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.

Previous Showing 11-16 of 16 results.

A084206 G.f. A(x) defined by: A(x)^6 consists entirely of integer coefficients between 1 and 6 (A083946); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, -2, 7, -27, 115, -510, 2343, -11029, 52896, -257457, 1268098, -6307546, 31633044, -159757597, 811708539, -4145882814, 21273287952, -109603172373, 566748274099, -2940175511195, 15297961574259, -79808998488751, 417373462315834
Offset: 0

Views

Author

Paul D. Hanna, May 20 2003

Keywords

Comments

Limit a(n)/a(n+1) -> r = -0.1815238859919 where A(r)=0.

Links

Crossrefs

Cf. A083946, A084202-A084205, A084207-A084212.

Programs

  • Mathematica
    kmax = 25;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^6 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1};
coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 6, 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 *)

A084208 G.f. A(x) defined by: A(x)^8 consists entirely of integer coefficients between 1 and 8 (A083948); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, -3, 15, -82, 484, -2992, 19110, -124979, 832234, -5621028, 38402783, -264858143, 1841221687, -12886279885, 90713376563, -641815393278, 4561172770669, -32542369727538, 232992967457839
Offset: 0

Views

Author

Paul D. Hanna, May 20 2003

Keywords

Comments

Limit a(n)/a(n+1) --> r = -0.131401689761435 where A(r)=0.

Crossrefs

Cf. A083948, A084202-A084207, A084209-A084212.

Programs

  • Mathematica
    kmax = 20;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^8 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1};
coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 8, 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 *)

A084209 G.f. A(x) defined by: A(x)^9 consists entirely of integer coefficients between 1 and 9 (A083949); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, -3, 15, -85, 523, -3367, 22371, -152104, 1052568, -7385756, 52410754, -375382683, 2709626768, -19688989762, 143885743077, -1056748051734, 7795106129504, -57723430872280, 428923406694402
Offset: 0

Views

Author

Paul D. Hanna, May 20 2003

Keywords

Comments

Limit a(n)/a(n+1) --> r = -0.126715878986521 where A(r)=0.

Crossrefs

Cf. A083949, A084202-A084208, A084210-A084212.

Programs

  • Mathematica
    kmax = 20;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^9 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1};
coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 9, 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 *)

A084210 G.f. A(x) defined by: A(x)^10 consists entirely of integer coefficients between 1 and 10 (A083950); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, -4, 25, -173, 1292, -10105, 81565, -673691, 5662878, -48263038, 415950272, -3617999891, 31714089336, -279828926113, 2483097203637, -22143011361045, 198317403322755
Offset: 0

Views

Author

Paul D. Hanna, May 20 2003

Keywords

Comments

Limit a(n)/a(n+1) --> r = -0.104430987675729 where A(r)=0.

Crossrefs

Cf. A083950, A084202-A084209, A084211, A084212.

Programs

  • Mathematica
    kmax = 20;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^10 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1};
coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 10, 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 *)

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

Views

Author

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

Keywords

Comments

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

Crossrefs

Cf. A083952, A111581, A084202, A108336.

Programs

  • PARI
    {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)))}

Formula

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.

A134780 The square root of A134779.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, -1, 6, -7, 19, -33, 73, -146, 311, -652, 1392, -2977, 6420, -13899, 30247, -66078, 144911, -318853, 703768, -1557718, 3456813, -7689531, 17142887, -38296408, 85715645, -192191445, 431647744, -970958480, 2187288804, -4934101775, 11144794835, -25203825094
Offset: 0

Views

Author

Paul D. Hanna & Robert G. Wilson v, Nov 11 2007

Keywords

Crossrefs

Cf. A084202, A134779.

Programs

  • Mathematica
    a[n_] := a[n] = Block[{k = a[n - 1] + 2, s = Sum[ a[i]*x^i, {i, 0, n - 1}]}, If[IntegerQ@ Last@ CoefficientList[ Series[ Sqrt[s + k*x^n], {x, 0, n}], x], k, k + 1]]; a[0] = 1; CoefficientList[ Series[ Sqrt[ Sum[ a[i]*x^i, {i, 0, 36}]], {x, 0, 36}], x]
Previous Showing 11-16 of 16 results.

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