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Centers of rows of the triangular array formed by the natural numbers.
5

%I #49 Aug 12 2018 13:21:51

%S 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 145,162,163,181,200,201,221,242,243,265,288,289,313,338,339,365,392,

%U 393,421,450,451,481,512,513,545,578,579,613,648,649,685,722,723

%N Centers of rows of the triangular array formed by the natural numbers.

%C 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 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 Union of A080827 and A000982. - _David James Sycamore_, Aug 09 2018

%H Reinhard Zumkeller, <a href="/A251599/b251599.txt">Table of n, a(n) for n = 1..10000</a>

%H David James Sycamore, <a href="/A251599/a251599.jpg">A080827 & A000982</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-2,0,-1,1).

%F 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 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 First ten terms (1,2,3,5,8,9,13,18,19,25) may be read down the center of the triangular formation:

%e 1

%e 2 3

%e 4 5 6

%e 7 8 9 10

%e 11 12 13 14 15

%e 16 17 18 19 20 21

%e 22 23 24 25 26 27 28

%p a:= n-> (m-> 2*(m+1)^2-[2*m+1, 0, -1][1+r])(iquo(n-1, 3, 'r')):

%p seq(a(n), n=1..100); # _Alois P. Heinz_, Dec 10 2014

%t LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {1, 2, 3, 5, 8, 9, 13}, 60] (* _Jean-François Alcover_, Jan 09 2016 *)

%o (Haskell)

%o a251599 n = a251599_list !! (n-1)

%o a251599_list = f 0 $ g 1 [1..] where

%o f i (us:vs:wss) = [head $ drop i us] ++ (take 2 $ drop i vs) ++

%o f (i + 1) wss

%o g k zs = ys : g (k + 1) xs where (ys,xs) = splitAt k zs

%o -- _Reinhard Zumkeller_, Dec 12 2014

%o (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 Cf. A000124, A000217, A001844.

%Y Cf. A092942 (first differences).

%Y Cf. A001105, A056106, A058331.

%Y Cf. A080827, A000982.

%K nonn

%O 1,2

%A _Dave Durgin_, Dec 05 2014