A125171 -id:A125171 - cp's OEIS frontend

A027994 a(n) = (F(2n+3) - F(n))/2 where F() = Fibonacci numbers A000045.

1, 2, 6, 16, 43, 114, 301, 792, 2080, 5456, 14301, 37468, 98137, 256998, 672946, 1761984, 4613239, 12078110, 31621701, 82787980, 216743836, 567446112, 1485598681, 3889356696, 10182482353, 26658108074, 69791870526, 182717549872, 478360854115, 1252365133866, 3278734743901, 8583839415648, 22472784017272

Offset: 0

Clark Kimberling

Substituting x*(1-x)/(1-2x) into x^2/(1-x^2) yields x^2*(g.f. of sequence).
The number of (s(0), s(1), ..., s(n+1)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n+1, s(0) = 2, s(n+1) = 3. - Herbert Kociemba, Jun 02 2004
Diagonal sums of triangle in A125171. - Philippe Deléham, Jan 14 2014

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1).

Cf. A000667, A059216, A059219, A059502, A027926.

[(Fibonacci(2*n+3)-Fibonacci(n))/2 : n in [0..40]]; // Vincenzo Librandi, Jan 01 2025

Table[(Fibonacci[2n+3]-Fibonacci[n])/2,{n,0,30}] (* or *) LinearRecurrence[{4,-3,-2,1},{1,2,6,16},30] (* Harvey P. Dale, Apr 28 2022 *)

PARI
```
a(n)=(fibonacci(2*n+3)-fibonacci(n))/2
```

G.f.: (1-x)^2/((1-x-x^2)*(1-3*x+x^2)). - Floor van Lamoen and N. J. A. Sloane, Jan 21 2001
a(n) = Sum_{k=0..n} T(n, k)*T(n, n+k), T given by A027926.
a(n) = 2*a(n-1) + Sum_{m < n-1} a(m) + F(n-1) = A059512(n+2) - F(n) where F(n) is the n-th Fibonacci number (A000045). - Floor van Lamoen, Jan 21 2001
a(n) = (2/5)*Sum_{k=1..4} sin(2*Pi*k/5)*sin(3*Pi*k/5)*(1+2*cos(Pi*k/5))^(n+1). - Herbert Kociemba, Jun 02 2004
a(-1-2n) = A056014(2n), a(-2n) = A005207(2n-1).
E.g.f.: exp(3*x/2)*cosh(sqrt(5)*x/2) + exp(x/2)*(2*exp(x) - 1)*sinh(sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, Jan 01 2025

A125172 Triangle T(n,k) with partial column sums of the even-indexed Fibonacci numbers.

Original entry on oeis.org

1, 3, 1, 8, 4, 1, 21, 12, 5, 1, 55, 33, 17, 6, 1, 144, 88, 50, 23, 7, 1, 377, 232, 138, 73, 30, 8, 1, 987, 609, 370, 211, 103, 38, 9, 1, 2584, 1596, 979, 581, 314, 141, 47, 10, 1

Offset: 1

Gary W. Adamson, Nov 22 2006

"Partial column sums" means the 1st column consists of the even-indexed Fibonacci numbers, the 2nd column shows the partial sums of the first column, the 3rd column the partial sums of the 2nd, etc. - R. J. Mathar, Sep 06 2011

Mirror of the fission triangle A193667, as in the Mathematica program below. - Clark Kimberling, Aug 11 2011

			First few rows of the triangle:
    1;
    3,  1;
    8,  4,  1;
   21, 12,  5,  1;
   55, 33, 17,  6,  1;
  144, 88, 50, 23,  7,  1;
  ...

Cf. A105693 (row sums), A125171, A193667.

z = 11;
p[n_, x_] := (x + 1)^n;
q[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193667 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* this sequence *)
(* Clark Kimberling, Aug 11 2011 *)

T(n,1) = A001906(n).
T(n,k) = T(n-1,k-1) + T(n-1,k), k > 1.
From R. J. Mathar, Sep 06 2011: (Start)
T(n,k) = A125171(n,k), i.e., A125171 without column k=0.
Conjecture: T(n,k) = T(n,k-1) - A121460(n+1,k). (End)

cp's OEIS Frontend

A027994 a(n) = (F(2n+3) - F(n))/2 where F() = Fibonacci numbers A000045.

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