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 1-3 of 3 results.

A055330 Number of rooted identity trees with n nodes and 5 leaves.

Original entry on oeis.org

3, 26, 116, 387, 1068, 2587, 5678, 11540, 22034, 39957, 69366, 116009, 187823, 295574, 453582, 680625, 1000952, 1445516, 2053343, 2873165, 3965216, 5403347, 7277330, 9695538, 12787847, 16708973, 21642067, 27802808, 35443793, 44859494, 56391551, 70434706
Offset: 10

Views

Author

Christian G. Bower, May 12 2000

Keywords

Links

Crossrefs

Column 5 of A055327.
Cf. A061347, A257145.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( x^10*(3 + 14*x+24*x^2+36*x^3+41*x^4+38*x^5+29*x^6+16*x^7+6*x^8+3*x^9)/((1-x)^3*(&*[1-x^j: j in [1..5]])) )); // G. C. Greubel, Nov 09 2023
  • Mathematica
    Drop[CoefficientList[Series[x^10*(3+14*x+24*x^2+36*x^3+41*x^4+38*x^5+29*x^6 +16*x^7+6*x^8+3*x^9)/((1-x)^3*Product[1-x^j, {j,5}]), {x,0,40}], x], 10] (* G. C. Greubel, Nov 09 2023 *)
  • SageMath
    def p(x): return 3 +14*x +24*x^2 +36*x^3 +41*x^4 +38*x^5 +29*x^6 +16*x^7 +6*x^8 +3*x^9
def A055330_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^10*p(x)/((1-x)^3*product(1-x^j for j in range(1,6))) ).list()
a=A055330_list(50); a[10:] # G. C. Greubel, Nov 09 2023

Formula

G.f.: x^10*(3 +14*x +24*x^2 +36*x^3 +41*x^4 +38*x^5 +29*x^6 +16*x^7 +6*x^8 +3*x^9)/((1-x)^9*(1+x)^3*(1+x^2)*(1+x+x^2)*(1+x+x^2+x^3+x^4)). - Colin Barker, Nov 07 2012
a(n) = (1/(8*10!))*(5303207 -25330590*n +28099260*n^2 -18286800*n^3 +7777980*n^4 -1990044*n^5 +286440*n^6 -21240*n^7 +630*n^8) -(-1)^n*(89 - 34*n +4*n^2)/2048 -(3/64)*(-1)^binomial(n+1,2) -A061347(n+1)/81 +A257145(n+2)/25. - G. C. Greubel, Nov 09 2023

A293292 Numbers with last digit less than 5 (in base 10).

Original entry on oeis.org

0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 40, 41, 42, 43, 44, 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 70, 71, 72, 73, 74, 80, 81, 82, 83, 84, 90, 91, 92, 93, 94, 100, 101, 102, 103, 104, 110, 111, 112, 113, 114, 120, 121, 122, 123, 124, 130
Offset: 1

Views

Author

Bruno Berselli, Oct 05 2017

Keywords

Comments

Equivalently, numbers k such that floor(k/5) = 2*floor(k/10).
After 0, partial sums of A010122 starting from the 2nd term.
The sequence differs from A007091 after a(25).
Also numbers k such that floor(k/5) is even. - Peter Luschny, Oct 05 2017

Links

Crossrefs

Cf. A010122, A239229, A257145, A293481 (complement).
Sequences of the type floor(n/d) = (10/d)*floor(n/10), where d is a factor of 10: A008592 (d=1), A197652 (d=2), this sequence (d=5), A001477 (d=10).
Sequences of the type n + r*floor(n/r): A005843 (r=1), A042948 (r=2), A047240 (r=3), A047476 (r=4), this sequence (r=5).

Programs

  • Magma
    [n: n in [0..130] | n mod 10 lt 5];
  • Magma
    [n: n in [0..130] | IsEven(Floor(n/5))];
  • Magma
    [n+5*Floor(n/5): n in [0..70]];
  • Maple
    select(k -> type(floor(k/5), even), [$0..130]); # Peter Luschny, Oct 05 2017
  • Mathematica
    Table[n + 5 Floor[n/5], {n, 0, 70}]
Reap[For[k = 0, k <= 130, k++, If[Floor[k/5] == 2*Floor[k/10], Sow[k]]]][[2, 1]] (* or *) LinearRecurrence[{1, 0, 0, 0, 1, -1}, {0, 1, 2, 3, 4, 10}, 66] (* Jean-François Alcover, Oct 05 2017 *)
  • PARI
    concat(0, Vec(x^2*(1 + x + x^2 + x^3 + 6*x^4) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)) + O(x^70))) \\ Colin Barker, Oct 05 2017
  • PARI
    select(k->floor(k/5) == 2*floor(k/10), vector(1000, k, k)) \\ Colin Barker, Oct 05 2017
  • Python
    [k for k in range(131) if (k//5) % 2 == 0] # Peter Luschny, Oct 05 2017
  • Python
    def A293292(n): return (n-1<<1)-(n-1)%5 # Chai Wah Wu, Oct 29 2024
  • Sage
    [k for k in (0..130) if 2.divides(floor(k/5))] # Peter Luschny, Oct 05 2017

Formula

G.f.: x^2*(1 + x + x^2 + x^3 + 6*x^4)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-5) - a(n-6).
a(n) = (n-1) + 5*floor((n-1)/5) = 10*floor((n-1)/5) + ((n-1) mod 5).
a(n) = A257145(n+2) - A239229(n-1). - R. J. Mathar, Oct 05 2017
a(n) = 2n-2-((n-1) mod 5). - Chai Wah Wu, Oct 29 2024

Extensions

Definition by David A. Corneth, Oct 05 2017

A266084 Expansion of (5 - x - x^2 - x^3 - x^4 + 4*x^5)/( x^6 - x^5 - x + 1).

Original entry on oeis.org

5, 4, 3, 2, 1, 10, 9, 8, 7, 6, 15, 14, 13, 12, 11, 20, 19, 18, 17, 16, 25, 24, 23, 22, 21, 30, 29, 28, 27, 26, 35, 34, 33, 32, 31, 40, 39, 38, 37, 36, 45, 44, 43, 42, 41, 50, 49, 48, 47, 46, 55, 54, 53, 52, 51, 60, 59, 58, 57, 56, 65, 64, 63, 62, 61, 70
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 21 2015

Keywords

Comments

Invert blocks of five in the sequence of natural numbers.

Links

Crossrefs

Cf. A000027, A113655, A113778.

Programs

  • Magma
    [5+5*Floor(n/5)-n mod 5: n in [0..70]]; // Vincenzo Librandi, Dec 21 2015
  • Mathematica
    Table[5 + 5 Floor[n/5] - Mod[n, 5], {n, 0, 50}]
CoefficientList[Series[(5 - x - x^2 - x^3 - x^4 + 4 x^5)/(x^6 - x^5 - x + 1), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)
Reverse/@Partition[Range[80],5]//Flatten (* or *) LinearRecurrence[ {1,0,0,0,1,-1},{5,4,3,2,1,10},80] (* Harvey P. Dale, Sep 02 2016 *)
  • PARI
    a(n) = 5 + 5*(n\5) - (n % 5); \\ Michel Marcus, Dec 21 2015
  • PARI
    x='x+O('x^100); Vec((5-x-x^2-x^3-x^4+4*x^5)/(x^6-x^5-x+1)) \\ Altug Alkan, Dec 21 2015

Formula

G.f.: (5 - x - x^2 - x^3 - x^4 + 4*x^5)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-5) - a(n-6) for n>5.
a(n) = 5 + 5*floor(n/5) - n mod 5.
a(n) = n+1+2*A257145(n+3). - R. J. Mathar, Apr 12 2019
Showing 1-3 of 3 results.

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