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 1-4 of 4 results.

A049235 Sum of balls on the lawn for the s=3 tennis ball problem.

Original entry on oeis.org

0, 6, 75, 708, 5991, 47868, 369315, 2783448, 20631126, 151026498, 1094965524, 7878119760, 56330252412, 400703095284, 2838060684483, 20027058300144, 140874026880204, 988194254587242, 6915098239841331, 48285969880645908, 336521149274459979
Offset: 0

Views

Author

N. J. A. Sloane, Jan 19 2003

Keywords

Links

Crossrefs

The four sequences T_n, Y_n, A_n, S_n for s=2 are A000108, A000302, A000346, A031970, for s=3, A001764, A006256, A075045, this sequence, for s=4, A002293, A078995, A078999, A078516.
Cf. A079486.

Programs

  • Maple
    T := (n,s)->binomial(s*n,n)/((s-1)*n+1); Y := (n,s)->add(binomial(s*k,k)*binomial(s*(n-k),n-k),k=0..n); A := (n,s)->Y(n+1,s)/2-(1/2)*((2*s-3)*n+2*s-2)*T(n+1,s); S := (n,s)->(1/2)*(s*n^2+(3*s-1)*n+2*s)*T(n+1,s)-Y(n+1,s)/2;
F := 3*(2-3*t)*t*((t-1)*(3*t-1))^(-3);  G := t*(t-1)^2;   Ginv := RootOf(G-x,t);
ogf := series(eval(F,t=Ginv), x=0, 20);
  • Mathematica
    a[n_] := a[n] = Switch[n, 0, 0, 1, 6, 2, 75, 3, 708, 4, 5991, _, -((1/(8*(2*(n-5)^2 + 25*(n-5) + 78)))*(-(531441*(n-5)^2* a[n-5]) + 196830*(n-5)^2*a[n-4] - 24057*(n-5)^2*a[n-3] + 1809*(n-5)^2*a[n-2] - 232*(n-5)^2*a[n-1] - 1594323*(n-5)*a[n-5] + 747954*(n-5)*a[n-4] - 120285*(n-5)*a[n-3] + 16362*(n-5)*a[n-2] - 2798*(n-5)*a[n-1] - 1180980*a[n-5] + 656100*a[n-4] - 131220*a[n-3] + 36825*a[n-2] - 8352*a[n-1]))];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 02 2023, after Robert Israel *)

Formula

a(n) is asymptotic to c*sqrt(n)*(27/4)^n with c=2.4... - Benoit Cloitre, Jan 26 2003, c = 81*sqrt(3/Pi)/32 = 2.4735502165085321... - Vaclav Kotesovec, Feb 07 2019
G.f.: F(G^(-1)(x)) where F = 3*(2-3*t)*t*((t-1)*(3*t-1))^(-3) and G = t*(t-1)^2. - Mark van Hoeij, Oct 30 2011
D-finite with recurrence (531441*n^2 + 1594323*n + 1180980)*a(n) + (-196830*n^2 - 747954*n - 656100)*a(n + 1) + (24057*n^2 + 120285*n + 131220)*a(n + 2) + (-1809*n^2 - 16362*n - 36825)*a(n + 3) + (232*n^2 + 2798*n + 8352)*a(n + 4) + (-16*n^2 - 200*n - 624)*a(n + 5) = 0. - Robert Israel, Jun 20 2019

A031970 Tennis ball problem: Balls 1 and 2 are thrown into a room; you throw one on lawn; then balls 3 and 4 are thrown in and you throw any of the 3 balls onto the lawn; then balls 5 and 6 are thrown in and you throw one of the 4 balls onto the lawn; after n turns, consider all possible collections on lawn and add all the values.

Original entry on oeis.org

0, 3, 23, 131, 664, 3166, 14545, 65187, 287060, 1247690, 5368670, 22917198, 97195968, 410030812, 1722027973, 7204620067, 30044212828, 124932768082, 518215690018, 2144815618522, 8859729437488, 36533517261412, 150410878895818, 618371102344846, 2538971850705064, 10412490129563556
Offset: 0

Views

Author

Louis Shapiro

Keywords

Links

Crossrefs

Cf. A049235, A078516, A079486, A000108.

Programs

  • GAP
    List([0..30], n-> (2*n^2+5*n+4)*Binomial(2*n+1, n)/(n+2) - 2^(2*n+1)); # G. C. Greubel, Apr 02 2019
  • Magma
    [(2*n^2+5*n+4)*Binomial(2*n+1, n)/(n+2) - 2^(2*n+1): n in [0..30]]; // G. C. Greubel, Apr 02 2019
  • Mathematica
    CoefficientList[Series[(1-9*x+20*x^2-(1-7*x+8*x^2)*Sqrt[1-4*x])/(2*x^2*(1 -8*x+16*x^2)), {x,0,30}],x] (* G. C. Greubel, Apr 02 2019 *)
  • PARI
    a(n) = (2*n^2 + 5*n + 4)*binomial(2*n+1, n)/(n+2) - 2^(2*n+1);
/* Joerg Arndt, Dec 04 2012 */
  • Sage
    [(2*n^2+5*n+4)*binomial(2*n+1, n)/(n+2) - 2^(2*n+1) for n in (0..30)] # G. C. Greubel, Apr 02 2019

Formula

a(n) = (2*n^2 + 5*n + 4)*binomial(2*n+1, n)/(n+2) - 2^(2*n+1). - Colin Mallows.
a(n) = Sum_{i=0..n-1} (4*n-4*i-1)*A028364(n,i), where A028364 is a Catalan triangle. e.g. for n=3 T[3..] = [5,7,9,14] then S(3) = 131 = 11*5 + 7*7 + 3*9. - David Scambler, Apr 27 2009
G.f.: (1-9*x+20*x^2-(1-7*x+8*x^2)*sqrt(1-4*x))/(2*x^2*(1-8*x+16*x^2)). - Vladimir Kruchinin, Apr 02 2019
D-finite with recurrence: (n+2)*a(n) +(-15*n-14)*a(n-1) +2*(40*n-3)*a(n-2) +8*(-22*n+25)*a(n-3) +64*(2*n-5)*a(n-4)=0. - R. J. Mathar, Jan 28 2020

Extensions

More terms from Joerg Arndt, Dec 04 2012

A079889 Primes indexed by A078515; i.e., primes which start record runs of consecutive primes with distinct first differences.

Original entry on oeis.org

2, 17, 83, 113, 491, 1367, 1801, 5869, 15919, 34883, 70639, 70657, 214867, 2515871, 3952733, 13010143, 30220163, 60155567, 69931991, 203674907, 1092101119, 1363592621, 1363592677, 2124140323, 23024158649, 30282104173, 196948778371
Offset: 1

Views

Author

N. J. A. Sloane, Jan 07 2003

Keywords

Examples

			2, 17 and 83 are in the sequence because the 3 consecutive primes 2,3,5 have distinct first differences 1,2, the 4 consecutive primes 17,19,23,29 have distinct differences 2,4,6, and the 5 consecutive primes 83,89,97,101,103 have distinct differences 6,8,4,2.

Crossrefs

Cf. A078515, A078516. Same as A079007 with duplicates removed.

Formula

a(n) = A078515(n)-th prime.

Extensions

More terms from Don Reble, Jan 15 2003
a(25)-a(27) from Donovan Johnson, Oct 23 2012

A268315 Decimal expansion of 256/27.

Original entry on oeis.org

9, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8
Offset: 1

Views

Author

Gheorghe Coserea, Feb 01 2016

Keywords

Examples

			9.481481481481481481481481481481...

Crossrefs

Exponential growth rate for A000260, A002293, A005810, A006633, A006634, A052203, A066381, A069271, A078516, A078995, A153396, A196678, A268316.

Programs

  • Magma
    [9] cat &cat[[4, 8, 1]^^45]; // Vincenzo Librandi, Feb 04 2016
  • Mathematica
    Join[{9}, PadRight[{}, 120, {4, 8, 1}]] (* Vincenzo Librandi, Feb 04 2016 *)
  • PARI
    1.0 * 256/27

Extensions

More digits from Jon E. Schoenfield, Mar 15 2018
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.