A168613 -id:A168613 - cp's OEIS frontend

A168610 a(n) = 3^n + 5.

6, 8, 14, 32, 86, 248, 734, 2192, 6566, 19688, 59054, 177152, 531446, 1594328, 4782974, 14348912, 43046726, 129140168, 387420494, 1162261472, 3486784406, 10460353208, 31381059614, 94143178832, 282429536486, 847288609448

Offset: 0

Vincenzo Librandi, Dec 01 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A168613.

I:=[6, 8]; [n le 2 select I[n] else 4*Self(n-1)-3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jul 06 2012

CoefficientList[Series[2*(3-8*x)/((1-x)*(1-3*x)),{x,0,40}],x] (* Vincenzo Librandi, Jul 06 2012 *)
LinearRecurrence[{4,-3}, {6, 8}, 25] (* G. C. Greubel, Jul 27 2016 *)

a(n)=3^n+5 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 3*a(n-1) - 10 with a(0)=6.
G.f.: 2*(3 - 8*x)/((1-x)*(1-3*x)). - Vincenzo Librandi, Jul 06 2012
a(n) = 4*a(n-1) -3*a(n-2). - Vincenzo Librandi, Jul 06 2012
a(n) = 2*A115098(n)+2. - Bruno Berselli, Jul 06 2012
E.g.f.: exp(3*x) + 5*exp(x). - G. C. Greubel, Jul 27 2016

Formula and examples edited to use correct offset by Jon E. Schoenfield, Jun 19 2010

A367606 Comma-successor to n working in base 3, but written in base 10, or -1 if n has no successor.

Original entry on oeis.org

5, 9, 4, -1, 12, 8, 11, 15, 10, 14, 19, 13, 17, 22, 16, 21, 25, 20, 24, 27, 23, -1, 30, 26, 29, 33, 28, 32, 36, 31, 35, 39, 34, 38, 42, 37, 41, 45, 40, 44, 48, 43, 47, 51, 46, 50, 55, 49, 53, 58, 52, 57, 61, 56, 60, 64, 59, 63, 67, 62, 66, 70, 65, 69, 73, 68, 72, 76, 71, 75, 79, 74, 78, 81, 77, -1, 84, 80, 83, 87, 82

Offset: 1

N. J. A. Sloane, Dec 11 2023

This is a base-3 analog of A367338.
It seems that the indices of the terms equal to -1 are in A168613. - Ivan N. Ianakiev, Dec 12 2023
This is true for A168613(n), n >= 2.  See proofs in A367341. - Michael S. Branicky, Dec 15 2023

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A121085, A168613, A367338, A367341, A367355, A367356, A367607, A367608, A367609.

from sympy.ntheory.factor_ import digits
def a(n):
    b = n + 3*(n%3)
    return next((b+y for y in [1, 2] if digits(b+y, 3)[1] == y), -1)
print([a(n) for n in range(1, 82)]) # Michael S. Branicky, Dec 11 2023

A277104 a(n) = 9*3^n - 15.

Original entry on oeis.org

12, 66, 228, 714, 2172, 6546, 19668, 59034, 177132, 531426, 1594308, 4782954, 14348892, 43046706, 129140148, 387420474, 1162261452, 3486784386, 10460353188, 31381059594, 94143178812, 282429536466, 847288609428, 2541865828314, 7625597484972, 22876792454946

Offset: 1

Emeric Deutsch, Nov 05 2016

a(n) is the first Zagreb index of the Hanoi graph H[n].
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices.  Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph.
The M-polynomial of the Hanoi graph H[n] is M(H[n],x,y) = 6*x^2*y^3 + (3/2)*(3^n - 5)*x^3*y^3.

E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
I. Gutman and K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.
Eric. W. Weisstein's World of Mathematics, Hanoi Graph
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A277105.

[9*3^n-15: n in [1..30]]; // Bruno Berselli, Nov 14 2016

Maple
```
seq(9*3^n-15, n = 1..30);
```

Table[9 3^n - 15, {n, 1, 30}] (* Bruno Berselli, Nov 14 2016 *)

a(n)=3^(n+2)-15 \\ Charles R Greathouse IV, Nov 14 2016

O.g.f.: 6*x*(2 + 3*x)/((1 - x)*(1 - 3*x)).
E.g.f.: 3*(1 - exp(x))*(2 - 3*exp(x) - 3*exp(2*x)). - Bruno Berselli, Nov 14 2016
a(n) = 3*A168613(n+1). - R. J. Mathar, Apr 07 2022

cp's OEIS Frontend

A168610 a(n) = 3^n + 5.

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A367606 Comma-successor to n working in base 3, but written in base 10, or -1 if n has no successor.

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Python

A277104 a(n) = 9*3^n - 15.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula