A114146 -id:A114146 - cp's OEIS frontend

A355902 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.

0, 3, 10, 26, 56, 112, 196, 331, 522, 790, 1138, 1615, 2204, 2975, 3910, 5041, 6388, 8047, 9958, 12262, 14894, 17920, 21346, 25347, 29796, 34875, 40522, 46854, 53826, 61716, 70274, 79883, 90380, 101875, 114346, 127981, 142612, 158737, 176086, 194827, 214852, 236717, 259906, 285124, 311970, 340588, 370990, 403819, 438440, 475556

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Sep 05 2022

nonn

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.

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

Scott R. Shannon, Image for a(4) = 56. Note in this and other images the entire 2xn array is shown so the number of cells is twice a(n).
Scott R. Shannon, Image for a(6) = 196.
Scott R. Shannon, Image for a(10) = 1138.
Scott R. Shannon, Image for a(15) = 5041.
N. J. A. Sloane, Illustration for a(2) = 10.
N. J. A. Sloane, Illustration for a(3) = 26.

Cf. A356790, A306302, A355798, A290131, A331452.

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

a(n) = A356790(2,n+2)/2 - 2.

cp's OEIS Frontend

A355902 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.

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