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

Author

Henry Bottomley, Jun 02 2000

Keywords

• nonn
• easy

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).

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.

Programs

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

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 *)

Maxima

a(n):=sum((binomial(n+3*m+1,n-m-1)),m,0,n-1); /* Vladimir Kruchinin, Nov 18 2020 */

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

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

Extensions

More terms from James Sellers, Jun 05 2000