A002783 a(n) = 2*(3^n - 2^n) + 1.
1, 3, 11, 39, 131, 423, 1331, 4119, 12611, 38343, 116051, 350199, 1054691, 3172263, 9533171, 28632279, 85962371, 258018183, 774316691, 2323474359, 6971471651, 20916512103, 62753730611, 188269580439, 564825518531, 1694510110023, 5083597438931, 15250926534519
Offset: 0
Examples
From _J. M. Bergot_ and _M. F. Hasler_, Oct 10 2012: (Start)
For n=3, the triangle with left and right border (1,3,3,1) and internal terms m(i,j) = m(i-1,j-1) + m(i-1,j) is
1
3 3
3 6 3
1 9 9 1
and the sum of all the elements is 39 = a(3). (End)
References
- H. Gupta, On a problem in parity, Indian J. Math., 11 (1969), 157-163.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- H. Gupta, On a problem in parity, Indian J. Math., 11 (1969), 157-163. [Annotated scanned copy]
- 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.
- Index entries for linear recurrences with constant coefficients, signature (6,-11,6).
Programs
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Magma
[2*(3^n - 2^n)+1 : n in [0..30]]; // Wesley Ivan Hurt, Jul 08 2014
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Maple
A002783:=n->2*(3^n - 2^n)+1: seq(A002783(n), n=0..30); # Wesley Ivan Hurt, Jul 08 2014
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Mathematica
CoefficientList[Series[(-1 + 3*x - 4*x^2)/((x - 1)*(3*x - 1)*(2*x - 1)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jul 08 2014 *) -
PARI
a(n)=2*(3^n-2^n)+1 \\ Charles R Greathouse IV, Oct 07 2015
Formula
G.f.: ( -1+3*x-4*x^2 ) / ( (x-1)*(3*x-1)*(2*x-1) ). - Simon Plouffe in his 1992 dissertation
a(n+1) - a(n) = 2*A027649(n). - R. J. Mathar, Oct 05 2012
E.g.f.: exp(x)*(1 - 2*exp(x) + 2*exp(2*x)). - Stefano Spezia, May 18 2024
Extensions
More terms from Wesley Ivan Hurt, Jul 08 2014
Comments