A055990 a(n) is its own 4th difference.

1, 4, 14, 50, 181, 657, 2385, 8657, 31422, 114051, 413966, 1502555, 5453761, 19795288, 71850128, 260791401, 946583628, 3435774958, 12470688498, 45264335853, 164294064481, 596331286321, 2164478699633, 7856317702310, 28515747394555, 103502414271126

Offset

1,2

Comments

Number of compositions of 4*n-2 into parts 1 and 4. - Seiichi Manyama, Feb 03 2024

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (5,-6,4,-1).

Formula

a(n) = 5*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) = a(n-1)+A055989(n) = A055991(n)-A055991(n-1) = A055988(n+1)-2*A055988(n)+A055988(n-1).

G.f.: x*(1-x)/(1-5*x+6*x^2-4*x^3+x^4). [Colin Barker, Apr 05 2012]

a(n) = Sum_{m=0..n-1} C(n+3m+1,n-m-1). - Vladimir Kruchinin, Nov 18 2020

Mathematica

CoefficientList[Series[(1-x)/(1-5*x+6*x^2-4*x^3+x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 06 2012 *)
LinearRecurrence[{5, -6, 4, -1}, {1, 4, 14, 50}, 30] (* Harvey P. Dale, Oct 18 2015 *)

Prog

(Magma) I:=[1, 4, 14, 50]; [n le 4 select I[n] else 5*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Apr 06 2012
(PARI) Vec((1-x)/(1-5*x+6*x^2-4*x^3+x^4)+O(x^99)) \\ Charles R Greathouse IV, Apr 06 2012
(Maxima)
a(n):=sum((binomial(n+3*m+1, n-m-1)), m, 0, n-1); /* Vladimir Kruchinin, Nov 18 2020 */

Crossrefs

Cf. A055988, A055989, A055991 for the other differences of a(n). See A000079, A001906, A052529 for examples of sequences which are respectively their own first, second and third differences.

Cf. A003269.

Sequence in context: A211304 A047065 A047008 * A211308 A087945 A051924

Adjacent sequences: A055987 A055988 A055989 * A055991 A055992 A055993

Keyword

nonn,easy

Author

Henry Bottomley, Jun 02 2000

Extensions

More terms from James A. Sellers, Jun 05 2000

Status

approved