A121177 -id:A121177 - cp's OEIS frontend

A125831 a(n) = (5^n - 1)/2.

0, 2, 12, 62, 312, 1562, 7812, 39062, 195312, 976562, 4882812, 24414062, 122070312, 610351562, 3051757812, 15258789062, 76293945312, 381469726562, 1907348632812, 9536743164062, 47683715820312, 238418579101562, 1192092895507812, 5960464477539062, 29802322387695312

Offset: 0

Zerinvary Lajos, Feb 03 2007

Number of compositions of odd numbers into n parts < 5. - Adi Dani, Jun 11 2011

Numbers whose base 5 representation is 22222...2 (n times).

			a(2)=12: there are 12 compositions of odd numbers into 2 parts < 5:
1: (0,1),(1,0);
3: (0,3),(3,0),(1,2),(2,1);
5: (1,4),(4,1),(2,3),(3,2);
7: (3,4),(4,3). - _Adi Dani_, Jun 11 2011

S. J. Cyvin, B. N. Cyvin, and J. Brunvoll. Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), pp. 55-70, eqs. (6) and (7) on p. 58.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Adi Dani, Restricted compositions of natural numbers.
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A003463, A024049, A121177 (same with different offset).

List([0..30], n-> (5^n-1)/2); # G. C. Greubel, Aug 03 2019

[(5^n-1)/2: n in [0..30]]; // Vincenzo Librandi, Jun 11 2011

Maple
```
seq((5^n-1)/2, n=0..30);
```

Table[(5^n -1)/2, {n, 0, 30}] (* Harvey P. Dale, Dec 03 2010 *)

a(n)=5^n\2 \\ Charles R Greathouse IV, Jun 11 2011

[(5^n-1)/2 for n in (0..30)] # G. C. Greubel, Aug 03 2019

a(n) = 5*a(n-1) + 2 for n > 0, a(0)=0. - Vincenzo Librandi, Sep 30 2010
From Colin Barker, May 16 2013: (Start)
a(n) = 6*a(n-1) - 5*a(n-2).
G.f.: 2*x/((1-x)*(1-5*x)). (End)
a(n) = 2*A003463(n). - Joerg Arndt, Aug 03 2019
From Elmo R. Oliveira, Dec 10 2023: (Start)
a(n) = A024049(n)/2.
E.g.f.: (1/2)*(exp(5*x) - exp(x)). (End)

Offset corrected by N. J. A. Sloane, Oct 02 2010

Major edit by Joerg Arndt, Jun 11 2011

A121101 Catapolyoctagons (see Cyvin et al. for precise definition).

Original entry on oeis.org

1, 1, 3, 9, 39, 169, 819, 3969, 19719, 97969, 489219, 2442969, 12211719, 61042969, 305199219, 1525917969, 7629511719, 38147167969, 190735449219, 953675292969, 4768374511719, 23841862792969, 119209304199219, 596046472167969, 2980232312011719, 14901161315917969, 74505806335449219

Offset: 1

N. J. A. Sloane, Aug 11 2006

nonn

From Petros Hadjicostas, Jul 24 2019: (Start)
The sequence (a(n): n >= 1) counts the isomers of unbranched alpha-4-catapoly-q-qons with alpha = 0 and q = 8. It appears in Table 21 (p. 12) in Brunvoll et al. (1997).
An unbranched alpha-4-catapoly-q-gon consists of alpha tetragons and  n - alpha q-gons (where q > 4). Thus, n is the total number of polygons in the unbranched catacondensed polygonal system. Since we have alpha = 0 and q = 8 for this sequence, n counts the octagons.
The formula for a(n) below follows from the "master formula" I_{ra} in Exhibit 4 (p. 13) in Brunvoll et al. (1997) with alpha = 0 and q = 8 provided that a binomial coefficient of the form binomial(k, s) with s < 0 is set to zero.
Amazingly, the empirical g.f. of Colin Barker below is correct and follows easily from the formula for a(n) given below (with a(1) = 1).
(End)

S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134(1) (1997), 55-70; see Table I (p. 58).

J. Brunvoll, S. J. Cyvin and B. N. Cyvin, Isomer enumeration of polygonal systems representing polyclic conjugated hydrocarbons: unbalanced catacondensed systems with tetragons and q-gons, J. Molec. Struct. (Theochem), 364 (1996), 1-13.

Cf. A121102, A121177.

[1] cat [(1/4)*(1+5^(n-2)+2*(2-(-1)^n)*5^((n div 2)-1)): n in [2..30]]; // Vincenzo Librandi, Jul 26 2019

Join[{1}, Table[(1/4) (1 + 5^(r - 2) + 2 (2 - (-1)^r) 5^(Floor[r/2] - 1)), {r, 2, 30}]] (* Vincenzo Librandi, Jul 26 2019 *)

def A121101_list(prec):
    P. = PowerSeriesRing(ZZ, default_prec=prec)
    def g(x): return x*(10*x^4-21*x^3+3*x^2+5*x-1)/((x-1)*(5*x-1)*(5*x^2-1))
    return P(g(x)).list()
print(A121101_list(27)) # Peter Luschny, Jul 26 2019

G.f.: x*(10*x^4 - 21*x^3 + 3*x^2 + 5*x - 1) / ((x - 1)*(5*x - 1)*(5*x^2 - 1)). - Colin Barker, Aug 29 2013

a(r) = (1/4) * (1 + 5^(r-2) + 2 * (2-(-1)^r) * 5^(floor(r/2) - 1)) for r >= 2. - Petros Hadjicostas, Jul 24 2019

More terms from Petros Hadjicostas, Jul 24 2019 using the "master formula" in the references.

cp's OEIS Frontend

A125831 a(n) = (5^n - 1)/2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions