A027949 a(n) = T(2n,n+1), T given by A027948.

1, 4, 25, 97, 309, 894, 2462, 6610, 17519, 46135, 121115, 317484, 831660, 2177872, 5702389, 14929789, 39087537, 102333450, 267913514, 701407870, 1836310955, 4807525939, 12586267895, 32951278872, 86267569944, 225851432284, 591286728337, 1548008754265

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-13,13,-6,1).

Crossrefs

Cf. A000045, A027948.

Programs

GAP

Concatenation([1], List([2..40], n-> Fibonacci(2*n+4) -(2*n^2 +3*n +3)) ); # G. C. Greubel, Sep 29 2019

Magma

[1] cat [Fibonacci(2*n+4) -(2*n^2 +3*n +3): n in [2..40]]; // G. C. Greubel, Sep 29 2019

Maple

with(combinat); seq(`if`(n=1, 1, fibonacci(2*n+4) -(2*n^2 +3*n +3)), n=1..40); # G. C. Greubel, Sep 29 2019

Mathematica

Join[{1},Table[Fibonacci[2n+4]-2n^2-3n-3,{n,2,40}]] (* or *) Join[ {1}, LinearRecurrence[{6,-13,13,-6,1}, {4,25,97,309,894}, 40]] (* Harvey P. Dale, Apr 20 2012 *)
CoefficientList[Series[(x^5-6x^4+14x^3-14x^2+2x-1)/((x-1)^3(x^2-3x+1)), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 20 2014 *)

PARI

Vec(x*(x^5-6*x^4+14*x^3-14*x^2+2*x-1)/((x-1)^3*(x^2-3*x+1)) + O(x^40)) \\ Colin Barker, Nov 19 2014

PARI

vector(40, n, if(n==1,1,fibonacci(2*n+4) -(2*n^2 +3*n +3)) ) \\ G. C. Greubel, Sep 29 2019

Sage

[1]+[fibonacci(2*n+4) -(2*n^2 +3*n +3) for n in (2..40)] # G. C. Greubel, Sep 29 2019

Formula

For n>1, a(n) = Fibonacci(2*n+4) - (2*n^2 + 3*n + 3).

a(1)=1, a(2)=4, a(3)=25, a(4)=97, a(5)=309, a(6)=894, a(n) = 6*a(n-1) - 13*a(n-2) +13*a(n-3) -6*a(n-4) +a(n-5). - Harvey P. Dale, Apr 20 2012

G.f.: x*(1 -2*x +14*x^2 -14*x^3 +6*x^4 -x^5)/((1-x)^3*(1-3*x+x^2)). - Colin Barker, Nov 19 2014

a(n) = Sum_{j=0..n-1} binomial(2*n-j, j+3), with a(1)=1. - G. C. Greubel, Sep 29 2019

Extensions

More terms from Harvey P. Dale, Apr 20 2012