A128209 Jacobsthal numbers(A001045) + 1.

1, 2, 2, 4, 6, 12, 22, 44, 86, 172, 342, 684, 1366, 2732, 5462, 10924, 21846, 43692, 87382, 174764, 349526, 699052, 1398102, 2796204, 5592406, 11184812, 22369622, 44739244, 89478486, 178956972, 357913942, 715827884, 1431655766, 2863311532, 5726623062

Offset: 0

Author

Paul Barry, Feb 19 2007

Keywords

• easy
• nonn

Comments

Row sums of A128208.

Essentially the same as A052953. - R. J. Mathar, Jun 14 2008

Let I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then, for n >= 1, a(n+1) is the number of different representations of matrix P^(-1)+I+P by sum of permutation matrices. - Vladimir Shevelev, Apr 12 2010

a(n) is the rank of Fibonacci(n+2) in row n of A049456 (regarded as an irregular triangle read by rows). - N. J. A. Sloane, Nov 23 2016

References

• V. S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, 3(1992),15-19. [From Vladimir Shevelev, Apr 12 2010]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• D. H. Lehmer, On Stern's Diatomic Series, Amer. Math. Monthly 36(1) 1929, pp. 59-67. [Annotated and corrected scanned copy]
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Crossrefs

Cf. A167030, A153643, A049456.

Programs

Magma

[1+2^n/3-(-1)^n/3: n in [0..40]]; // Vincenzo Librandi, Aug 11 2011

Mathematica

CoefficientList[Series[(1-3*x^2)/(1-2*x-x^2+2*x^3),{x,0,40}],x] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2012 *)

PARI

a(n)=2^n\/3+1 \\ Charles R Greathouse IV, Jan 31 2012

Python

def A128209(n): return ((1<Chai Wah Wu, Apr 17 2025

Formula

a(n) = 1 + 2^n/3 - (-1)^n/3.

G.f.: (1-3*x^2)/(1 - 2*x - x^2 + 2*x^3).