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 1-6 of 6 results.

A105872 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-3*k, n).

Original entry on oeis.org

1, 2, 6, 21, 75, 273, 1009, 3770, 14202, 53846, 205216, 785460, 3017106, 11624580, 44905518, 173863965, 674506059, 2621371005, 10203609597, 39773263035, 155231706951, 606554343495, 2372544034143, 9289131196485, 36401388236461
Offset: 0

Views

Author

Paul Barry, Apr 23 2005

Keywords

Links

Crossrefs

Cf. A144904, A360150, A360151, A360152, A360153.

Programs

  • Mathematica
    Table[Sum[Binomial[2n-3k,n],{k,0,Floor[n/2]}],{n,0,30}] (* Harvey P. Dale, Jan 13 2015 *)
  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n-3*k, n)); \\ Seiichi Manyama, Jan 28 2023

Formula

G.f.: 2/(4*x^2+sqrt(1-4*x)*(3*x+1)-5*x+1). - Vladimir Kruchinin, May 24 2014
Conjecture: -3*(n+1)*(7*n-2)*a(n) +6*(7*n+5)*(2*n-1)*a(n-1) -(n+1)*(7*n-2)*a(n-2) +2*(7*n+5)*(2*n-1)*a(n-3)=0. - R. J. Mathar, Nov 28 2014
a(n) ~ 2^(2*n+3) / (7*sqrt(Pi*n)). - Vaclav Kotesovec, Jan 28 2023
a(n) = [x^n] 1/((1-x^3) * (1-x)^(n+1)). - Seiichi Manyama, Apr 08 2024

Extensions

Erroneous title changed by Paul Barry, Apr 14 2010
Name corrected by Seiichi Manyama, Jan 28 2023

A360150 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-k,n-3*k).

Original entry on oeis.org

1, 2, 6, 21, 77, 288, 1090, 4159, 15964, 61557, 238221, 924597, 3597290, 14024341, 54770176, 214218966, 838959762, 3289471537, 12910910288, 50720828034, 199422778415, 784672001097, 3089564308849, 12172411084432, 47984843655991, 189260578353602
Offset: 0

Views

Author

Seiichi Manyama, Jan 28 2023

Keywords

Crossrefs

Cf. A105872, A144904, A360151, A360152, A360153, A360168.
Cf. A000108.

Programs

  • Maple
    A360150 := proc(n)
    add(binomial(2*n-k,n-3*k),k=0..n/3) ;
end proc:
seq(A360150(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • Mathematica
    a[n_] := Sum[Binomial[2*n - k, n - 3*k], {k, 0, Floor[n/3]}]; Array[a, 26, 0] (* Amiram Eldar, Jan 28 2023 *)
  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n-k, n-3*k));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x^3*(2/(1+sqrt(1-4*x)))^5)))

Formula

G.f.: 1 / ( sqrt(1-4*x) * (1 - x^3 * c(x)^5) ), where c(x) is the g.f. of A000108.
a(n) ~ 2^(2*n+1) / sqrt(Pi*n). - Vaclav Kotesovec, Jan 28 2023
D-finite with recurrence n*a(n) +2*(-7*n+6)*a(n-1) +2*(36*n-61)*a(n-2) +4*(-41*n+103)*a(n-3) +(161*n-530)*a(n-4) +(-71*n+278)*a(n-5) +6*(2*n-9)*a(n-6)=0. - R. J. Mathar, Mar 12 2023
a(n) = [x^n] 1/(((1-x)^2-x^3) * (1-x)^(n-1)). - Seiichi Manyama, Apr 09 2024

A360153 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-6*k,n-3*k).

Original entry on oeis.org

1, 2, 6, 21, 72, 258, 945, 3504, 13128, 49565, 188260, 718560, 2753721, 10588860, 40835160, 157871241, 611669250, 2374441380, 9233006541, 35956933050, 140220970200, 547490880981, 2140055896770, 8373651697800, 32795094564081, 128550662334522
Offset: 0

Views

Author

Seiichi Manyama, Jan 28 2023

Keywords

Crossrefs

Cf. A105872, A144904, A360150, A360151, A360152, A360168.
Cf. A006134, A106188.

Programs

  • Maple
    A360153 := proc(n)
    add(binomial(2*n-6*k,n-3*k),k=0..n/3) ;
end proc:
seq(A360153(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • Mathematica
    a[n_] := Sum[Binomial[2*n - 6*k, n - 3*k], {k, 0, Floor[n/3]}]; Array[a, 26, 0] (* Amiram Eldar, Jan 28 2023 *)
  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n-6*k, n-3*k));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x^3)))

Formula

G.f.: 1 / ( sqrt(1-4*x) * (1 - x^3) ).
a(n) ~ 2^(2*n + 6) / (63 * sqrt(Pi*n)). - Vaclav Kotesovec, Jan 28 2023
a(n)-a(n-3) = A000984(n). - R. J. Mathar, Mar 12 2023
D-finite with recurrence n*a(n) +2*(-2*n+1)*a(n-1) -n*a(n-3) +2*(2*n-1)*a(n-4)=0. - R. J. Mathar, Mar 12 2023

A360151 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-4*k,n-3*k).

Original entry on oeis.org

1, 2, 6, 21, 74, 267, 981, 3648, 13690, 51744, 196699, 751237, 2880345, 11080081, 42743148, 165291569, 640563158, 2487083484, 9672626600, 37674470433, 146937686295, 573781535775, 2243050091905, 8777451670102, 34379401083017, 134770951530840
Offset: 0

Views

Author

Seiichi Manyama, Jan 28 2023

Keywords

Crossrefs

Cf. A105872, A144904, A360150, A360152, A360153.
Cf. A000108.

Programs

  • Maple
    A360151 := proc(n)
    add(binomial(2*n-4*k,n-3*k),k=0..n/3) ;
end proc:
seq(A360151(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • Mathematica
    a[n_] := Sum[Binomial[2*n - 4*k, n - 3*k], {k, 0, Floor[n/3]}]; Array[a, 26, 0] (* Amiram Eldar, Jan 28 2023 *)
  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n-4*k, n-3*k));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x^3*(2/(1+sqrt(1-4*x)))^2)))

Formula

G.f.: 1 / ( sqrt(1-4*x) * (1 - x^3 * c(x)^2) ), where c(x) is the g.f. of A000108.
a(n) ~ 2^(2*n+4) / (15*sqrt(Pi*n)). - Vaclav Kotesovec, Jan 28 2023
D-finite with recurrence +2*n*a(n) +(-11*n+6)*a(n-1) +(19*n-24)*a(n-2) +2*(-16*n+33)*a(n-3) +2*(11*n-36)*a(n-4) +(-25*n+78)*a(n-5) +6*(n-3)*a(n-6) +4*(-2*n+9)*a(n-7)=0. - R. J. Mathar, Mar 12 2023

A360168 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n,n-3*k).

Original entry on oeis.org

1, 2, 6, 21, 78, 297, 1145, 4447, 17358, 68001, 267141, 1051767, 4148281, 16385111, 64797543, 256515731, 1016368078, 4030114641, 15990813773, 63485616391, 252175202373, 1002136689071, 3984080489263, 15844839393411, 63036297959993, 250855287692647
Offset: 0

Views

Author

Seiichi Manyama, Jan 28 2023

Keywords

Crossrefs

Cf. A105872, A144904, A360150, A360151, A360152, A360153.
Cf. A000108, A032443, A114121.
Cf. A002450, A371777.

Programs

  • Maple
    A360168 := proc(n)
    add(binomial(2*n,n-3*k),k=0..n/3) ;
end proc:
seq(A360168(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • Mathematica
    a[n_] := Sum[Binomial[2*n, n - 3*k], {k, 0, Floor[n/3]}]; Array[a, 26, 0] (* Amiram Eldar, Jan 28 2023 *)
  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n, n-3*k));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x^3*(2/(1+sqrt(1-4*x)))^6)))

Formula

G.f.: 1 / ( sqrt(1-4*x) * (1 - x^3 * c(x)^6) ), where c(x) is the g.f. of A000108.
D-finite with recurrence n*a(n) +2*(-4*n+3)*a(n-1) +8*(2*n-3)*a(n-2) +3*(-n+2)=0. - R. J. Mathar, Mar 12 2023
a(n) = [x^n] 1/(((1-x)^3-x^3) * (1-x)^(n-2)). - Seiichi Manyama, Apr 10 2024

A360212 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-5*k,n-3*k).

Original entry on oeis.org

1, 2, 6, 19, 67, 242, 890, 3310, 12423, 46959, 178526, 681893, 2614698, 10059000, 38807021, 150080294, 581649776, 2258469988, 8783966719, 34214789901, 133450049457, 521134066663, 2037313708685, 7972641631438, 31228124666374, 122421230120657
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2023

Keywords

Crossrefs

Cf. A191993, A307354, A360186.
Cf. A000108, A360152.

Programs

  • Maple
    A360212 := proc(n)
    add((-1)^k*binomial(2*n-5*k,n-3*k),k=0..n/3) ;
end proc:
seq(A360212(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • PARI
    a(n) = sum(k=0, n\3, (-1)^k*binomial(2*n-5*k, n-3*k));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1+2*x^3/(1+sqrt(1-4*x)))))

Formula

G.f.: 1 / ( sqrt(1-4*x) * (1 + x^3 * c(x)) ), where c(x) is the g.f. of A000108.
D-finite with recurrence 2*n*a(n) +4*(-2*n+1)*a(n-1) +(3*n-4)*a(n-2) +2*(-6*n+11)*a(n-3) +(n-4)*a(n-4) +2*(-n+9)*a(n-5) +4*(-2*n+1)*a(n-6) +(n-4)*a(n-7) +2*(-2*n+9)*a(n-8)=0. - R. J. Mathar, Mar 12 2023
Showing 1-6 of 6 results.

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