A002783 a(n) = 2*(3^n - 2^n) + 1. (Formerly M2892 N1159)

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

0,2

Comments

Binomial transform of A095121. - R. J. Mathar, Oct 05 2012

Create a triangle having its left and right border both equal to the n-th row of Pascal's triangle, and internal terms m(i,j) = m(i-1,j-1) + m(i-1,j). Then the sum of all elements equals a(n). - J. M. Bergot, Oct 07 2012, edited by M. F. Hasler, Oct 10 2012

First differences of A090326 (with offset 1). - Wesley Ivan Hurt, Jul 08 2014

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).

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

Example

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)

Maple

A002783:=n->2*(3^n - 2^n)+1: seq(A002783(n), n=0..30); # Wesley Ivan Hurt, Jul 08 2014

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 *)

Prog

(Magma) [2*(3^n - 2^n)+1 : n in [0..30]]; // Wesley Ivan Hurt, Jul 08 2014
(PARI) a(n)=2*(3^n-2^n)+1 \\ Charles R Greathouse IV, Oct 07 2015

Crossrefs

Cf. A027649, A090326, A095121.

Sequence in context: A089579 A227638 A166336 * A289834 A007482 A134760

Adjacent sequences: A002780 A002781 A002782 * A002784 A002785 A002786

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Wesley Ivan Hurt, Jul 08 2014

Status

approved