cp's OEIS Frontend

A251599 Centers of rows of the triangular array formed by the natural numbers.

%I A251599 #49 Aug 12 2018 13:21:51
%S A251599 1,2,3,5,8,9,13,18,19,25,32,33,41,50,51,61,72,73,85,98,99,113,128,129,
%T A251599 145,162,163,181,200,201,221,242,243,265,288,289,313,338,339,365,392,
%U A251599 393,421,450,451,481,512,513,545,578,579,613,648,649,685,722,723
%N A251599 Centers of rows of the triangular array formed by the natural numbers.
%C A251599 Forms a cascade of 3-number triangles down the center of the triangle array. Related to A000124 (left/west bank of same triangular array), A000217 (right/east bank) and A001844 (center column).
%C A251599 Sums of the mentioned cascading triangles: a(3*n-2) + a(3*n-1) + a(3*n) = A058331(n) + A001105(n) + A001844(n-1) = 2*A056106(n) = 2*(3*n^2-n+1). - _Reinhard Zumkeller_, Dec 13 2014
%C A251599 Union of A080827 and A000982. - _David James Sycamore_, Aug 09 2018
%H A251599 Reinhard Zumkeller, <a href="/A251599/b251599.txt">Table of n, a(n) for n = 1..10000</a>
%H A251599 David James Sycamore, <a href="/A251599/a251599.jpg">A080827 & A000982</a>
%H A251599 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-2,0,-1,1).
%F A251599 Terms for n=1 (mod 3): 2m^2+2m+1, for n=2 (mod 3): 2m^2+4m+2, for n=0 (mod 3): 2m^2+4m+3, where m = floor((n-1)/3).
%F A251599 G.f.: -x*(x^2+1)*(x^4-x^3+x+1)/((x^2+x+1)^2*(x-1)^3). - _Alois P. Heinz_, Dec 10 2014
%e A251599 First ten terms (1,2,3,5,8,9,13,18,19,25) may be read down the center of the triangular formation:
%e A251599                1
%e A251599              2   3
%e A251599            4   5   6
%e A251599          7   8   9  10
%e A251599       11  12  13  14  15
%e A251599     16  17  18  19  20  21
%e A251599   22  23  24  25  26  27  28
%p A251599 a:= n-> (m-> 2*(m+1)^2-[2*m+1, 0, -1][1+r])(iquo(n-1, 3, 'r')):
%p A251599 seq(a(n), n=1..100);  # _Alois P. Heinz_, Dec 10 2014
%t A251599 LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {1, 2, 3, 5, 8, 9, 13}, 60] (* _Jean-François Alcover_, Jan 09 2016 *)
%o A251599 (Haskell)
%o A251599 a251599 n = a251599_list !! (n-1)
%o A251599 a251599_list = f 0 $ g 1 [1..] where
%o A251599    f i (us:vs:wss) = [head $ drop i us] ++ (take 2 $ drop i vs) ++
%o A251599                      f (i + 1) wss
%o A251599    g k zs = ys : g (k + 1) xs where (ys,xs) = splitAt k zs
%o A251599 -- _Reinhard Zumkeller_, Dec 12 2014
%o A251599 (PARI) Vec(-x*(x^2+1)*(x^4-x^3+x+1)/((x^2+x+1)^2*(x-1)^3) + O(x^80)) \\ _Michel Marcus_, Jan 09 2016
%Y A251599 Cf. A000124, A000217, A001844.
%Y A251599 Cf. A092942 (first differences).
%Y A251599 Cf. A001105, A056106, A058331.
%Y A251599 Cf. A080827, A000982.
%K A251599 nonn
%O A251599 1,2
%A A251599 _Dave Durgin_, Dec 05 2014