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%I #40 Jul 27 2024 23:53:59
%S 0,3,10,26,56,112,196,331,522,790,1138,1615,2204,2975,3910,5041,6388,
%T 8047,9958,12262,14894,17920,21346,25347,29796,34875,40522,46854,
%U 53826,61716,70274,79883,90380,101875,114346,127981,142612,158737,176086,194827,214852,236717,259906,285124,311970,340588,370990,403819,438440,475556
%N Start with a 2 X n array of squares, join every vertex on top edge to every vertex on bottom edge; a(n) = one-half the number of cells.
%C Note that this figure can be obtained by drawing an "equatorial" line through the middle of the strip of n adjacent rectangles in A306302. This cuts each of the 2n "equatorial" cells in A306302 in two. It follows that 2*a(n) = A306302(n) + 2*n, i.e. that a(n) = A306302(n)/2 + n. Note that there is an explicit formula for A306302(n) in terms of n. - _Scott R. Shannon_, Sep 06 2022.
%C This means the present sequence is one more member of the large class of sequences which are essentially the same as A115004 (see Cross-References). - _N. J. A. Sloane_, Sep 06 2022
%H Scott R. Shannon, <a href="/A355902/a355902.jpg">Image for a(4) = 56</a>. Note in this and other images the entire 2xn array is shown so the number of cells is twice a(n).
%H Scott R. Shannon, <a href="/A355902/a355902_1.jpg">Image for a(6) = 196</a>.
%H Scott R. Shannon, <a href="/A355902/a355902_2.jpg">Image for a(10) = 1138</a>.
%H Scott R. Shannon, <a href="/A355902/a355902_3.jpg">Image for a(15) = 5041</a>.
%H N. J. A. Sloane, <a href="/A355902/a355902.png">Illustration for a(2) = 10.</a>
%H N. J. A. Sloane, <a href="/A355902/a355902_1.png">Illustration for a(3) = 26.</a>
%F a(n) = A356790(2,n+2)/2 - 2.
%Y Cf. A356790, A306302, A355798, A290131, A331452.
%Y The following nine sequences are all essentially the same. The simplest is A115004(n), which we denote by z(n). Then A088658(n) = 4*z(n-1); A114043(n) = 2*z(n-1)+2*n^2-2*n+1; A114146(n) = 2*A114043(n); A115005(n) = z(n-1)+n*(n-1); A141255(n) = 2*z(n-1)+2*n*(n-1); A290131(n) = z(n-1)+(n-1)^2; A306302(n) = z(n)+n^2+2*n; A355902(n) = n + A306302(n)/2. - _N. J. A. Sloane_, Sep 06 2022
%K nonn
%O 0,2
%A _Scott R. Shannon_ and _N. J. A. Sloane_, Sep 05 2022