|
%I #28 Sep 08 2022 08:45:34
%S 1,2,4,8,14,20,26,32,38,44,50,56,62,68,74,80,86,92,98,104,110,116,122,
%T 128,134,140,146,152,158,164,170,176,182,188,194,200,206,212,218,224,
%U 230,236,242,248,254,260,266,272,278,284,290,296,302,308,314,320,326
%N Binomial transform of [1, 1, 1, 1, -1, -1, 5, -11, 19, -29, 41, ...].
%C Sum of antidiagonal terms of the following arithmetic array:
%C 1, 1, 1, 1, 1, 1, 1, 1, ...
%C 1, 2, 3, 4, 5, 6, 7, 8, ...
%C 1, 3, 5, 7, 9, 11, 13, 15, ...
%C 1, 4, 7, 10, 13, 16, 19, 22, ...
%C 1, 5, 9, 13, 17, 21, 25, 29, ...
%C 1, 6, 11, 16, 21, 26, 31, 36, ...
%C 1, 7, 13, 19, 25, 31, 37, 43, ...
%C 1, 8, 15, 22, 29, 36, 43, 50, ...
%C ...
%C For [1, 1, 1, 1, -1, -1, 5, -11, 19, -29, 41, -55, ...], see A??????.
%H G. C. Greubel, <a href="/A140164/b140164.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F Binomial transform of [1, 1, 1, 1, -1, -1, 5, -11, 19, -29, 41, -55,...]; where A028387 = (1, 5, 11, 19, 29, 41,...), such that A028387(n) = (2*T(n) - 1).
%F From _R. J. Mathar_, May 03 2010: (Start)
%F G.f.: x*(1+x^2+2*x^3+2*x^4)/(1-x)^2. [G.f. amended by _Georg Fischer_, May 12 2019]
%F a(n) = A016933(n-2), n>2. (End)
%F a(n) = 2*(3n-5), n >= 3, if offset is 0 instead of 1. - _Daniel Forgues_, Jul 17 2016
%e a(4) = 8 = (1, 3, 3, 1) dot (1, 1, 1, 1) = (1 + 3 + 3 + 1).
%e a(5) = 14 = (4 + 5 + 4 + 1).
%p From _R. J. Mathar_, May 03 2010: (Start)
%p A028387 := proc(n) option remember; if n <= 2 then op(n+1,[1,5,11]) ; else 3*procname(n-1)-3*procname(n-2)+procname(n-3) ; end if; end proc:
%p read("transforms") ; L := [1,1,1,1,-1, seq((-1)^(n+1)*A028387(n), n=0..60)]; BINOMIAL(L) ; (End)
%t Table[If[n < 4, 2^(n - 1), 6 n - 16], {n, 60}] (* or *)
%t Rest@CoefficientList[Series[x*(1+x^2+2x^3+2x^4)/(1-x)^2, {x, 0, 60}], x] (* _Michael De Vlieger_, Jul 18 2016 *)
%o (PARI) a(n)=if(n<4,2^(n-1),6*n-16) \\ _Charles R Greathouse IV_, Jul 17 2016
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( x*(1+x^2+2*x^3+2*x^4)/(1-x)^2 )); // _G. C. Greubel_, May 12 2019
%o (Sage) (x*(1+x^2+2*x^3+2*x^4)/(1-x)^2).series(x, 60).coefficients(x, sparse=False) # _G. C. Greubel_, May 12 2019
%o (GAP) Concatenation([1,2,4], List([4..60], n-> 6*n-16)) # _G. C. Greubel_, May 12 2019
%Y Cf. A028387.
%K nonn,easy
%O 1,2
%A _Gary W. Adamson_, May 10 2008
%E More terms from _R. J. Mathar_, May 03 2010
|
