A102285 G.f. (1-x)/(7*x^2-6*x+1).

1, 5, 23, 103, 457, 2021, 8927, 39415, 174001, 768101, 3390599, 14966887, 66067129, 291634565, 1287337487, 5682582967, 25084135393, 110726731589, 488771441783, 2157541529575, 9523849084969, 42040303802789

Offset: 0

Author

Creighton Dement, Feb 19 2005

Keywords

• nonn
• easy

Comments

A floretion-generated sequence relating to the second binomial transform of Pell numbers A000129.

Floretion Algebra Multiplication Program, FAMP Code: (a(n)) = jesforseq[ + .5'i + .5i' + 2'jj' + .5'ij' + .5'ji' ]; A000004 = vesforseq.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-7).

Crossrefs

Cf. A086351, A027649, A007070 (inverse binomial transform), A081179, A163350 (binomial transform).

Programs

Magma

[Floor(((1+Sqrt(2))*(3+Sqrt(2))^n+(1-Sqrt(2))*(3-Sqrt(2))^n)/2): n in [0..30]]; // Vincenzo Librandi, Oct 12 2011

Mathematica

CoefficientList[Series[(1-x)/(7x^2-6x+1),{x,0,30}],x] (* or *) LinearRecurrence[{6,-7},{1,5},30] (* Harvey P. Dale, Dec 10 2017 *)

Formula

a(n) = A086351(n+1) - 3*A086351(n) (FAMP result); Inversion gives A027649 (SuperSeeker result); Inverse binomial transform of A007070 (SuperSeeker result);

From Al Hakanson (hawkuu(AT)gmail.com), Jul 25 2009: (Start)

a(n) = ((1+sqrt(2))*(3+sqrt(2))^n + (1-sqrt(2))*(3-sqrt(2))^n)/2 offset 0.

Third binomial transform of 1,2,2,4,4. (End)

a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0)=1, a(1)=5. - Philippe Deléham, Sep 19 2009

a(n) = A081179(n) + A086351(n). - Joseph M. Shunia, Sep 09 2019

a(n) = A081179(n+1)-A081179(n). - R. J. Mathar, Sep 11 2019