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

1,2

Comments

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

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

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).

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

Example

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

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 *)

Prog

(PARI) x = 'x + O('x^50); Vec(x*(1+6*x^2)/(1-x)^3) \\ G. C. Greubel, Feb 23 2017
(Magma) [(7*n^2 - 17*n + 12)/2 : n in [1..60]]; // Wesley Ivan Hurt, Oct 10 2021

Crossrefs

Cf. A000217, A007318, A140064, A140066.

Sequence in context: A083539 A237426 A066643 * A294418 A308669 A115549

Adjacent sequences: A140062 A140063 A140064 * A140066 A140067 A140068

Keyword

nonn,easy

Author

Gary W. Adamson, May 03 2008

Extensions

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

More terms from Vladimir Joseph Stephan Orlovsky, Oct 25 2008

Status

approved