A093129 -id:A093129 - cp's OEIS frontend

A260304 a(n) = 5a(n-1) - 5a(n-2) for n>1, a(0)=2, a(1)=3.

2, 3, 5, 10, 25, 75, 250, 875, 3125, 11250, 40625, 146875, 531250, 1921875, 6953125, 25156250, 91015625, 329296875, 1191406250, 4310546875, 15595703125, 56425781250, 204150390625, 738623046875, 2672363281250, 9668701171875, 34981689453125, 126564941406250

Offset: 0

Ilya Gutkovskiy, Nov 21 2015

Lim_{n -> infinity} a(n + 1)/a(n) = 2 + phi = 3.6180339887..., where phi is the golden ratio (A001622).

Index entries for linear recurrences with constant coefficients, signature (5,-5).

Cf. A020876, A030191, A098111.

Cf. A093129: initial values 1,2; A081567: initial values 1,3.

[n le 2 select n+1 else 5*Self(n-1)-5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 23 2015

Table[((5 + 2 Sqrt[5]) ((5 - Sqrt[5])/2)^n + (5 - 2 Sqrt[5]) ((5 + Sqrt[5])/2)^n)/5, {n, 0, 30}]
RecurrenceTable[{a[0] == 2, a[1] == 3, a[n] == 5 a[n - 1] - 5 a[n - 2]}, a, {n, 0, 30}] (* Bruno Berselli, Nov 23 2015 *)

a(n)=([0,1; -5,5]^n*[2;3])[1,1] \\ Charles R Greathouse IV, Jul 26 2016

G.f.: (2 - 7*x)/(1 - 5*x + 5*x^2).
a(n) = ((5 + 2*sqrt(5))*((5 - sqrt(5))/2)^n + (5 - 2*sqrt(5))*((5 + sqrt(5))/2)^n)/5.
a(n) = 2*A030191(n) - 7*A030191(n-1). - Bruno Berselli, Nov 23 2015

Edited by Bruno Berselli, Nov 23 2015

A320101 Table read by rows: T(n,k) is the number of no-leaf subgraphs of the n X k grid where 1 <= k <= n.

Original entry on oeis.org

1, 1, 2, 1, 5, 43, 1, 15, 463, 16372, 1, 50, 5193, 583199, 65485654, 1, 175, 58653, 20788249, 7354266811, 2602065897364, 1, 625, 663203, 741026781, 825905301851

Offset: 1

Peter Kagey, Oct 05 2018

			Three of the T(4,3) = 463 subgraphs of the 4 X 3 grid with no leaf vertices are
  +   +---+      +   +   +      +   +---+
      |   |                         |   |
  +---+---+      +---+---+      +   +---+
  |   |    ,     |   |   |, and          .
  +---+   +      +   +---+      +---+   +
  |   |          |   |          |   |
  +---+   +      +---+   +      +---+   +
Table begins:
  n\k|    1    2     3        4          5             6
  ---+---------------------------------------------------
   1 |    1
   2 |    1    2
   3 |    1    5    43
   4 |    1   15   463    16372
   5 |    1   50  5193   583199   65485654
   6 |    1  175 58653 20788249 7354266811 2602065897364

Cf. A093129, A301976, A320097, A320099.

cp's OEIS Frontend

A260304 a(n) = 5a(n-1) - 5a(n-2) for n>1, a(0)=2, a(1)=3.

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A320101 Table read by rows: T(n,k) is the number of no-leaf subgraphs of the n X k grid where 1 <= k <= n.

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cp's OEIS Frontend

A260304 a(n) = 5*a(n-1) - 5*a(n-2) for n>1, a(0)=2, a(1)=3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A320101 Table read by rows: T(n,k) is the number of no-leaf subgraphs of the n X k grid where 1 <= k <= n.

Views

Author

Keywords

Examples

Crossrefs

A260304 a(n) = 5a(n-1) - 5a(n-2) for n>1, a(0)=2, a(1)=3.