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URL: https://oeis.org/A006138

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A006138
a(n) = a(n-1) + 3*a(n-2).
15
1, 2, 5, 11, 26, 59, 137, 314, 725, 1667, 3842, 8843, 20369, 46898, 108005, 248699, 572714, 1318811, 3036953, 6993386, 16104245, 37084403, 85397138, 196650347, 452841761, 1042792802, 2401318085, 5529696491, 12733650746, 29322740219, 67523692457, 155491913114
OFFSET
0,2
COMMENTS
The binomial transform of a(n) is b(n) = A006190(n+1), which satisfies b(n) = 3*b(n-1) + b(n-2). - Paul Barry, May 21 2006
Partial sums of A105476. - Paul Barry, Feb 02 2007
An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 69, 261, 321 and 324, lead to this sequence. For the central square these vectors lead to the companion sequence A105476 (without the first leading 1). - Johannes W. Meijer, Aug 15 2010
Equals the INVERTi transform of A063782: (1, 3, 10, 32, 104, ...). - Gary W. Adamson, Aug 14 2010
Pisano period lengths: 1, 3, 1, 6, 24, 3, 24, 6, 3, 24, 120, 6, 156, 24, 24, 12, 16, 3, 90, 24, ... - R. J. Mathar, Aug 10 2012
The sequence is the INVERT transform of A016116: (1, 1, 2, 2, 4, 4, 8, 8, ...). - Gary W. Adamson, Jul 17 2015
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
N. T. Gridgeman, A new look at Fibonacci generalization, Fib,. Quart., 11 (1973), 40-55.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
FORMULA
a(n) = Sum_{k=0..n+1} A122950(n+1,k)*2^(n+1-k). - Philippe Deléham, Jan 04 2008
G.f.: (1+x)/(1-x-3*x^2). - Paul Barry, May 21 2006
a(n) = Sum_{k=0..n} C(floor((2n-k)/2),n-k)*3^floor(k/2). - Paul Barry, Feb 02 2007
a(n) = A006130(n) + A006130(n-1). - R. J. Mathar, Jun 22 2011
a(n) = (i*sqrt(3))^(n-1)*(i*sqrt(3)*ChebyshevU(n, 1/(2*i*sqrt(3))) + ChebyshevU(n-1, 1/(2*i*sqrt(3)))), where i=sqrt(-1). - G. C. Greubel, Nov 19 2019
MAPLE
A006138:=-(1+z)/(-1+z+3*z**2); # Simon Plouffe in his 1992 dissertation
MATHEMATICA
CoefficientList[Series[(1+z)/(1-z-3*z^2), {z, 0, 40}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
Table[(I*Sqrt[3])^(n-1)*(I*Sqrt[3]*ChebyshevU[n, 1/(2*I*Sqrt[3])] + ChebyshevU[n-1, 1/(2*I*Sqrt[3])]), {n, 0, 40}]//Simplify (* G. C. Greubel, Nov 19 2019 *)
LinearRecurrence[{1, 3}, {1, 2}, 40] (* Harvey P. Dale, May 29 2025 *)
PROG
(PARI) main(size)={my(v=vector(size), i); v[1]=1; v[2]=2; for(i=3, size, v[i]=v[i-1]+3*v[i-2]); return(v); } /* Anders Hellström, Jul 17 2015 */
(Magma) [n le 2 select n else Self(n-1)+3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Sep 15 2016
(SageMath)
def A006138_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P((1+x)/(1-x-3*x^2)).list()
A006138_list(40) # G. C. Greubel, Nov 19 2019
(GAP) a:=[1, 2];; for n in [3..40] do a[n]:=a[n-1]+3*a[n-2]; od; a; # G. C. Greubel, Nov 19 2019
CROSSREFS
KEYWORD
nonn,easy
EXTENSIONS
Typo in formula corrected by Johannes W. Meijer, Aug 15 2010
STATUS
approved