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.

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A146885 a(n) = 8*Sum_{k=0..n} 7^k.

Original entry on oeis.org

8, 64, 456, 3200, 22408, 156864, 1098056, 7686400, 53804808, 376633664, 2636435656, 18455049600, 129185347208, 904297430464, 6330082013256, 44310574092800, 310174018649608, 2171218130547264, 15198526913830856
Offset: 0

Views

Author

Roger L. Bagula, Nov 02 2008

Keywords

Links

Crossrefs

Cf. A005843, A023000, A068156, A100774, A132583, A146882, A146883, A146885.

Programs

  • Magma
    [n le 2 select 8^n else 8*Self(n-1) -7*Self(n-2): n in [1..41]]; // G. C. Greubel, Oct 12 2022
  • Mathematica
    a[n_]:= Sum[8*7^m, {m,0,n}]; Table[a[n], {n,0,30}]
LinearRecurrence[{8,-7}, {8,64}, 41] (* G. C. Greubel, Oct 12 2022 *)
  • SageMath
    [(4/3)*(7^(n+1)-1) for n in range(41)] # G. C. Greubel, Oct 12 2022

Formula

From G. C. Greubel, Oct 12 2022: (Start)
a(n) = (4/3)*(7^(n+1) - 1).
a(n) = 8*A023000(n+1).
a(n) = 8*a(n-1) - 7*a(n-2).
G.f.: 8/((1-x)*(1-7*x)).
E.g.f.: (4/3)*(7*exp(7*x) - exp(x)). (End)

A277874 a(n) is the last number in the (2n+1)-element alternating sequence of x/2 and (3x+1) iterations starting with A277215(n).

Original entry on oeis.org

0, 40, 16, 52, 160, 9232, 18952, 4372, 13120, 39364, 118096, 2480056, 5314408, 35075104, 9565936, 28697812, 86093440, 1807962280, 8523250756, 2324522932, 6973568800, 20920706404, 62762119216, 188286357652, 564859072960, 11862040532200, 25418658283288, 15251194969972, 45753584909920, 960825283108360
Offset: 0

Views

Author

Hartmut F. W. Hoft, Nov 03 2016

Keywords

Comments

a(n) has the form 2*(q*3^n - 1) where q is the smallest odd number so that the alternating Collatz sequence of 2n+1 elements starting at 2*(q*2^n - 1) ends at the maximum of its Collatz trajectory.
Subsequence of a(n) when q=1 is a subsequence of A100774.
Conjecture: this sequence is infinite.

Examples

			a(0) = 0 = 2*(1*3^0 - 1) since it is the start and end of the first alternating sequence of 1 element and the maximum of its trajectory.
a(5) = 9232 = 2*(19*3^5 - 1) is the last element in the first alternating sequence of 11 elements [1214, 607, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232] that ends in the trajectory maximum.

Crossrefs

Cf. A100774, A277215.

Programs

  • Mathematica
    (* we use function altdata[] from A277215 *)
a277874[n_]:=Map[#[[4]]&, altdata[2,n]]
Join[{0,40}, a277874[29]] (*sequence data*)

A373293 Number of edge covers of the fan graph F_{n,3}.

Original entry on oeis.org

16, 154, 1240, 9202, 66016, 466954, 3283240, 23026402, 161316016, 1129605754, 7908421240, 55362491602, 387548070016, 2712868376554, 18990174295240, 132931507044802, 930521410248016, 6513652454539354, 45595574930185240, 319169047756526002
Offset: 1

Views

Author

Feryal Alayont, Jun 22 2024

Keywords

Comments

Label vertices of F_{n,3} with v1, ..., v_n, a, b, c, with b adjacent to both a and c. An edge cover is a subset of the edges so that each vertex is the endpoint of at least one edge. Each of the n vertices has 7 ways to connect to vertices a, b, c. Once we connect all v_i's, we then have 4 options for the edges ab and bc to exist or not, giving 4*7^m options. But not all of these will be an edge cover. For example, if all v_i's connected to a and b only, we have to add edge bc in the second step. So 5*3^m are removed. But we removed 3 cases where all v_i's connected to only a, or only b, or only c too many times.

Links

Crossrefs

Cf. A100774 (in F_{n,2}).

Programs

  • Mathematica
    A373293[n_] := 4*7^n - 5*3^n + 3; Array[A373293, 25] (* or *)
LinearRecurrence[{11, -31, 21}, {16, 154, 1240}, 25] (* Paolo Xausa, Jun 24 2024 *)
  • Python
    def a_n(n):
    return 4 * 7**n - 5 * 3**n + 3

Formula

a(n) = 4*7^n - 5*3^n + 3.
From Stefano Spezia, Jun 24 2024: (Start)
G.f.: 2*x*(8 - 11*x + 21*x^2)/((1 - x)*(1 - 3*x)*(1 - 7*x)).
E.g.f.: 4*exp(7*x) - 5*exp(3*x) + 3*exp(x) - 2. (End)

A374450 Number of edge covers of the fan graph F_{n,4}.

Original entry on oeis.org

59, 1289, 23123, 376913, 5875499, 89719769, 1357012163, 20434006433, 307062808859, 4609813953449, 69174320548403, 1037804612461553, 15568397893099019, 233535269569297529, 3503094152437895843, 52546868050923710273, 788206211120541289979, 11823115499323514984009, 177346888817516282750483
Offset: 1

Views

Author

Marshall Nicholson, Jul 08 2024

Keywords

Comments

Label vertices of F_{n,4} with v1, ..., v_n, a, b, c, d with be adjacent to a and c and d adjacent to c. An edge cover is a subset of the edges so that each vertex is the endpoint of at least one edge. Each of the n vertices has 15 ways to connect to vertices a, b, c, d. Once we connect all v_i's, we then have 8 options for the edges ab, bc, and cd to exist or not, giving 8*15^n options. But not all of these will be an edge cover. For example, if all v_i's connect to only b and c, we have to add edges ab and cd in the second step. So, 12*7^n options are removed. However, we removed too many options here. This line of reasoning continues, following the Inclusion-Exclusion Principle, to obtain the remaining pieces of the formula for a(n).

Links

Crossrefs

Cf. A100774, A373293, A374096.

Programs

  • Mathematica
    A374450[n_] := 8*15^n - 12*7^n + 9*3^n - 4; Array[A374450, 20] (* or *)
LinearRecurrence[{26, -196, 486, -315}, {59, 1289, 23123, 376913}, 20] (* Paolo Xausa, Jan 22 2025 *)
  • Python
    def a_n(n):
    return 8 * 15**n - 12*7^n + 9 * 3**n + 3

Formula

a(n) = 8*15^n - 12*7^n + 9*3^n - 4.
From Stefano Spezia, Jul 08 2024: (Start)
G.f.: x*(59 - 245*x + 1173*x^2 - 315*x^3)/((1 - x)*(1 - 3*x)*(1 - 7*x)*(1 - 15*x)).
E.g.f.: 8*exp(15*x) - 12*exp(7*x) + 9*exp(3*x) - 4*exp(x) - 1. (End)
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