A172447 a(n) = (-1 + 5*2^(2*n + 1) - 3*n)/9.

1, 4, 17, 70, 283, 1136, 4549, 18202, 72815, 291268, 1165081, 4660334, 18641347, 74565400, 298261613, 1193046466, 4772185879, 19088743532, 76354974145, 305419896598, 1221679586411, 4886718345664, 19546873382677, 78187493530730, 312749974122943, 1250999896491796

Offset: 0

Author

Paul Curtz, Feb 03 2010

Keywords

• nonn
• easy

Comments

a(n) mod 10 gives the 10-periodic sequence 1, 4, 7, 0, 3, 6, 9, 2, 5, 8 (and repeat, A131579 shifted, A144468 reversed) which contains all ten digits, that has a "palindromic" symmetry: 1 + 8 = 4 + 5 = 7 + 2 = 0 + 9 = 3 + 6 = 9.

The inverse binomial transform gives 1, 3, 10, 30, 90, ... (A062107 shifted). - R. J. Mathar, Feb 11 2010

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Crossrefs

Cf. A072197 (first differences).

Programs

Magma

[(-1+5*2^(2*n+1)-3*n)/9: n in [0..30]]; // Vincenzo Librandi, Aug 05 2011

Mathematica

LinearRecurrence[{6, -9, 4}, {1, 4, 17}, 30] (* Harvey P. Dale, Mar 25 2016 *)
((-1 + 5 2^(2# + 1) - 3#)/9  &) /@ Range[0, 29] (* Alonso del Arte, Apr 25 2020 *)

PARI

a(n)=(10*4^n-3*n)\9 \\ Charles R Greathouse IV, Jul 21 2015

Scala

val powerOf2: LazyList[BigInt] = LazyList.iterate(1: BigInt)(_ * 2)
(0 to 29).map(n => (-1 + 5 * powerOf2(2 * n + 1) - 3 * n)/9) // Alonso del Arte, Apr 25 2020

Formula

a(n) = 6*a(n - 1) - 9*a(n - 2) + 4*a(n - 3).

a(n + 1) - 4*a(n) = n.

a(n) = A172416(2n + 1).

G.f.: (1 - 2*x + 2*x^2)/((1 - 4*x) * (x - 1)^2). - R. J. Mathar, Feb 11 2010

E.g.f.: (10*exp(4*x) - (1 + 3*x)*exp(x))/9. - G. C. Greubel, Nov 02 2018

Extensions

Definition replaced by closed formula by R. J. Mathar, Feb 11 2010