

Offset: 1

N. J. A. Sloane, following a suggestion from Paul D. Hanna, Jun 30 2005

nonn

Robert G. Wilson v, Table of n, a(n) for n = 1..1356

Cf. A083952, A108337, A108338. See A111363 for another version.

a[n_] := a[n] = Block[{s = Sum[a[i]*x^i, {i, 0, n - 1}]}, If[ IntegerQ@ Last@ CoefficientList[ Series[ Sqrt[s + x^n], {x, 0, n}], x], 1, 2]]; Union@ Table[ If[ a[n] == 1, n, 0], {n, 0, 300}] (* Robert G. Wilson v, Nov 25 2006 *)

A108783_upto(N=200)=[k-1 | k<-select(t->t==1, A083952_upto(N),1)] \\ M. F. Hasler, Jan 27 2025

A108340 A083952 read mod 2.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Jul 03 2005

nonn

A110627 Bisection of A083952 such that the self-convolution is congruent modulo 4 to A083952, which consists entirely of 1's and 2's.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 2

Offset: 0

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

nonn

Congruent modulo 2 to A084202 and A108336; the self-convolution of A084202 equals A083952.

Cf. A083952, A111581, A084202, A108336.

{a(n)=local(p=2,A,C,X=x+x*O(x^(p*n)));if(n==0,1, A=sum(i=0,n-1,a(i)*x^(p*i))+p*x*((1-x^(p-1))/(1-X))/(1-X^p); for(k=1,p,C=polcoeff((A+k*x^(p*n))^(1/p),p*n); if(denominator(C)==1,return(k);break)))}

a(n) = A083952(2*n) for n>=0. G.f. satisfies: A(x^2) = G(x) - 2*x/(1-x^2), where G(x) is the g.f. of A083952. G.f. satisfies: A(x)^2 = A(x^2) + 2*x/(1-x^2) + 4*x^2*H(x) where H(x) is the g.f. of A111581.

A113737 Decimal expansion of the constant given by Sum_{n>=0} A083952(n)/2^(n+1).

Original entry on oeis.org

1, 3, 6, 6, 5, 7, 7, 1, 4, 0, 3, 9, 7, 5, 6, 6, 2, 7, 3, 7, 3, 4, 8, 0, 7, 0, 7, 4, 1, 0, 2, 8, 0, 3, 8, 8, 1, 0, 2, 6, 3, 9, 9, 2, 3, 9, 1, 9, 5, 5, 5, 9, 6, 1, 7, 4, 1, 4, 7, 8, 3, 0, 1, 7, 7, 7, 2, 7, 4, 2, 8, 7, 4, 7, 0, 4, 0, 5, 5, 2, 5, 1, 4, 4, 8, 7, 4, 4, 3, 4, 0, 5, 7, 4, 3, 3, 8, 6, 3, 5, 1, 2, 5, 3, 1, 3

Offset: 1

Robert G. Wilson v, Nov 08 2005

			1.3665771403975662737348070741028038810263992391955596174147830177...

Cf. A083952.

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]; t = Table[a[n], {n, 0, 370}]; RealDigits[ FromDigits[{t - 1, 0}, 2], 10, 111][[1]]

Name changed and comments removed by Paul D. Hanna, Apr 19 2013

A111355 Consider the sequence defined in A083952 as a binary string of the digits 1 and 2. Then a(n) is the beginning position of the first occurrence of exactly 2n-1 consecutive 2's.

Original entry on oeis.org

2, 4, 222, 154, 754, 204, 14, 246, 1300, 3642

Offset: 1

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

nonn

Strings of ones are always isolated and at an odd position.

Cf. A083952.

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]; p = Flatten[ Position[ Table[ a[n], {n, 0, 4150}], 1]]; f[n_] := Block[{k = n, m = 1}, While[m < Length[p] && p[[m + 1]] - p[[m]] != n, m++ ]; If[m < Length[p] - k, p[[m]] + 1, 0]]; Table[ f[2n], {n, 10}]

A111372 Consider the sequence defined in A083952 as a string of the digits 1 and 2. Then a(n) is the beginning position of the first occurrence of just n consecutive 1's at every other position.

Original entry on oeis.org

6, 0, 198, 170, 384, 992, 542, 648, 762, 5070

Offset: 0

Robert G. Wilson v, Nov 08 2005

nonn

			a(0) = 6 because A083952(6) is an isolated 1,
a(1) = 0 because A083952(0) = A083952(2) = 1 but A083952(4) = 2,
a(2) = 198 because A083952(198) = A083952(200) = A083952(202) = 1, but A083952(196) = A083952(204) = 2, etc.

Cf. A083952, A111363.

a[n_] := a[n] = Block[{s = Sum[ a[i]*x^i, {i, 0, n - 1}]}, If[ IntegerQ@ Last@ CoefficientList[ Series[ Sqrt[s + x^n], {x, 0, n}], x], 1, 2]]; t = Union@ Table[ If[ a[n] == 1, n, 0], {n, 0, 5506}]; f[n_] := Block[{k = n, m = 2}, If[n == 1, 0, While[m < Length@t && t[[m - 1]] + 2 == t[[m]] || t[[m]] + 2k != t[[m + k]] || t[[m + k]] + 2 == t[[m + k + 1]], m++ ]; If[m < Length@t - k, t[[m]], 0]]]; Table[ f[n], {n, 0, 9}]

Corrected and extended by Robert G. Wilson v, Nov 27 2006

A113738 Continued fraction expansion of the constant given by Sum_{n>=0} A083952(n)/2^(n+1).

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 12, 22, 1, 1, 4, 4, 1, 2, 1, 3, 2, 2, 11, 1, 202, 1, 5, 1, 2, 1, 5, 1, 2, 1, 1, 1, 1, 3, 2, 1, 2, 2, 5, 2, 1, 1, 2, 1, 3, 1, 1, 15, 2, 1, 12, 1, 2, 2, 1, 1, 1, 2, 1, 3, 1, 2, 10, 1, 6, 5, 1, 2, 1, 2, 6, 1, 1, 2, 8, 2, 19, 3, 1, 1, 1, 2, 1, 1, 1, 5, 1, 2, 2, 1, 3, 1, 6, 12, 2, 4, 1, 5, 1, 2, 4

Offset: 0

Robert G. Wilson v, Nov 08 2005

Cf. A083952, A113737.

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]; t = Table[a[n], {n, 0, 370}]; ContinuedFraction[ FromDigits[{t - 1, 0}, 2]]

Entry revised by Paul D. Hanna, Apr 19 2013

A109626 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 lower left to upper right.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 3, 2, 1, 1, 5, 2, 1, 2, 1, 1, 6, 5, 4, 3, 2, 1, 1, 7, 3, 5, 3, 3, 1, 1, 1, 8, 7, 2, 5, 4, 3, 2, 1, 1, 9, 4, 7, 3, 1, 4, 3, 2, 1, 1, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 11, 5, 3, 2, 7, 6, 5, 1, 3, 1, 1, 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 13, 6, 11, 10, 9, 4, 1, 3, 5

Offset: 1

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

			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

G. C. Greubel, Antidiagonals n = 1..100, flattened

Rows: A000012, A083952, A083953, A083954, A083945, A083946, A083947, A083948, A083949, A083950, A084066, A084067.
Columns: A000012, A111627, A026741, A051176, A111607, A060791, A111608, A106608, A111609, A111610, A111611, A106612, A106614, A106618, A106620.
Diagonals: A000027 (main), A111614 (first upper), A111627 (2nd), A111615 (3rd), A111618 (first lower), A111623 (2nd).
Other diagonals: A005408 (T(2*n-1, n)), A111626, A111627, A111628, A111629, A111630.
Cf. A111603, A111604.

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}]];
T[n_, m_]:= f[n][[m]];
Flatten[Table[T[i,n-i], {n,15}, {i,n-1,1,-1}]]

A109626_row(n, len=40)={my(A=1, m); vector(len, k, if(k>m=1, while(denominator(polcoeff(sqrtn(O(x^k)+A+=x^(k-1), n), k-1))>1, m++); m, 1))} \\ M. F. Hasler, Jan 27 2025

When m is prime, column m is T(n,m) = n/gcd(m, n) = numerator of n/(n+m). - M. F. Hasler, Jan 27 2025

cp's OEIS Frontend

A084202 G.f. A(x) defined by: A(x)^2 consists entirely of integer coefficients between 1 and 2 (A083952); A(x) is the unique power series solution with A(0)=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A108336 Unique sequence of 1's and 0's such that (Sum_{n >= 0} a(n)*x^n)^2 mod 4 has coefficients which are all 1's and 2's (A083952).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A108783 Positions of 1's in A083952, where A083952 gives the integer coefficients a(n) of A(x), where a(n) = 1 or 2 for all n, such that A(x)^(1/2) has only integer coefficients.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A108340 A083952 read mod 2.

Views

Author

Keywords

A110627 Bisection of A083952 such that the self-convolution is congruent modulo 4 to A083952, which consists entirely of 1's and 2's.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

A113737 Decimal expansion of the constant given by Sum_{n>=0} A083952(n)/2^(n+1).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions

A111355 Consider the sequence defined in A083952 as a binary string of the digits 1 and 2. Then a(n) is the beginning position of the first occurrence of exactly 2n-1 consecutive 2's.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A111372 Consider the sequence defined in A083952 as a string of the digits 1 and 2. Then a(n) is the beginning position of the first occurrence of just n consecutive 1's at every other position.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions

A113738 Continued fraction expansion of the constant given by Sum_{n>=0} A083952(n)/2^(n+1).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Extensions

A109626 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 lower left to upper right.

Views

Author