A053404 Expansion of 1/((1+3*x)*(1-4*x)).

1, 1, 13, 25, 181, 481, 2653, 8425, 40261, 141361, 624493, 2320825, 9814741, 37664641, 155441533, 607417225, 2472715621, 9761722321, 39434309773, 156574977625, 629786694901, 2508686426401, 10066126765213, 40170363882025

Offset: 0

Author

Barry E. Williams, Jan 07 2000

Keywords

• easy
• nonn

Comments

Hankel transform is := 1,12,0,0,0,... - Philippe Deléham, Nov 02 2008

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=2, 13*a(n-2) equals the number of 13-colored compositions of n with all parts >=2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
• F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
• A. K. Whitford, Binet's formula generalized, Fib. Quart., 15 (1977), pp. 21, 24, 29.
• Index entries for linear recurrences with constant coefficients, signature (1,12).

Crossrefs

Cf. A000045, A001045, A006130, A006131, A015440 - A015443, A015445 - A015447, A053404, A053428, A168579, A350468, A350469.

Programs

Magma

[((4^(n+1)) - (-3)^(n+1))/7: n in [0..30]]; // G. C. Greubel, Jan 16 2018

Maple

seq(simplify(hypergeom([1/2 - (1/2)*n, -(1/2)*n], [-n], -48)), n = 1..40); # Peter Bala, Jul 05 2025

Mathematica

CoefficientList[Series[1/((1 + 3 x) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 06 2014 *)

PARI

a(n)=([0,1; 12,1]^n*[1;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

Sage

[lucas_number1(n,1,-12) for n in range(1, 25)] # Zerinvary Lajos, Apr 22 2009

Formula

a(n) = ((4^(n+1))-(-3)^(n+1))/7.

a(n) = a(n-1) + 12*a(n-2), n > 1; a(0)=1, a(1)=1.

From Paul Barry, Jul 30 2004: (Start)

Convolution of 4^n and (-3)^n.

G.f.: 1/((1+3x)(1-4x)); a(n) = Sum_{k=0..n, 4^k*(-3)^(n-k)} = Sum_{k=0..n, (-3)^k*4^(n-k)}. (End)

a(n) = Sum_{k, 0<=k<=n} A109466(n,k)*(-12)^(n-k). - Philippe Deléham, Oct 26 2008

a(n) = (sum_{1<=k<=n+1, k odd} C(n+1,k)*7^(k-1))/2^n. - Vladimir Shevelev, Feb 05 2014

From Peter Bala, Jun 27 2025: (Start)

a(n) = hypergeom([1/2 - (1/2)*n, -(1/2)*n], [-n], -48) for n >= 1.

The following products telescope:

Product_{k >= 0} (1 + 12^k/a(2*k+1)) = 8.

Product_{k >= 1} (1 - 12^k/a(2*k+1)) = 4/25.

Product_{k >= 0} (1 + (-12)^k/a(2*k+1)) = 8/7.

Product_{k >= 1} (1 - (-12)^k/a(2*k+1)) = 28/25. (End)

Extensions

More terms from James Sellers, Feb 02 2000