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

Author

Zerinvary Lajos, Feb 03 2007

Keywords

• easy
• nonn

Comments

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).

Examples

			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
		

References

• 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.

Links

• 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).

Crossrefs

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

Programs

GAP

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

Magma

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

Maple

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

Mathematica

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

PARI

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

Sage

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

Formula

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)

Extensions

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

Major edit by Joerg Arndt, Jun 11 2011