A377851 Smallest multiplier which can complete the square for n-polygonal numbers, together with a constant offset.

8, 1, 24, 8, 40, 3, 56, 16, 72, 5, 88, 24, 104, 7, 120, 32, 136, 9, 152, 40, 168, 11, 184, 48, 200, 13, 216, 56, 232, 15, 248, 64, 264, 17, 280, 72, 296, 19, 312, 80, 328, 21, 344, 88, 360, 23, 376, 96, 392, 25, 408, 104, 424, 27, 440, 112, 456, 29, 472

Offset: 3

Author

Jonathan Dushoff, Nov 09 2024

Keywords

• nonn
• easy

Comments

This smallest multiplier is also the only multiplier that is relatively prime to the offset.

The n-polygonal numbers, indexed by x, are P(n,x) = (n-2)*(x-1)*x/2 + x = A139601(n-3,x).

S(x) = P(n,x)*a(n) + A181318(n-4) completes the square in that quadratic, ensuring S(x) is a square for all x.

Examples

			For n=7, the heptagonal numbers are h(x) = x*(5*x-3)/2 and with multiplier a(7) = 40 and offset A181318(7-4) = 9 become 40*h(x)+9 = (10*x - 3)^2.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 3..10000
• Wikipedia, Completing the square.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Crossrefs

Cf. A181318 (offsets).

Programs

Mathematica

Table[8*(n - 2)/GCD[n, 4]^2, {n, 3, 100}] (* Paolo Xausa, Dec 07 2024 *)

PARI

a(n) = 8*(n-2)/gcd(n,4)^2 \\ Andrew Howroyd, Nov 10 2024

Formula

a(n) = 8*(n-2)/gcd(n,4)^2. - Andrew Howroyd, Nov 10 2024

From Stefano Spezia, Nov 13 2024: (Start)

G.f.: x^3*(8 + x + 24*x^2 + 8*x^3 + 24*x^4 + x^5 + 8*x^6)/(1 - x^4)^2.

E.g.f.: (4 + 32*x + 6*cos(x) + 2*(16*x - 5)*cosh(x) + 3*x*sin(x) + (5*x - 64)*sinh(x))/4. (End)