A027976 -id:A027976 - cp's OEIS frontend

A027960 'Lucas array': triangular array T read by rows.

1, 1, 3, 1, 1, 3, 4, 4, 1, 1, 3, 4, 7, 8, 5, 1, 1, 3, 4, 7, 11, 15, 13, 6, 1, 1, 3, 4, 7, 11, 18, 26, 28, 19, 7, 1, 1, 3, 4, 7, 11, 18, 29, 44, 54, 47, 26, 8, 1, 1, 3, 4, 7, 11, 18, 29, 47, 73, 98, 101, 73, 34, 9, 1, 1, 3, 4, 7, 11, 18, 29, 47, 76, 120, 171, 199, 174, 107, 43, 10, 1

Offset: 0

Clark Kimberling

The k-th row contains 2k+1 numbers.
Columns in the right half consist of convolutions of the Lucas numbers with the natural numbers.
T(n,k) = number of strings s(0),...,s(n) such that s(n)=n-k. s(0) in {0,1,2}, s(1)=1 if s(0) in {1,2}, s(1) in {0,1,2} if s(0)=0 and for 1 <= i <= n, s(i) = s(i-1)+d, with d in {0,2} if s(i)=2i, in {0,1,2} if s(i)=2i-1, in {0,1} if 0 <= s(i) <= 2i-2.

			                           1
                       1,  3,  1
                   1,  3,  4,  4,  1
               1,  3,  4,  7,  8,  5,   1
           1,  3,  4,  7, 11, 15, 13,   6,  1
        1, 3,  4,  7, 11, 18, 26, 28,  19,  7,  1
     1, 3, 4,  7, 11, 18, 29, 44, 54,  47, 26,  8, 1
  1, 3, 4, 7, 11, 18, 29, 47, 73, 98, 101, 73, 34, 9, 1

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Central column is the Lucas numbers without initial 2: A000204.
Columns in the right half include A027961, A027962, A027963, A027964, A053298.
Bisection triangles are in A026998 and A027011.
Row sums: A036563, A153881 (alternating sign).
Sums include: A023537, A027973, A027974, A027975, A027976, A027978, A027979, A027980, A027981, A027982, A027984, A027985, A027986, A027987, A053298.
Diagonals of the form T(n, 2*n-p): A000012 (p=0), A000027 (p=1), A034856 (p=2), A027965 (p=3), A027966 (p=4), A027967 (p=5), A027968 (p=6), A027969 (p=7), A027970 (p=8), A027971 (p=9), A027972 (p=10).
Diagonals of the form T(n, n+p): A000032 (p=0), A027961 (p=1), A023537 (p=2), A027963 (p=3), A027964 (p=4), A053298 (p=5), A027002 U A027018 (p=6), A027007 U A027014 (p=7), A027003 U A027019 (p=8).
Cf. A000032, A004396, A026998, A027011, A027977, A075193, A168616.

function T(n,k) // T = A027960
      if k le n then return Lucas(k+1);
      elif k gt 2*n then return 0;
      else return T(n-1, k-2) + T(n-1, k-1);
      end if;
end function;
[T(n,k): k in [0..2*n], n in [0..12]]; // G. C. Greubel, Jun 08 2025

T:=proc(n,k)option remember:if(k=0 or k=2*n)then return 1:elif(k=1)then return 3:else return T(n-1,k-2) + T(n-1,k-1):fi:end:
for n from 0 to 6 do for k from 0 to 2*n do print(T(n,k));od:od: # Nathaniel Johnston, Apr 18 2011

(* First program *)
t[, 0] = 1; t[, 1] = 3; t[n_, k_] /; (k == 2*n) = 1; t[n_, k_] := t[n, k] = t[n-1, k-2] + t[n-1, k-1]; Table[t[n, k], {n, 0, 8}, {k, 0, 2*n}] // Flatten (* Jean-François Alcover, Dec 27 2013 *)
(* Second program *)
f[n_, k_]:= f[n,k]= Sum[Binomial[2*n-k+j,j]*LucasL[2*(k-n-j)], {j,0,k-n-1}];
A027960[n_, k_]:= LucasL[k+1] - f[n,k]*Boole[k>n];
Table[A027960[n,k], {n,0,12}, {k,0,2*n}]//Flatten (* G. C. Greubel, Jun 08 2025 *)

T(r,n)=if(r<0||n>2*r,return(0)); if(n==0||n==2*r,return(1)); if(n==1,3,T(r-1,n-1)+T(r-1,n-2)) /* Ralf Stephan, May 04 2005 */

@CachedFunction
def T(n, k): # T = A027960
    if (k>2*n): return 0
    elif (kG. C. Greubel, Jun 01 2019; Jun 08 2025

T(n, k) = Lucas(k+1) for k <= n, otherwise the (2n-k)th coefficient of the power series for (1+2*x)/{(1-x-x^2)*(1-x)^(k-n)}.
Recurrence: T(n, 0)=T(n, 2n)=1 for n >= 0; T(n, 1)=3 for n >= 1; and for n >= 2, T(n, k) = T(n-1, k-2) + T(n-1, k-1) for 2 <= k <= 2*n-1.
From G. C. Greubel, Jun 08 2025: (Start)
T(n, k) = A000032(k+1) - f(n, k)*[k > n], where f(n, k) = Sum_{j=0..k-n-1} binomial(2*n -k+j, j)*A000032(2*(k-n-j)).
Sum_{k=0..A004396(n+1)} T(n-k, k) = A027975(n).
Sum_{k=0..n} T(n, k) = A027961(n).
Sum_{k=0..2*n} T(n, k) = A168616(n+2) + 2.
Sum_{k=n+1..2*n} (-1)^k*T(n, k) = A075193(n-1), n >= 1. (End)

Edited by Ralf Stephan, May 04 2005

A004695 a(n) = floor(Fibonacci(n)/2).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 6, 10, 17, 27, 44, 72, 116, 188, 305, 493, 798, 1292, 2090, 3382, 5473, 8855, 14328, 23184, 37512, 60696, 98209, 158905, 257114, 416020, 673134, 1089154, 1762289, 2851443, 4613732, 7465176, 12078908, 19544084, 31622993, 51167077, 82790070, 133957148

Offset: 0

N. J. A. Sloane, Dec 11 1996

Column sums of:
1 1 2 3 5  8 13 21 34 55...
      1 1  2  3  5  8 13...
              1  1  2  3...
                       1...
---------------------------
1 1 2 4 6 10 17 27 44 72...
This sequence counts partially ordered partitions of (n-3) into parts no greater than 3, where the position of the 1's and 2's is important. Alternatively, the position of the 3's is unimportant. (see example below). - David Neil McGrath, Apr 26 2015
Also the matching and vertex cover number of the (n-2)-Fibonacci cube graph. - Eric W. Weisstein, Sep 06 2017

			Partial Order of 6 into parts (1,2,3) with position of 3 unimportant. a(9)=17 These are (33),(321=231=213),(312=132=123),(3111=1311=1131=1113),(222),(2211),(2121),(2112),(1212),(1122),(1221),(21111),(12111),(11211),(11121),(11112),(111111). - _David Neil McGrath_, Apr 26 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
H. Matsui et al., Problem B-1019, Fibonacci Quarterly, Vol. 45, Number 2; 2007; p. 182. [A related sequence.]
Eric Weisstein's World of Mathematics, Fibonacci Cube Graph
Eric Weisstein's World of Mathematics, Matching Number
Eric Weisstein's World of Mathematics, Vertex Cover Number
Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1,-1).

Cf. A036605, A027976, A081410.

[Floor(Fibonacci(n)/2): n in [0..60]]; // Vincenzo Librandi, Apr 23 2011

seq(iquo(fibonacci(n),2),n=0..36); # Zerinvary Lajos, Apr 20 2008
f:=proc(n) option remember; local t1; if n <= 2 then RETURN(1); fi: if n mod 3 = 1 then t1:=1 else t1:=0; fi: f(n-1)+f(n-2)+t1; end; [seq(f(n), n=1..100)]; # N. J. A. Sloane, May 25 2008

CoefficientList[Series[x^3 / ((1 - x^3) (1 - x - x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 08 2013 *)
Floor[Fibonacci[Range[0, 50]]/2] (* Harvey P. Dale, Feb 15 2015 *)
LinearRecurrence[{1, 1, 1, -1, -1}, {0, 0, 0, 1, 1}, 50] (* Harvey P. Dale, Feb 15 2015 *)
Floor[Fibonacci[Range[0, 20]]/2] (* Eric W. Weisstein, Sep 06 2017 *)

PARI
```
a(n)=fibonacci(n)\2
```

G.f.: x^3/((1-x^3)*(1-x-x^2)). - Ralf Stephan, Jul 22 2003, corrected by Paul Barry
a(n) = Fibonacci(n)/2 - (1-cos(2Pi*n/3))/3. - Paul Barry, Oct 06 2003
From Paul Barry, Jan 14 2005: (Start)
a(n+2) = Sum_{k=0..floor(n/3)} F(n-3*k).
a(n+2) = Sum_{k=0..n} if(mod(n-k, 3)=0, F(k), 0). (End)
a(n+2) = Sum_{k=0..n} F(k)*(cos(2*Pi*(n-k)/3+Pi/3)/3+sqrt(3)*sin(2*Pi*(n-k)/3+Pi/3)/3+1/3). - Paul Barry, Apr 16 2005
a(n) = a(n-1)+a(n-2)+1 if n mod 3 = 0, else a(n) = a(n-1)+a(n-2). - Gary Detlefs, Dec 05 2010
a(n) = Fibonacci(n-2)+floor(Fibonacci(n-3)/2). - Gary Detlefs, Mar 28 2011
a(n) = (A000045(n) - A011655(n))/2.
a(n) = a(n-1)+a(n-2)+a(n-3)-a(n-4)-a(n-5), a(0)=0, a(1)=0, a(2)=0, a(3)=1, a(4)=1. - Carl Najafi, May 06 2014

cp's OEIS Frontend

A027960 'Lucas array': triangular array T read by rows.

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