A329402 Number of rectangles (w X h, w <= h) with integer side lengths w and h having area = n * perimeter.

2, 3, 5, 4, 5, 8, 5, 5, 8, 8, 5, 11, 5, 8, 14, 6, 5, 13, 5, 11, 14, 8, 5, 14, 8, 8, 11, 11, 5, 23, 5, 7, 14, 8, 14, 18, 5, 8, 14, 14, 5, 23, 5, 11, 23, 8, 5, 17, 8, 13, 14, 11, 5, 18, 14, 14, 14, 8, 5, 32, 5, 8, 23, 8, 14, 23, 5, 11, 14, 23, 5, 23, 5, 8, 23

Offset

1,1

Comments

This sequence was inspired by a generalization of the Isis Problem.

Number of ways of writing 1/n as a sum of inverses of consecutive triangular numbers (possibly infinite). For example a(2) = 3 because there are 3 ways of writing 1/2 as a sum of inverses of consecutive triangular numbers: 1/2 = 1/C(4,2) + ... + 1/C(12,2) = 1/C(3,2) + 1/C(4,2) = 1/C(5,2) + 1/C(6,2) + ... - Ludovic Schwob, Jul 25 2023

Links

Robert Israel, Table of n, a(n) for n = 1..10000

B. Greer, D. De Bock and W. Van Dooren, The Isis Problem as an Experimental probe and teaching resource, Journal of Mathematical Behavior, 28, (2009), 237-246.

Formula

a(n) = A038548(4*n^2). - Peter Bala, Mar 03 2020

Example

a(1) = 2 because there are two rectangles which have area = perimeter: 4 X 4 and 3 X 6.
a(2) = 3 because there are 3 rectangles for which area = 2 * perimeter: 8 X 8, 6 X 12, and 5 X 20.
a(3) = 5 because there are 5 rectangles for which area = 3 * perimeter: 12 X 12, 10 X 15, 9 X 18, 8 X 24, and 7 X 42.

Maple

f:= n -> (numtheory:-tau(4*n^2)+1)/2;
map(f, [$1..100]); # Robert Israel, Mar 31 2020

Mathematica

a[n_] := Ceiling[DivisorSigma[0, 4 n^2]/2]; Array[a, 75] (* Giovanni Resta, Mar 29 2020 *)

Prog

(Python 3.5.3)
numbers=[]
for n in range(500):
    c=int(0)
    n=int(n+1)
    for x in range(2*n+1, 4*n+1):
        y=(2*n*x)/(x-2*n)
        if y==y//1:
            y=int(y)
            c=c+1
    numbers.append(c)
print(numbers)

Crossrefs

Cf. A038548.

Sequence in context: A262507 A151679 A077664 * A339263 A179475 A262564

Adjacent sequences: A329399 A329400 A329401 * A329403 A329404 A329405

Keyword

nonn

Author

Nicolas Haverhals, Feb 28 2020

Status

approved