A084202 -id:A084202 - cp's OEIS frontend

A108337 Positions of the odd terms in A084202, which gives the coefficients b(n) such that (B(x))^2 has coefficients 1 and 2.

0, 1, 3, 5, 6, 13, 15, 16, 18, 25, 26, 28, 30, 31, 63, 65, 66, 68, 75, 76, 80, 82, 83, 85, 86, 87, 88, 90, 92, 96, 97, 99, 100, 101, 107, 108, 110, 113, 114, 115, 117, 118, 120, 121, 122, 130, 131, 132, 136, 137, 139, 141, 143, 144, 145, 147, 149, 150

Offset: 0

N. J. A. Sloane, Jul 02 2005

nonn

Also positions of 1's in A108336, which gives the coefficients of b(n) such that (B(x))^2 has coefficients 1 or 2 mod 4.

Equals A108783(n)/2.

Additional comments from Nadia Heninger, Jul 14 2007

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

N. J. A. Sloane, Jul 02 2005

Cf. A083952, A084202, A108336.

A083952 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

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

Offset: 0

Paul D. Hanna, May 09 2003

More generally, the sequence "integer coefficients of A(x), where 1<=a(n)<=m, such that A(x)^(1/m) consists entirely of integer coefficients", appears to have a unique solution for all m. [That is true - see Theorem 17 of Heninger-Rains-Sloane (2006). - N. J. A. Sloane, Aug 27 2015]

Is this sequence periodic? [It is not periodic for m = 2 or 3. Larger cases remain open. - N. J. A. Sloane, Aug 27 2015]

Robert G. Wilson v, 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; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A084202 (A(x)^(1/2)), A108335 (A084202 mod 4), A108336 (A084202 mod 2), A108340 (a(n) mod 2). Positions of 1's: A108783.

Cf. A083953, A083954, A083945, A083946.

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, 104}] (* Robert G. Wilson v, Nov 25 2006 *)
s = 0; a[n_] := a[n] = Block[{}, If[IntegerQ@ Last@ CoefficientList[ Series[ Sqrt[s + x^n], {x, 0, n}], x], s = s + x^n; 1, s = s + 2 x^n; 2]]; Table[ a@n, {n, 0, 104}] (* Robert G. Wilson v, Sep 08 2007 *)

A083952_upto(N=99)=vector(N+1, n, if(n>1, (denominator(polcoeff(sqrt(O(x^n)+N+=x^(n-1)),n-1))>1 && N+=x^(n-1))+1, N=1)) \\ M. F. Hasler, Jan 27 2025

More terms from N. J. A. Sloane, Jul 02 2005

A084212 G.f. A(x) defined by: A(x)^12 consists entirely of integer coefficients between 1 and 12 (A084067); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, -5, 37, -310, 2795, -26352, 256257, -2548875, 25793149, -264579518, 2743935678, -28716005918, 302812817148, -3213908529802, 34301475340630, -367873673112308, 3962187547336323

Offset: 0

Paul D. Hanna, May 20 2003

sign

Limit a(n)/a(n+1) --> r = -0.086930814654238 where A(r)=0.

Cf. A084067, A084202-A084211.

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 *)

A084203 G.f. A(x) defined by: A(x)^3 consists entirely of integer coefficients between 1 and 3 (A083953); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, 0, 0, 1, -1, 2, -2, 2, 0, -4, 12, -24, 38, -46, 33, 29, -176, 443, -827, 1222, -1310, 433, 2488, -8814, 19528, -33599, 44928, -37805, -17916, 168049, -463252, 921694, -1446018, 1679053, -808620, -2598482, 10515127, -24690122, 44515322, -62719429, 58496244, 10670109, -213311788, 632128236

Offset: 0

Paul D. Hanna, May 19 2003

sign

Does limit_{n ->infinity} a(n)/a(n+1) exist?

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. A083953, A084202, A084204-A084212.

kmax = 45;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^3 + O[x]^kmax , x];
r = {};
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 3, a[k-1], Integers] // ToRules]; coes = coes /. r, {k, 1, kmax}];
Array[a, kmax, 0] /. r (* Jean-François Alcover, Jul 26 2018 *)

A084204 G.f. A(x) defined by: A(x)^4 consists entirely of integer coefficients between 1 and 4 (A083954); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, -1, 3, -7, 20, -58, 177, -554, 1769, -5739, 18866, -62684, 210146, -709882, 2413743, -8253995, 28366316, -97916761, 339326189, -1180068800, 4116957243, -14404398636, 50530280752, -177684095927, 626181400993, -2211215950469, 7823025701314, -27724997048327

Offset: 0

Paul D. Hanna, May 20 2003

sign

Limit a(n)/a(n+1) -> r = -0.269562488839799 where A(r)=0.

Robert Israel, Table of n, a(n) for n = 0..1000
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. A083954, A084202, A084203, A084205-A084212.

g:= 1: a[0]:= 1:
for n from 1 to 50 do
  a[n]:= -floor((coeff(g^4,x,n)-1)/4);
  g:= g + a[n]*x^n;
od:
seq(a[n],n=0..50); # Robert Israel, Sep 04 2019

kmax = 30;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^4 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1}; coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 4, a[k-1], Integers] // ToRules];
coes = coes /. r, {k, 3, kmax + 1}];
Table[a[k], {k, 0, kmax}] /. r (* Jean-François Alcover, Jul 26 2018 *)

A084205 G.f. A(x) defined by: A(x)^5 consists entirely of integer coefficients between 1 and 5 (A083945); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, -1, 3, -8, 24, -76, 252, -854, 2950, -10343, 36706, -131570, 475576, -1731357, 6342042, -23356185, 86421603, -321111661, 1197586539, -4481348585, 16819759474, -63302097780, 238835017492, -903165412289, 3422512973645, -12994514592311, 49425252955926

Offset: 0

Paul D. Hanna, May 20 2003

sign

Limit a(n)/a(n+1) -> r = -0.2512525316047635 where A(r)=0.

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. A083945, A084202-A084204, A084206-A084212.

kmax = 30;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^5 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1};
coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 5, a[k-1], Integers] // ToRules];
coes = coes /. r, {k, 3, kmax+1}];
Table[a[k], {k, 0, kmax}] /. r (* Jean-François Alcover, Jul 26 2018 *)

A084207 G.f. A(x) defined by: A(x)^7 consists entirely of integer coefficients between 1 and 7 (A083947); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, -2, 8, -34, 158, -768, 3858, -19851, 104023, -552974, 2973832, -16146688, 88376636, -487034106, 2699839758, -15043262970, 84197804254, -473140314356, 2668221663736, -15095165871964, 85645090974518, -487190919969502

Offset: 0

Paul D. Hanna, May 20 2003

sign

Limit a(n)/a(n+1) -> r = -0.166670835025545 where A(r)=0.

Cf. A083947, A084202-A084206, A084208-A084212.

kmax = 25;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^7 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1};
coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 7, a[k-1], Integers] // ToRules];
coes = coes /. r, {k, 3, kmax + 1}];
Table[a[k], {k, 0, kmax}] /. r (* Jean-François Alcover, Jul 26 2018 *)

A084211 G.f. A(x) defined by: A(x)^11 consists entirely of integer coefficients between 1 and 11 (A084066); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, -4, 26, -189, 1479, -12106, 102224, -883031, 7761209, -69153920, 623018880, -5664270185, 51892998965, -478521450110, 4437418074830, -41350439060725, 386983852716405

Offset: 0

Paul D. Hanna, May 20 2003

sign

Limit a(n)/a(n+1) --> r = -0.100000000011 where A(r)=0.

Cf. A084066, A084202-A084210, A084212.

kmax = 20;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^11 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1};
coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 11, a[k-1], Integers] // ToRules];
coes = coes /. r, {k, 3, kmax + 1}];
Table[a[k], {k, 0, kmax}] /. r (* Jean-François Alcover, Jul 26 2018 *)

cp's OEIS Frontend

A108337 Positions of the odd terms in A084202, which gives the coefficients b(n) such that (B(x))^2 has coefficients 1 and 2.

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Extensions

A108335 A084202 read mod 4.

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A083952 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.

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A084212 G.f. A(x) defined by: A(x)^12 consists entirely of integer coefficients between 1 and 12 (A084067); A(x) is the unique power series solution with A(0)=1.

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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).

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A084203 G.f. A(x) defined by: A(x)^3 consists entirely of integer coefficients between 1 and 3 (A083953); A(x) is the unique power series solution with A(0)=1.

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A084204 G.f. A(x) defined by: A(x)^4 consists entirely of integer coefficients between 1 and 4 (A083954); A(x) is the unique power series solution with A(0)=1.

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Maple

Mathematica

A084205 G.f. A(x) defined by: A(x)^5 consists entirely of integer coefficients between 1 and 5 (A083945); A(x) is the unique power series solution with A(0)=1.

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Mathematica

A084207 G.f. A(x) defined by: A(x)^7 consists entirely of integer coefficients between 1 and 7 (A083947); A(x) is the unique power series solution with A(0)=1.

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Mathematica

A084211 G.f. A(x) defined by: A(x)^11 consists entirely of integer coefficients between 1 and 11 (A084066); A(x) is the unique power series solution with A(0)=1.

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