|
%I M2892 N1159 #65 Oct 20 2023 21:12:14
%S 1,3,11,39,131,423,1331,4119,12611,38343,116051,350199,1054691,
%T 3172263,9533171,28632279,85962371,258018183,774316691,2323474359,
%U 6971471651,20916512103,62753730611,188269580439,564825518531,1694510110023,5083597438931,15250926534519
%N a(n) = 2*(3^n - 2^n) + 1.
%C Binomial transform of A095121. - _R. J. Mathar_, Oct 05 2012
%C 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
%C First differences of A090326 (with offset 1). - _Wesley Ivan Hurt_, Jul 08 2014
%D H. Gupta, On a problem in parity, Indian J. Math., 11 (1969), 157-163.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A002783/b002783.txt">Table of n, a(n) for n = 0..1000</a>
%H H. Gupta, <a href="/A002783/a002783_1.pdf">On a problem in parity</a>, Indian J. Math., 11 (1969), 157-163. [Annotated scanned copy]
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6).
%F G.f.: ( -1+3*x-4*x^2 ) / ( (x-1)*(3*x-1)*(2*x-1) ). - _Simon Plouffe_ in his 1992 dissertation
%F a(n+1) - a(n) = 2*A027649(n). - _R. J. Mathar_, Oct 05 2012
%e From _J. M. Bergot_ and _M. F. Hasler_, Oct 10 2012: (Start)
%e 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
%e 1
%e 3 3
%e 3 6 3
%e 1 9 9 1
%e and the sum of all the elements is 39 = a(3). (End)
%p A002783:=n->2*(3^n - 2^n)+1: seq(A002783(n), n=0..30); # _Wesley Ivan Hurt_, Jul 08 2014
%t 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 *)
%o (Magma) [2*(3^n - 2^n)+1 : n in [0..30]]; // _Wesley Ivan Hurt_, Jul 08 2014
%o (PARI) a(n)=2*(3^n-2^n)+1 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y Cf. A027649, A090326, A095121.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Wesley Ivan Hurt_, Jul 08 2014
|
