A127147 Q(n,5), where Q(m,k) is defined in A127080 and A127137.

43, 28, 15, 4, -5, -12, -17, -20, -21, -20, -17, -12, -5, 4, 15, 28, 43, 60, 79, 100, 123, 148, 175, 204, 235, 268, 303, 340, 379, 420, 463, 508, 555, 604, 655, 708, 763, 820, 879, 940, 1003, 1068, 1135, 1204, 1275, 1348, 1423, 1500, 1579, 1660, 1743, 1828, 1915, 2004, 2095, 2188

Offset: 0

Author

N. J. A. Sloane, Mar 24 2007

Keywords

• sign
• easy

Comments

Numbers m such that m + 21 is a square. The product of two consecutive terms belongs to the sequence, see formula. - Klaus Purath, Oct 30 2022

References

• V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

A row of A127080.

Programs

GAP

List([0..60], n-> (n-8)^2 -21); # G. C. Greubel, Aug 12 2019

Magma

[n^2-16*n+43: n in [0..60]]; // Vincenzo Librandi, Nov 12 2014

Maple

seq((n-8)^2 -21, n=0..60); # G. C. Greubel, Aug 12 2019

Mathematica

CoefficientList[Series[(60x^2 -101x +43)/(1-x)^3, {x,0,60}], x] (* Vincenzo Librandi, Nov 12 2014 *)
(Range[0,60] -8)^2 -21 (* G. C. Greubel, Aug 12 2019 *)

PARI

Vec(-(60*x^2-101*x+43)/(x-1)^3 + O(x^60)) \\ Colin Barker, Nov 12 2014

Sage

[(n-8)^2 -21 for n in (0..60)] # G. C. Greubel, Aug 12 2019

Formula

a(n) = n^2 - 16*n + 43.

From Colin Barker, Nov 12 2014: (Start)

a(n) = 3*a(n-1) -3*a(n-2) +a(n-3).

G.f.: (43 -101*x + 60*x^2)/(1-x)^3. (End)

E.g.f.: (43 - 15*x + x^2)*exp(x). - G. C. Greubel, Aug 12 2019

From Klaus Purath, Oct 30 2022: (Start)

According to the formula a(n) = n^2 - 16*n + 43 when expanded to negative indices, a(n)*a(n+1) = a(a(n)+n) = (a(n)+n)*(a(n+1)-(n+1)) + 43.

a(n) = 2*a(n-1) - a(n-2) + 2. (End)