A084569 Partial sums of A084570.

1, 3, 9, 21, 44, 82, 142, 230, 355, 525, 751, 1043, 1414, 1876, 2444, 3132, 3957, 4935, 6085, 7425, 8976, 10758, 12794, 15106, 17719, 20657, 23947, 27615, 31690, 36200, 41176, 46648, 52649, 59211, 66369, 74157, 82612, 91770, 101670, 112350, 123851

Offset: 0

Author

Paul Barry, May 31 2003

Keywords

• easy
• nonn

Comments

Conjecture: a(n) is the number of perimeter-magic (hollow) squares of order 3 with magic sum n+3. Order 3 means each of the 4 edges has 3 elements >=1; the square has 8 elements. The elements do not need to be distinct, and squares obtained by rotations are counted only once. The square (read ccw) for magic sum 3 has elements 1 1 1 1 1 1 1 1. The 3 squares with magic sum 4 are 1 1 2 1 1 1 2 1, 1 1 2 1 1 2 1 2 and 1 2 1 2 1 2 1 2. - R. J. Mathar, Mar 08 2025

Links

• R. J. Mathar, Generating perimeter-magic polygons, C++ (2025)
• Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Crossrefs

Cf. A116701.

Programs

Mathematica

Accumulate[LinearRecurrence[{3,-2,-2,3,-1},{1,2,6,12,23},50]] (* or *) LinearRecurrence[{4,-5,0,5,-4,1},{1,3,9,21,44,82},50] (* Harvey P. Dale, Nov 12 2014 *)

Formula

a(n) = (-1)^n/8 + (n^4 + 6*n^3 + 17*n^2 + 30*n + 21)/24.

a(n) = Sum_{k=0..n} Sum_{j=0..k} Sum_{i=0..j} (i + (-1)^i).

G.f.: ( -1+x-2*x^2 ) / ( (1+x)*(x-1)^5 ). - R. J. Mathar, Mar 08 2025

a(n)+a(n+1) = A116701(n+3)-1. - R. J. Mathar, Mar 08 2025