A140065 a(n) = (7*n^2 - 17*n + 12)/2.

1, 3, 12, 28, 51, 81, 118, 162, 213, 271, 336, 408, 487, 573, 666, 766, 873, 987, 1108, 1236, 1371, 1513, 1662, 1818, 1981, 2151, 2328, 2512, 2703, 2901, 3106, 3318, 3537, 3763, 3996, 4236, 4483, 4737, 4998, 5266, 5541, 5823, 6112, 6408, 6711, 7021, 7338, 7662

Offset: 1

Author

Gary W. Adamson, May 03 2008

Keywords

• nonn
• easy
• changed

Comments

Binomial transform of [1, 2, 7, 0, 0, 0, ...].

This sequence together with 1, 6, 18, 37, 63, 96, ... with signature (3,-3,1) [not yet in OEIS] contain all numbers k such that 56*k - 47 is a square. - Klaus Purath, Oct 21 2021

Examples

			a(4) = 28 = (1, 3, 3, 1) * (1, 2, 7, 0) = (1 + 6 + 21 + 0).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A007318, A140064, A140066.

Programs

Magma

[(7*n^2 - 17*n + 12)/2 : n in [1..60]]; // Wesley Ivan Hurt, Oct 10 2021

Maple

seq((12-17*n+7*n^2)*1/2, n=1..40); # Emeric Deutsch, May 07 2008

Mathematica

Table[(7 n^2 - 17 n + 12)/2, {n, 1, 50}] (* Bruno Berselli, Mar 12 2015 *)
LinearRecurrence[{3,-3,1},{1,3,12},50] (* Harvey P. Dale, May 28 2017 *)

PARI

x = 'x + O('x^50); Vec(x*(1+6*x^2)/(1-x)^3) \\ G. C. Greubel, Feb 23 2017

Formula

A007318 * [1, 2, 7, 0, 0, 0, ...].

a(n) = A000217(n) + 6*A000217(n-2) = (A140064(n) + A140066(n))/2. - R. J. Mathar, May 06 2008

O.g.f.: x*(1+6*x^2)/(1-x)^3. - Alexander R. Povolotsky, May 06 2008

a(n) = 7*n + a(n-1) - 12 for n > 1, a(1)=1. - Vincenzo Librandi, Jul 08 2010

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 4. - Klaus Purath, Oct 21 2021

E.g.f.: exp(x)*(6 - 5*x + 7*x^2/2) - 6. - Elmo R. Oliveira, Oct 31 2024

Extensions

More terms from R. J. Mathar and Emeric Deutsch, May 06 2008

More terms from Vladimir Joseph Stephan Orlovsky, Oct 25 2008