A027948 -id:A027948 - cp's OEIS frontend

A027935 Triangular array T read by rows: T(n,k)=t(n,2k), t given by A027926; 0 <= k <= n, n >= 0.

1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 5, 7, 1, 1, 2, 5, 12, 11, 1, 1, 2, 5, 13, 26, 16, 1, 1, 2, 5, 13, 33, 51, 22, 1, 1, 2, 5, 13, 34, 79, 92, 29, 1, 1, 2, 5, 13, 34, 88, 176, 155, 37, 1, 1, 2, 5, 13, 34, 89, 221, 365, 247, 46, 1, 1, 2, 5, 13, 34, 89, 232, 530, 709, 376, 56, 1

Offset: 0

Clark Kimberling

			Triangle begins:
  1;
  1, 1;
  1, 2, 1;
  1, 2, 4,  1:
  1, 2, 5,  7,  1;
  1, 2, 5, 12, 11,  1;
  1, 2, 5, 13, 26, 16, 1;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

The row sums of this bisection of the "Fibonacci array" A027926 are powers of 2, see A027948 for the other bisection.

Flat(List([0..12], n-> List([0..n], k-> Sum([0..Int((2*n-2*k+1)/2) ], j-> Binomial(n-j, 2*(n-k-j))) ))); # G. C. Greubel, Sep 27 2019

T:= func< n,k | &+[Binomial(n-j, 2*(n-k-j)) : j in [0..Floor((2*n -2*k+1)/2)]] >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 27 2019

T:= proc(n, k) option remember;
     add( binomial(n-j, 2*(n-k-j)), j=0..floor((2*n - 2*k+1)/2))
   end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Sep 27 2019

T[n_, k_]:= Sum[Binomial[n-j, 2*(n-k-j)], {j,0,Floor[(2*n-2*k+1)/2]}]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 27 2019 *)

T(n,k) = sum(j=0,(2*n-2*k+1)\2, binomial(n-j, 2*(n-k-j)));
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Sep 27 2019

[[ sum(binomial(n-j, 2*(n-k-j)) for j in (0..floor((2*n-2*k+1)/2)) ) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Sep 27 2019

T(n,k) = Sum_{j=0..floor((2*n-2*k-1)/2)} binomial(n-j, 2*(n-k-j)). - G. C. Greubel, Sep 27 2019

A027953 a(0)=1, a(n) = Fibonacci(2n+4) - (2n+3).

Original entry on oeis.org

1, 3, 14, 46, 133, 364, 972, 2567, 6746, 17690, 46345, 121368, 317784, 832011, 2178278, 5702854, 14930317, 39088132, 102334116, 267914255, 701408690, 1836311858, 4807526929, 12586268976, 32951280048, 86267571219, 225851433662

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-8,5,-1).

Cf. A000045, A027948.

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

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

with(combinat); seq(`if`(n=0,1, fibonacci(2*n+4) -(3 +2*n)), n=0..40); # G. C. Greubel, Sep 29 2019

Join[{1},Table[Fibonacci[2n+4]-(2n+3),{n,30}]] (* or *) LinearRecurrence[ {5,-8,5,-1},{1,3,14,46,133},30] (* Harvey P. Dale, Oct 04 2017 *)

vector(40, n, my(m=n-1); if(m==0, 1, fibonacci(2*m+4) -(3 +2*m)) ) \\ G. C. Greubel, Sep 29 2019

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

a(n) = T(2n+1, n+1), T given by A027948.
G.f.: (1-2*x+7*x^2-5*x^3+x^4)/((1-3*x+x^2)*(1-x)^2). - Vladeta Jovovic, Mar 27 2003
a(n) = Sum_{j=0..n} binomial(2*n-j+1, j+2), with a(0)=1. - G. C. Greubel, Sep 29 2019

More terms from Vladeta Jovovic, Mar 27 2003

cp's OEIS Frontend

A027935 Triangular array T read by rows: T(n,k)=t(n,2k), t given by A027926; 0 <= k <= n, n >= 0.

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