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%I #33 Sep 08 2022 08:45:30
%S 1,2,2,4,6,12,22,44,86,172,342,684,1366,2732,5462,10924,21846,43692,
%T 87382,174764,349526,699052,1398102,2796204,5592406,11184812,22369622,
%U 44739244,89478486,178956972,357913942,715827884,1431655766,2863311532,5726623062
%N Jacobsthal numbers(A001045) + 1.
%C Row sums of A128208.
%C Essentially the same as A052953. - _R. J. Mathar_, Jun 14 2008
%C 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
%C 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
%D 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]
%H Vincenzo Librandi, <a href="/A128209/b128209.txt">Table of n, a(n) for n = 0..1000</a>
%H D. H. Lehmer, <a href="/A002487/a002487_1.pdf">On Stern's Diatomic Series</a>, Amer. Math. Monthly 36(1) 1929, pp. 59-67. [Annotated and corrected scanned copy]
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2).
%F a(n) = 1 + 2^n/3 - (-1)^n/3.
%F G.f.: (1-3*x^2)/(1 - 2*x - x^2 + 2*x^3).
%t 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 *)
%o (Magma) [1+2^n/3-(-1)^n/3: n in [0..40]]; // _Vincenzo Librandi_, Aug 11 2011
%o (PARI) a(n)=2^n\/3+1 \\ _Charles R Greathouse IV_, Jan 31 2012
%Y Cf. A167030, A153643, A049456.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Feb 19 2007
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