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 1-8 of 8 results.

A257290 Number of 3-Motzkin paths of length n with no level steps at even level.

Original entry on oeis.org

1, 0, 1, 3, 11, 39, 140, 504, 1823, 6621, 24144, 88380, 324699, 1197045, 4427565, 16427385, 61129025, 228103185, 853399640, 3200710680, 12032399045, 45332769075, 171148151095, 647412581643, 2453529142471, 9314461044639, 35419207688050, 134894888442714, 514506926871927
Offset: 0

Views

Author

José Luis Ramírez Ramírez, Apr 20 2015

Keywords

Examples

			For n=3 we have 3 paths: UH1D, UH2D, UH3D.

Links

Crossrefs

Cf. A090345, A025266.

Programs

  • Mathematica
    CoefficientList[Series[(1-3*x-Sqrt[(1-3*x)*(1-3*x-4*x^2)])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 21 2015 *)
  • PARI
    x='x+O('x^50); Vec((1-3*x-sqrt((1-3*x)*(1-3*x-4*x^2)))/(2*x^2)) \\ G. C. Greubel, Feb 14 2017

Formula

a(n) = Sum_{i=0..floor(n/2)} 3^(n-2i)*C(i)*binomial(n-i-1,n), where C(i) is the n-th Catalan number A000108.
G.f.: (1 - 3*z - sqrt((1-3*z)*(1-3*z-4*z^2)))/(2*z^2).
a(n) ~ sqrt(5) * 4^n / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015
Conjecture: (n+2)*a(n) +3*(-2*n-1)*a(n-1) +5*(n-1)*a(n-2) +6*(2*n-5)*a(n-3)=0. - R. J. Mathar, Sep 24 2016

A346503 G.f. A(x) satisfies A(x) = 1 + x^3 * A(x)^2 / (1 - x).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 3, 5, 7, 14, 26, 43, 79, 148, 264, 483, 903, 1664, 3080, 5771, 10795, 20209, 38059, 71799, 135569, 256762, 487310, 925981, 1762841, 3361897, 6419595, 12275301, 23505143, 45061424, 86485016, 166176499, 319630115, 615387675, 1185940209, 2287527119, 4416083429
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 21 2021

Keywords

Crossrefs

Cf. A002212, A023426, A023431, A090345, A216604 (partial sums), A346504, A366588, A376490.

Programs

  • Mathematica
    nmax = 40; A[] = 0; Do[A[x] = 1 + x^3 A[x]^2/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = a[2] = 0; a[n_] := a[n] = a[n - 1] + Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 40}]

Formula

a(0) = 1, a(1) = a(2) = 0; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).
a(n) ~ 2^(n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 30 2021
From Seiichi Manyama, Sep 26 2024: (Start)
G.f.: 2/(1 + sqrt(1 - 4*x^3/(1 - x))).
a(n) = Sum_{k=0..floor(n/3)} binomial(2*k,k) * binomial(n-2*k-1,n-3*k) / (k+1). (End)

A346504 G.f. A(x) satisfies: A(x) = 1 + x + x^3 * A(x)^2 / (1 - x).

Original entry on oeis.org

1, 1, 0, 1, 3, 4, 6, 14, 28, 49, 95, 196, 386, 754, 1524, 3102, 6258, 12700, 26032, 53440, 109772, 226457, 468863, 972300, 2020274, 4208530, 8784556, 18365322, 38461110, 80682740, 169501696, 356579216, 751138916, 1584281062, 3345404514, 7072055268, 14965933024, 31702754496
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 21 2021

Keywords

Crossrefs

Cf. A023426, A023431, A025243, A090345, A248100, A257290, A346503.

Programs

  • Mathematica
    nmax = 37; A[] = 0; Do[A[x] = 1 + x + x^3 A[x]^2/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[2] = 0; a[n_] := a[n] = a[n - 1] + Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 37}]
CoefficientList[Series[(1 - x)*(1 - Sqrt[(1 - x - 4*x^3 - 4*x^4)/(1 - x)]) / (2*x^3), {x, 0, 40}], x] (* Vaclav Kotesovec, Sep 27 2023 *)

Formula

a(0) = a(1) = 1, a(2) = 0; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).
G.f.: (1-x)*(1 - sqrt((1 - x - 4*x^3 - 4*x^4)/(1-x))) / (2*x^3). - Vaclav Kotesovec, Sep 27 2023

A185087 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*A000108(k+1).

Original entry on oeis.org

1, 1, 3, 5, 12, 24, 55, 119, 272, 612, 1411, 3247, 7565, 17667, 41561, 98099, 232696, 553784, 1322813, 3169065, 7614583, 18342921, 44294991, 107200829, 259983346, 631718606, 1537737567, 3749440151, 9156561590, 22394270034, 54845701243
Offset: 0

Views

Author

Paul Barry, Feb 18 2011

Keywords

Comments

Essentially identical to A090345 (=1,0,1,1,3,5,12,24...). - Joerg Arndt, Mar 18 2011
Hankel transform is A008619(n+1) (counting numbers doubled).

Links

Programs

  • Mathematica
    CoefficientList[Series[(1 - x - 2*x^2 - Sqrt[1 - 2 x - 3 x^2 + 4 x^3])/(2*x^4), {x,0,50}], x] (* G. C. Greubel, Jun 22 2017 *)
  • PARI
    x='x+O('x^50); Vec((1-x-2*x^2-sqrt(1-2*x-3*x^2+4*x^3))/(2*x^4)) \\ G. C. Greubel, Jun 22 2017

Formula

G.f.: (1-x-2x^2-sqrt(1-2x-3x^2+4x^3))/(2x^4).
G.f.: 1/(1-x-2x^2/(1-(1/2)x^2/(1-x-(3/2)x^2/(1-(2/3)x^2/(1-x-(4/3)x^2/(1-(3/4)x^2/(1-... (continued fraction).
a(n)=sum{k=0..n, sum{j=0..n, binomial(k-j,n-k-j)*binomial(k,j)*if(n-k-j>=0, A001006(n-k-j),0)}}.
a(n)=A090345(n+2).
Conjecture: (n+4)*a(n) -(2*n+5)*a(n-1) -3*(n+1)*a(n-2) +2*(2*n-1)*a(n-3)=0. - R. J. Mathar, Nov 16 2011

A257390 Number of 4-Motzkin paths of length n with no level steps at even level.

Original entry on oeis.org

1, 0, 1, 4, 18, 80, 357, 1596, 7150, 32096, 144362, 650568, 2937316, 13286368, 60205805, 273290988, 1242639446, 5659468736, 25816338046, 117945079736, 539646216188, 2472638868960, 11345220210658, 52124831171544, 239792244636876, 1104495824173376
Offset: 0

Views

Author

José Luis Ramírez Ramírez, Apr 21 2015

Keywords

Links

Crossrefs

Cf. A090345, A025266, A257290.

Programs

  • Maple
    rec:= a(n) = ((-16*n + 40)*a(n-3) + (-12*n+12)*a(n-2) +(8*n+4)*a(n-1))/(n+2):
f:= gfun:-rectoproc({rec,a(0)=1,a(1)=0,a(2)=1},a(n),remember):
seq(f(i),i=0..100); # Robert Israel, Apr 22 2015
  • Mathematica
    CoefficientList[Series[(1-4*x-Sqrt[(1-4*x)*(1-4*x-4*x^2)])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 22 2015 *)
  • PARI
    x='x+O('x^50); Vec((1-4*x-sqrt((1-4*x)*(1-4*x-4*x^2)))/(2*x^2)) \\ G. C. Greubel, Apr 08 2017

Formula

a(n) = Sum_{i=0..floor(n/2)}4^(n-2i)*C(i)*binomial(n-i-1,n), where C(i) is the i-th Catalan number A000108.
G.f.: (1-4*x-sqrt((1-4*x)*(1-4*x-4*x^2)))/(2*x^2).
a(n) ~ 2^(n+3/4) * (1+sqrt(2))^(n+1/2) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 22 2015
a(n) = ((-16*n + 40)*a(n-3) + (-12*n+12)*a(n-2) +(8*n+4)*a(n-1))/(n+2). - Robert Israel, Apr 22 2015

A257516 Number of 3-generalized Motzkin paths of length n with no level steps H=(3,0) at even level.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 5, 4, 15, 15, 48, 57, 162, 218, 570, 842, 2070, 3284, 7709, 12922, 29299, 51255, 113220, 204781, 443574, 823554, 1757947, 3331818, 7035054, 13552699, 28387680, 55401396, 115369417, 227501256, 471780468, 938107057, 1939727280, 3883120002
Offset: 0

Views

Author

José Luis Ramírez Ramírez, Apr 27 2015

Keywords

Examples

			For n=6 we have 5 paths: UDUDUD, UUDDUD, UDUUDD, UUUDDD and UUDUDD

Crossrefs

Cf. A002212, A090345.

Programs

  • Mathematica
    CoefficientList[Series[(1-x^3-Sqrt[(1-x^3)*(1-4*x^2-x^3)])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 27 2015 *)

Formula

G.f.: (1-x^3-sqrt((1-x^3)*(1-4*x^2-x^3)))/(2*x^2).

A357307 a(0) = 1, a(1) = 0, a(2) = 1; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).

Original entry on oeis.org

1, 0, 1, 2, 2, 4, 8, 13, 25, 49, 91, 177, 349, 681, 1349, 2693, 5377, 10806, 21820, 44163, 89721, 182868, 373616, 765341, 1571551, 3233690, 6667242, 13772469, 28498419, 59065838, 122606998, 254865837, 530507839, 1105663034, 2307131590, 4819623077, 10079039819, 21099213611, 44211213545
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 23 2022

Keywords

Crossrefs

Cf. A000245, A023426, A023431, A090345, A346503, A346504, A357308.

Programs

  • Mathematica
    a[0] = 1; a[1] = 0; a[2] = 1; a[n_] := a[n] = a[n - 1] + Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 38}]
nmax = 38; A[] = 0; Do[A[x] = 1 + x^2 (1 + x A[x]^2)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 + x^2 * (1 + x * A(x)^2) / (1 - x).

A039984 An example of a d-perfect sequence.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Image, under the coding a,c,d -> 1, b -> 0, of the fixed point, starting with a, of the morphism a -> ab, b -> cd, c -> cd, d -> bb. - Jeffrey Shallit, May 15 2016

Links

  • D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions, in Sequences and their Applications, C. Ding, T. Helleseth, and H. Niederreiter, eds., Proceedings of SETA'98 (Singapore, 1998), 308-317, 1999. DOI: 10.1007/978-1-4471-0551-0_23

Formula

a(n) = A090345(n) mod 2. - Christian G. Bower, Jun 12 2005

Extensions

More terms from Christian G. Bower, Jun 12 2005
Showing 1-8 of 8 results.

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

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