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.

A299915 a(n) = A299914(2n).

Original entry on oeis.org

0, 1, 9, 69, 513, 3789, 27945, 206037, 1518993, 11198493, 82558521, 608644773, 4487100705, 33080169069, 243876313161, 1797924789621, 13254807348657, 97718168662461, 720405829778265, 5311034444054853, 39154440039154497, 288657547023732237, 2128064642743736169
Offset: 0

Views

Author

N. J. A. Sloane, Mar 10 2018

Keywords

References

  • Murat Sahin and Elif Tan, Conditional (strong) divisibility sequences, Fib. Q., 56 (No. 1, 2018), 18-31.

Links

Crossrefs

Cf. A299914.

Programs

  • Magma
    I:=[0,1]; [n le 2 select I[n] else 9*Self(n-1)-12*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 11 2018
  • Maple
    a:= n-> (<<0|1>, <-12|9>>^n)[1, 2]:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 10 2018
  • Mathematica
    CoefficientList[Series[x/(12 x^2 - 9 x + 1), {x, 0, 21}], x] (* Michael De Vlieger, Mar 10 2018 *)
LinearRecurrence[{9, -12}, {0, 1}, 30] (* Vincenzo Librandi, Mar 11 2018 *)
  • PARI
    concat(0, Vec(x / (12*x^2-9*x+1) + O(x^30))) \\ Colin Barker, Mar 12 2018

Formula

G.f.: x/(12*x^2-9*x+1). - Alois P. Heinz, Mar 10 2018
From Colin Barker, Mar 12 2018: (Start)
a(n) = (-((9-sqrt(33))/2)^n + ((9+sqrt(33))/2)^n) / sqrt(33).
a(n) = 9*a(n-1) - 12*a(n-2) for n>1.
(End)
E.g.f.: 2*exp(9*x/2)*sinh(sqrt(33)*x/2)/sqrt(33). - Stefano Spezia, Dec 24 2021

Extensions

More terms from Altug Alkan, Mar 10 2018

A299916 a(n) = A299914(2n+1).

Original entry on oeis.org

1, 6, 42, 306, 2250, 16578, 122202, 900882, 6641514, 48963042, 360969210, 2661166386, 19618866954, 144635805954, 1066295850138, 7861032979794, 57953746616490, 427251323790882, 3149816954720058, 23221336706989938, 171194226906268746, 1262092001672539458
Offset: 0

Views

Author

N. J. A. Sloane, Mar 10 2018

Keywords

Comments

a(n) is the number of holes shaped like six-pointed stars, in descending size, found in the cross-section, in the shape of a regular hexagon, of a Menger Sponge. - Albert Säfström, Jul 25 2018

References

  • Murat Sahin and Elif Tan, Conditional (strong) divisibility sequences, Fib. Q., 56 (No. 1, 2018), 18-31.

Links

Crossrefs

Cf. A299914.

Programs

  • Magma
    I:=[1,6]; [n le 2 select I[n] else 9*Self(n-1)-12*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 11 2018
  • Maple
    a:= n-> (<<0|1>, <-12|9>>^n. <<1, 6>>)[1, 1]:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 10 2018
  • Mathematica
    CoefficientList[Series[-(3 x - 1)/(12 x^2 - 9 x + 1), {x, 0, 20}], x] (* Michael De Vlieger, Mar 10 2018 *)
LinearRecurrence[{9, -12}, {1, 6}, 30] (* Vincenzo Librandi, Mar 11 2018 *)
  • PARI
    Vec((1 - 3*x) / (1 - 9*x + 12*x^2) + O(x^30)) \\ Colin Barker, Mar 12 2018

Formula

G.f.: -(3*x-1)/(12*x^2-9*x+1). - Alois P. Heinz, Mar 10 2018
From Colin Barker, Mar 12 2018: (Start)
a(n) = 2^(-1-n)*((9-sqrt(33))^n*(-3+sqrt(33)) + (3+sqrt(33))*(9+sqrt(33))^n) / sqrt(33).
a(n) = 9*a(n-1) - 12*a(n-2) for n>1.
(End)

Extensions

More terms from Altug Alkan, Mar 10 2018

A299913 a(n) = a(n-1) + 2*a(n-2) if n even, or 3*a(n-1) + 4*a(n-2) if n odd, starting with 0, 1.

Original entry on oeis.org

0, 1, 1, 7, 9, 55, 73, 439, 585, 3511, 4681, 28087, 37449, 224695, 299593, 1797559, 2396745, 14380471, 19173961, 115043767, 153391689, 920350135, 1227133513, 7362801079, 9817068105, 58902408631, 78536544841, 471219269047, 628292358729, 3769754152375, 5026338869833
Offset: 0

Views

Author

N. J. A. Sloane, Mar 10 2018

Keywords

References

  • Murat Sahin and Elif Tan, Conditional (strong) divisibility sequences, Fib. Q., 56 (No. 1, 2018), 18-31.

Links

Crossrefs

Bisections give A023001, A083068.
Cf. A299914, A299915, A299916.

Programs

  • Maple
    a:= n-> (<<0|1|0>, <0|0|1>, <8|8|-1>>^n. <<0, 1, 1>>)[1,1]:
seq(a(n), n=0..35);  # Alois P. Heinz, Mar 10 2018
  • Mathematica
    Fold[Append[#1, Inner[Times, 2 Boole[OddQ@ #2] + {1, 2}, {#1[[-1]], #1[[-2]]}, Plus]] &, {0, 1}, Range[2, 30]] (* or *)
CoefficientList[Series[-x (2 x + 1)/((x + 1) (8 x^2 - 1)), {x, 0, 30}], x] (* Michael De Vlieger, Mar 10 2018 *)
nxt[{n_,a_,b_}]:={n+1,b,If[OddQ[n],b+2a,3b+4a]}; NestList[nxt,{1,0,1},30][[;;,2]] (* Harvey P. Dale, Mar 02 2025 *)
  • PARI
    concat(0, Vec(x*(1 + 2*x) / ((1 + x)*(1 - 8*x^2)) + O(x^40))) \\ Colin Barker, Mar 11 2018

Formula

G.f.: -x*(2*x+1)/((x+1)*(8*x^2-1)). - Alois P. Heinz, Mar 10 2018
From Colin Barker, Mar 11 2018: (Start)
a(n) = (2^(3*n/2) - 1) / 7 for n even.
a(n) = 3*2^((3*(n-1))/2+1)/7 + 1/7 for n odd.
a(n) = -a(n-1) + 8*a(n-2) + 8*a(n-3) for n>2.
(End)

Extensions

More terms from Altug Alkan, Mar 10 2018
Showing 1-3 of 3 results.

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

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