A082044 Main diagonal of A082043: a(n) = n^4 + 2*n^2 + 1.

1, 4, 25, 100, 289, 676, 1369, 2500, 4225, 6724, 10201, 14884, 21025, 28900, 38809, 51076, 66049, 84100, 105625, 131044, 160801, 195364, 235225, 280900, 332929, 391876, 458329, 532900, 616225, 708964, 811801, 925444, 1050625, 1188100

Offset: 0

Author

Paul Barry, Apr 03 2003

Keywords

• easy
• nonn

Comments

a(n) = longest side b of all integer-sided triangles with sides a <= b <= c and inradius n >= 1. Triangle has sides (n^2+2, n^4+2*n^2+1, n^4+3*n^2+1).

Examples

			G.f. = 1 + 4*x + 25*x^2 + 100*x^3 + 289*x^4 + 676*x^5 + 1369*x^6 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000217, A002061 A002522, A059826, A082043, A082047.

See A120062 for sequences related to integer-sided triangles with integer inradius n.

Programs

Magma

[(n^2+1)^2: n in [0..40]]; // G. C. Greubel, Dec 24 2022

Maple

seq(fibonacci(3,n)^2,n=0..33); # Zerinvary Lajos, Apr 09 2008

Mathematica

Fibonacci[3,Range[0,40]]^2 (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,4,25,100,289},40] (* Harvey P. Dale, Feb 27 2015 *)

PARI

a(n) = n^4+2*n^2+1; \\ Michel Marcus, Jan 22 2016

SageMath

[(n^2+1)^2 for n in range(41)] # G. C. Greubel, Dec 24 2022

Formula

a(n) = n^4 + 2*n^2 + 1.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Feb 27 2015

a(n) = (4*A000217(n-1)^2 + 2*A002061(n))^2 / a(n-1). - Bruce J. Nicholson, Apr 17 2017

a(n) = A002522(n)^2 = (n^2 + 1)^2 = a(-n) for all n in Z. - Michael Somos, Apr 17 2017

G.f.: (1 - x + 15*x^2 + 5*x^3 + 4*x^4) / (1 - x)^5. - Michael Somos, Apr 17 2017

From Amiram Eldar, Nov 02 2021: (Start)

Sum_{n>=0} 1/a(n) = Pi^2*csch(Pi)^2/4 + Pi*coth(Pi)/4 + 1/2.

Sum_{n>=0} (-1)^n/a(n) = Pi^2*csch(Pi)*coth(Pi)/4 + Pi*csch(Pi)/4 + 1/2. (End)

E.g.f.: (1 + 3*x + 9*x^2 + 6*x^3 + x^4)*exp(x). - G. C. Greubel, Dec 24 2022