A131068 Binomial transform of [1, 1, 7, 7, 7, ...].

1, 2, 10, 32, 82, 188, 406, 848, 1738, 3524, 7102, 14264, 28594, 57260, 114598, 229280, 458650, 917396, 1834894, 3669896, 7339906, 14679932, 29359990, 58720112, 117440362, 234880868, 469761886, 939523928, 1879048018, 3758096204

Offset: 0

Author

Gary W. Adamson, Jun 13 2007

Keywords

• nonn
• easy

Comments

Row sums of triangle A131067. - Emeric Deutsch, Jun 20 2007

Examples

			a(3) = 32 = sum of row 4 terms of triangle A131067: (1 + 15 + 15 + 1).
a(3) = 32 = (1, 3, 3, 1) dot (1, 1, 7, 7) = (1 + 3 + 21 + 7).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Crossrefs

Cf. A131060, A131061, A131063, A131064, A131065, A131066, A131067.

Programs

Magma

I:=[1, 2, 10]; [n le 3 select I[n] else 4*Self(n-1)-5*Self(n-2) + 2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 05 2012

Maple

a := proc (n) options operator, arrow; 7*2^n-6*n-6 end proc: seq(a(n), n = 0 .. 30); # Emeric Deutsch, Jun 20 2007

Mathematica

Table[7*2^n -6*(n+1), {n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Nov 10 2008; modified by G. C. Greubel, Mar 12 2020 *)
CoefficientList[Series[(1-2x+7x^2)/((1-2x)(1-x)^2),{x,0,40}],x] (* Vincenzo Librandi, Jul 05 2012 *)

PARI

vector(31, n, my(m=n-1); 7*2^m -6*(m+1) ) \\ G. C. Greubel, Mar 12 2020

Sage

[7*2^n -6*(n+1) for n in (0..30)] # G. C. Greubel, Mar 12 2020

Formula

From Emeric Deutsch, Jun 20 2007: (Start)

a(n) = 7*2^n - 6*(n + 1).

G.f.: (1 - 2*x + 7*x^2)/((1-2*x)*(1-x)^2). (End)

a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Vincenzo Librandi, Jul 05 2012

E.g.f.: 7*exp(2*x) - 6*(1+x)*exp(x). - G. C. Greubel, Mar 12 2020

Extensions

Corrected and extended by Emeric Deutsch, Jun 20 2007