A027974 -id:A027974 - cp's OEIS frontend

A101220 a(n) = Sum_{k=0..n} Fibonacci(n-k)*n^k.

0, 1, 3, 14, 91, 820, 9650, 140601, 2440317, 49109632, 1123595495, 28792920872, 816742025772, 25402428294801, 859492240650847, 31427791175659690, 1234928473553777403, 51893300561135516404, 2322083099525697299278

Offset: 0

Ross La Haye, Dec 14 2004

In what follows a(i,j,k) denotes a three-dimensional array, the terms a(n) are defined as a(n,n,n) in that array. - Joerg Arndt, Jan 03 2021
Previous name was: Three-dimensional array: a(i,j,k) = expansion of x*(1 + (i-j)*x)/((1-j*x)*(1-x-x^2)), read by a(n,n,n).
a(i,j,k) = the k-th value of the convolution of the Fibonacci numbers (A000045) with the powers of i = Sum_{m=0..k} a(i-1,j,m), both for i = j and i > 0; a(i,j,k) = a(i-1,j,k) + a(j,j,k-1), for i,k > 0; a(i,1,k) = Sum_{m=0..k} a(i-1,0,m), for i > 0. With F = Fibonacci and L = Lucas, then a(1,1,k) = F(k+2) - 1; a(2,1,k) = F(k+3) - 2; a(3,1,k) = L(k+2) - 3; a(4,1,k) = 4*F(k+1) + F(k) - 4; a(1,2,k) = 2^k - F(k+1); a(2,2,k) = 2^(k+1) - F(k+3); a(3,2,k) = 3(2^k - F(k+2)) + F(k); a(4,2,k) = 2^(k+2) - F(k+4) - F(k+2); a(1,3,k) = (3^k + L(k-1))/5, for k > 0; a(2,3,k) = (2 * 3^k - L(k)) /5, for k > 0; a(3,3,k) = (3^(k+1) - L(k+2))/5; a(4,3,k) = (4 * 3^k - L(k+2) - L(k+1))/5, etc..

			a(1,3,3) = 6 because a(1,3,0) = 0, a(1,3,1) = 1, a(1,3,2) = 2 and 4*2 - 2*1 - 3*0 = 6.

G. C. Greubel, Table of n, a(n) for n = 0..385
Eric Weisstein's World of Mathematics, Fibonacci Number
Eric Weisstein's World of Mathematics, Lucas Number

a(0, j, k) = A000045(k).
a(1, 2, k+1) - a(1, 2, k) = A099036(k).
a(3, 2, k+1) - a(3, 2, k) = A104004(k).
a(4, 2, k+1) - a(4, 2, k) = A027973(k).
a(1, 3, k+1) - a(1, 3, k) = A099159(k).
a(i, 0, k) = A109754(i, k).
a(i, i+1, 3) = A002522(i+1).
a(i, i+1, 4) = A071568(i+1).
a(2^i-2, 0, k+1) = A118654(i, k), for i > 0.
Sequences of the form a(n, 0, k): A000045(k+1) (n=1), A000032(k) (n=2), A000285(k-1) (n=3), A022095(k-1) (n=4), A022096(k-1) (n=5), A022097(k-1) (n=6), A022098(k-1) (n=7), A022099(k-1) (n=8), A022100(k-1) (n=9), A022101(k-1) (n=10), A022102(k-1) (n=11), A022103(k-1) (n=12), A022104(k-1) (n=13), A022105(k-1) (n=14), A022106(k-1) (n=15), A022107(k-1) (n=16), A022108(k-1) (n=17), A022109(k-1) (n=18), A022110(k-1) (n=19), A088209(k-2) (n=k-2), A007502(k) (n=k-1), A094588(k) (n=k).
Sequences of the form a(1, n, k): A000071(k+2) (n=1), A027934(k-1) (n=2), A098703(k) (n=3).
Sequences of the form a(2, n, k): A001911(k) (n=1), A008466(k+1) (n=2), A106517(k-1) (n=3).
Sequences of the form a(3, n, k): A027961(k) (n=1), A094688(k) (n=3).
Sequences of the form a(4, n, k): A053311(k-1) (n=1), A027974(k-1) (n=2).
Cf. A001622, A001891, A001924, A023537, A104161, A106517.

A101220:= func< n | (&+[n^k*Fibonacci(n-k): k in [0..n]]) >;
[A101220(n): n in [0..30]]; // G. C. Greubel, Jun 01 2025

Join[{0}, Table[Sum[Fibonacci[n-k]*n^k, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Jan 03 2021 *)

a(n)=sum(k=0,n,fibonacci(n-k)*n^k) \\ Joerg Arndt, Jan 03 2021

def A101220(n): return sum(n^k*fibonacci(n-k) for k in range(n+1))
print([A101220(n) for n in range(31)]) # G. C. Greubel, Jun 01 2025

a(i, j, 0) = 0, a(i, j, 1) = 1, a(i, j, 2) = i+1; a(i, j, k) = ((j+1)*a(i, j, k-1)) - ((j-1)*a(i, j, k-2)) - (j*a(i, j, k-3)), for k > 2.
a(i, j, k) = Fibonacci(k) + i*a(j, j, k-1), for i, k > 0.
a(i, j, k) = (Phi^k - (-Phi)^-k + i(((j^k - Phi^k) / (j - Phi)) - ((j^k - (-Phi)^-k) / (j - (-Phi)^-1)))) / sqrt(5), where Phi denotes the golden mean/ratio (A001622).
i^k = a(i-1, i, k) + a(i-2, i, k+1).
A104161(k) = Sum_{m=0..k} a(k-m, 0, m).
a(i, j, 0) = 0, a(i, j, 1) = 1, a(i, j, 2) = i+1, a(i, j, 3) = i*(j+1) + 2; a(i, j, k) = (j+2)*a(i, j, k-1) - 2*j*a(i, j, k-2) - a(i, j, k-3) + j*a(i, j, k-4), for k > 3. a(i, j, 0) = 0, a(i, j, 1) = 1; a(i, j, k) = a(i, j, k-1) + a(i, j, k-2) + i * j^(k-2), for k > 1.
G.f.: x*(1 + (i-j)*x)/((1-j*x)*(1-x-x^2)).
a(n, n, n) = Sum_{k=0..n} Fibonacci(n-k) * n^k. - Ross La Haye, Jan 14 2006
Sum_{m=0..k} binomial(k,m)*(i-1)^m = a(i-1,i,k) + a(i-2,i,k+1), for i > 1. - Ross La Haye, May 29 2006
From Ross La Haye, Jun 03 2006: (Start)
a(3, 3, k+1) - a(3, 3, k) = A106517(k).
a(1, 1, k) = A001924(k) - A001924(k-1), for k > 0.
a(2, 1, k) = A001891(k) - A001891(k-1), for k > 0.
a(3, 1, k) = A023537(k) - A023537(k-1), for k > 0.
Sum_{j=0..i+1} a(i-j+1, 0, j) - Sum_{j=0..i} a(i-j, 0, j) = A001595(i). (End)
a(i,j,k) = a(j,j,k) + (i-j)*a(j,j,k-1), for k > 0.
a(n) ~ n^(n-1). - Vaclav Kotesovec, Jan 03 2021

New name from Joerg Arndt, Jan 03 2021

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

Original entry on oeis.org

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

A142585 Inverse binomial transform of A014217.

Original entry on oeis.org

1, 0, 1, 0, -1, 5, -14, 35, -81, 180, -389, 825, -1726, 3575, -7349, 15020, -30561, 61965, -125294, 252795, -509161, 1024100, -2057549, 4130225, -8284926, 16609455, -33282989, 66669660, -133507081, 267285605, -535010414, 1070731475, -2142612801, 4287086100

Offset: 0

Paul Curtz, Sep 21 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-3,-1,2).

Cf. A000032, A000045, A014217, A027974.

[1] cat [(-1)^n*( Lucas(n) - 2^(n-1) ): n in [1..40]]; // G. C. Greubel, Apr 14 2021

Table[(-1)^n*(LucasL[n] -2^(n-1)) - Boole[n==0]/2, {n,0,40}] (* G. C. Greubel, Apr 14 2021 *)

[1]+[(-1)^n*( lucas_number2(n,1,-1) - 2^(n-1) ) for n in (1..40)] # G. C. Greubel, Apr 14 2021

a(n) = (-1)^(n+1) * A027974(n-4) for n > 4.
G.f.: (1+3*x+2*x^2+x^3)/((1+2*x)*(1+x-x^2)). - R. J. Mathar, Sep 22 2008
a(n) = (-1)^(n+1)*(2^(n-1) -F(n+1) -F(n-1)), where F(n) = A000045(n), for n>=1, with a(0)=1. - Johannes W. Meijer, Aug 15 2010
a(n) = -3*a(n-1) - a(n-2) + 2*a(n-3). - Wesley Ivan Hurt, Oct 06 2017

Edited and extended by R. J. Mathar, Sep 22 2008

A175660 Eight bishops and one elephant on a 3 X 3 chessboard. a(n) = 2^(n+2) - 3*F(n+2).

Original entry on oeis.org

1, 2, 7, 17, 40, 89, 193, 410, 859, 1781, 3664, 7493, 15253, 30938, 62575, 126281, 254392, 511745, 1028281, 2064314, 4141171, 8302637, 16638112, 33329357, 66744685, 133628474, 267482023, 535328225, 1071245704, 2143444841

Offset: 0

Johannes W. Meijer, Aug 06 2010, Aug 10 2010

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7, 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a bishop on the eight side and corner squares but on the central square the bishop turns into a raging elephant, see A175654.

The sequence above corresponds to four A[5] vectors with decimal values 171, 174, 234 and 426. These vectors lead for the side squares to A000079 and for the central square to A175661 (a(n) = 2^(n+2) - 3*F(n+1)).

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

Cf. A008466 (2^n-F(n+2)), A027934 (2^n-F(n+1)), A027974 (2^(n+3)-F(n+5)-F(n+3)), A074878 (3*2^n-2*F(n+2)), A142585 ((-1)^(n+1)*(2^(n-1)-F(n+1)-F(n-1))), A175661 (2^(n+2)-3*F(n+1)), A179610 (convolution of (-4)^n and F(n+1)).

nmax:=29; m:=1; A[5]:= [0,1,0,1,0,1,0,1,1]: A:=Matrix([[0,0,0,0,1,0,0,0,1], [0,0,0,1,0,1,0,0,0], [0,0,0,0,1,0,1,0,0], [0,1,0,0,0,0,0,1,0], A[5], [0,1,0,0,0,0,0,1,0], [0,0,1,0,1,0,0,0,0], [0,0,0,1,0,1,0,0,0], [1,0,0,0,1,0,0,0,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

Table[2^(n+2)-3Fibonacci[n+2],{n,0,30}] (* or *) LinearRecurrence[ {3,-1,-2},{1,2,7},30] (* Harvey P. Dale, Dec 28 2012 *)

G.f.: (1 - x + 2*x^2)/(1 - 3*x + x^2 + 2*x^3).
a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3) with a(0)=1, a(1)=2 and a(2)=7.
a(n) = 2^(n+2) - 3*F(n+2) with F(n)=A000045(n).

A227200 a(n) = a(n-1) + a(n-2) - 2^(n-1) with a(0)=a(2)=0, a(1)=-a(3)=1, a(4)=-5.

Original entry on oeis.org

0, 1, 0, -1, -5, -14, -35, -81, -180, -389, -825, -1726, -3575, -7349, -15020, -30561, -61965, -125294, -252795, -509161, -1024100, -2057549, -4130225, -8284926, -16609455, -33282989, -66669660, -133507081, -267285605, -535010414, -1070731475

Offset: 0

Chandrakant N Phadte, Sep 18 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. N. Phadte and S. P. Pethe, On Second Order Non-homogeneous recurrence relation, Annales Mathematicae et informaticae, 41 (2013), pp. 205-210.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Cf. versions with different signs: A027974, A142585.

Cf. A000032, A000045.

LET N=0
LET L=0
LET M=1
PRINT L
PRINT M
FOR I=1 TO 30
LET N=M+L-(2)^(I-1)
PRINT N
LET L=M
LET M=N
NEXT I
END

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((1-3*x)/((1-2*x)*(1-x-x^2)))); // Bruno Berselli, Oct 03 2013

I:=[0,1,0,-1,-5]; [n le 5 select I[n] else Self(n-1)+Self(n-2)-2^(n-3): n in [1..35]]; // Vincenzo Librandi, Oct 05 2013

Table[LucasL[n + 1] - 2^n, {n, 0, 30}] (* Bruno Berselli, Oct 03 2013 *)
CoefficientList[Series[x (1 - 3 x)/((1 - 2 x) (1 - x - x^2)), {x, 0, 40}], x](* Vincenzo Librandi, Oct 05 2013 *)

a(n)=fibonacci(n)+fibonacci(n+2)-2^n \\ Charles R Greathouse IV, Oct 03 2013

G.f.: x*(1-3*x)/((1-2*x)*(1-x-x^2)).
a(n) = -(-1)^n*A142585(n+1) = A000032(n+1) - 2^n. [Bruno Berselli, Oct 03 2013]
a(n) = 3*a(n-1) -a(n-2) -2*a(n-3). [Bruno Berselli, Oct 03 2013]

More terms from Bruno Berselli, Oct 03 2013

A237498 Riordan array (1/(1-x-x^2), x/(1+2*x)).

Original entry on oeis.org

1, 1, 1, 2, -1, 1, 3, 4, -3, 1, 5, -5, 10, -5, 1, 8, 15, -25, 20, -7, 1, 13, -22, 65, -65, 34, -9, 1, 21, 57, -152, 195, -133, 52, -11, 1, 34, -93, 361, -542, 461, -237, 74, -13, 1, 55, 220, -815, 1445, -1464, 935, -385, 100, -15, 1, 89, -385, 1850, -3705

Offset: 0

Philippe Deléham, Feb 08 2014

First column: Fibonacci numbers A000045(n+1).

			Triangle begins:
   1;
   1,    1;
   2,   -1,    1;
   3,    4,   -3,    1;
   5,   -5,   10,   -5,   1;
   8,   15,  -25,   20,  -7,   1;
  13,  -22,   65,  -65,  34,  -9,  1;
  ...
Production matrix is:
   1,  1;
   1, -2,  1;
   2,  0, -2,  1;
   4,  0,  0, -2,  1;
   8,  0,  0,  0, -2,  1;
  16,  0,  0,  0,  0, -2,  1;
  32,  0,  0,  0,  0,  0, -2,  1;
  64,  0,  0,  0,  0,  0,  0, -2,  1;
  ...

Indranil Ghosh, Rows 0..100, flattened

Columns: A000045, A084179.

nmax=10;Flatten[CoefficientList[Series[CoefficientList[Series[(1 + 2*x) / ((1 + 2*x - y*x) * (1 - x - x^2)), {x, 0, nmax }], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 15 2017 *)

Sum_{k=0..n} T(n,k)*x^k = A000045(n+1), A098600(n), A000032(n+1), A027961(n+1), A027974(n) for x = 0, 1, 2, 3, 4 respectively.
T(n,k) = T(n-1,k-1) - T(n-1,k) + 3*T(n-2,k) - T(n-2,k-1) + 2*T(n-3,k) - T(n-3,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = 2, T(2,1) = -1, T(n,k) = 0 if k<0 or if k>n.
T(n,0) = T(n-1,0) + T(n-2,0) with T(0,0) = T(1,0) = 1, T(n,k) = T(n-1,k-1) - 2*T(n-1,k) for k>=1.
G.f.: (1+2*x)/((1+2*x-y*x)*(1-x-x^2)).

cp's OEIS Frontend

A027983 Duplicate of A027974.

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A101220 a(n) = Sum_{k=0..n} Fibonacci(n-k)*n^k.

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

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A142585 Inverse binomial transform of A014217.

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A175660 Eight bishops and one elephant on a 3 X 3 chessboard. a(n) = 2^(n+2) - 3*F(n+2).

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A227200 a(n) = a(n-1) + a(n-2) - 2^(n-1) with a(0)=a(2)=0, a(1)=-a(3)=1, a(4)=-5.

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A237498 Riordan array (1/(1-x-x^2), x/(1+2*x)).

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