A075193 -id:A075193 - cp's OEIS frontend

A075298 Inverted (definition in A075193) generalized tribonacci numbers A001644.

1, 1, -5, 5, 1, -11, 15, -3, -23, 41, -21, -43, 105, -83, -65, 253, -271, -47, 571, -795, 177, 1189, -2161, 1149, 2201, -5511, 4459, 3253, -13223, 14429, 2047, -29699, 42081, -10335, -61445, 113861, -62751, -112555, 289167, -239363, -162359, 690889, -767893, -85355, 1544137, -2226675, 597183, 3173629

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Sep 13 2002

a(n) = -C(n+1), C(n)=reflected generalized tribonacci numbers A073145.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Curtis Cooper, S. Miller, Peter J. C. Moses, M. Sahin, and T. Thanatipanonda, On Identities of Ruggles, Horadam, Howard, and Young, Preprint 2016.
Index entries for linear recurrences with constant coefficients, signature (-1,-1,1).

Cf. A001644, A073145, A075193.

a:=[1,1,-5];; for n in [4..50] do a[n]:=-a[n-1]-a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 09 2019

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1+2*x-3*x^2)/(1+x+x^2-x^3) )); // G. C. Greubel, Apr 09 2019

CoefficientList[Series[(1+2x-3x^2)/(1+x+x^2-x^3), {x, 0, 50}], x]

my(x='x+O('x^50)); Vec((1+2*x-3*x^2)/(1+x+x^2-x^3)) \\ G. C. Greubel, Apr 09 2019

((1+2*x-3*x^2)/(1+x+x^2-x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Apr 09 2019

a(n) = -a(n-1) - a(n-2) + a(n-3), a(0)=1, a(1)=1, a(2)=-5.
G.f.: (1+2*x-3*x^2)/(1+x+x^2-x^3).
a(n) = A078046(n) + 3*A078046(n-1). - R. J. Mathar, Sep 20 2020

A075269 Product of Lucas numbers and inverted Lucas numbers: a(n)=A000032(n)*A075193(n).

Original entry on oeis.org

2, -3, 12, -28, 77, -198, 522, -1363, 3572, -9348, 24477, -64078, 167762, -439203, 1149852, -3010348, 7881197, -20633238, 54018522, -141422323, 370248452, -969323028, 2537720637, -6643838878, 17393796002, -45537549123, 119218851372, -312119004988, 817138163597

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Sep 11 2002

Index entries for linear recurrences with constant coefficients, signature (-2,2,1).

Cf. A000032, A002878, A075193.

CoefficientList[Series[(2 + x + 2x^2)/(1 + 2x - 2x^2 - x^3), {x, 0, 30}], x]
LinearRecurrence[{-2,2,1},{2,-3,12},30] (* Harvey P. Dale, Jun 30 2022 *)

a(n)=1+(-1)^n*(fibonacci(2*n)+fibonacci(2*n+2))

a(n) = 1 + (-1)^n*A002878(n).
From Michael Somos, Apr 07 2003: (Start)
G.f.: (2+x+2x^2)/((1+3x+x^2)(1-x)).
a(n) = -3a(n-1) - a(n-2)+5 = -2a(n-1) + 2a(n-2) + a(n-3) = a(-1-n). (End)
Sum_{n>=0} 1/a(n) = sqrt(5)/10. - Amiram Eldar, Jan 15 2022
a(n) = (-1)^n*A215602(n). - R. J. Mathar, Jul 09 2024
a(n) - a(n-1) = (-1)^n* A054888(n), n>0. - R. J. Mathar, Jul 09 2024

A075270 Sum of Lucas numbers and inverted Lucas numbers: a(n) = A000032(n)*A075193(n).

Original entry on oeis.org

3, -2, 7, -3, 18, -7, 47, -18, 123, -47, 322, -123, 843, -322, 2207, -843, 5778, -2207, 15127, -5778, 39603, -15127, 103682, -39603, 271443, -103682, 710647, -271443, 1860498, -710647, 4870847, -1860498, 12752043, -4870847, 33385282, -12752043, 87403803, -33385282, 228826127, -87403803

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Sep 12 2002

a(n) = (1 + (-1)^n)*L(n) + ((-1)^n)*L(n-1), L(n) Lucas numbers.

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Cf. A000032, A075193.

CoefficientList[Series[(3-2x-2x^2+3x^3)/(1-3x^2+x^4), {x, 0, 40}], x]

a(n) = 3*a(n-2) - a(n-4); a(0)=3, a(1)=-2, a(2)=7, a(3)=-3.

O.g.f. (3-2*x-2*x^2+3*x^3)/(1-3*x^2+x^4).

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

A186679 First differences of A116697.

Original entry on oeis.org

0, -3, 4, -4, 7, -14, 22, -33, 54, -90, 145, -232, 376, -611, 988, -1596, 2583, -4182, 6766, -10945, 17710, -28658, 46369, -75024, 121392, -196419, 317812, -514228, 832039, -1346270, 2178310, -3524577, 5702886, -9227466, 14930353, -24157816, 39088168, -63245987, 102334156, -165580140

Offset: 0

Reinhard Zumkeller, Feb 25 2011

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

Cf. A000045, A075193, A081714, A116697, A128533.

A186679:= func< n | (-1)^n*Fibonacci(n+2) - (-1)^Floor(n/2) >;
[A186679(n): n in [0..40]]; // G. C. Greubel, Aug 24 2025

Table[(-1)^n*Fibonacci[n+2] -(-1)^Floor[n/2], {n,0,40}] (* G. C. Greubel, Aug 24 2025 *)

def A186679(n): return (-1)**n*fibonacci(n+2) -(-1)**(n//2)
print([A186679(n) for n in range(41)]) # G. C. Greubel, Aug 24 2025

a(n) = A116697(n+1) - A116697(n).
a(2*n) = A128533(n).
a(2*n+1) = A081714(n+1).
a(n+2) = A075193(n+2) - a(n).
G.f.: x*(-3+x)/((1+x-x^2)*(1+x^2)). - Colin Barker, Sep 08 2012
From G. C. Greubel, Aug 24 2025: (Start)
a(n) = (-1)^n*Fibonacci(n+2) - (-1)^floor(n/2).
E.g.f.: exp(-x/2)*(cosh(p*x) - (3/sqrt(5))*sinh(p*x)) - cos(x) - sin(x), where 2*p = sqrt(5). (End)

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A075298 Inverted (definition in A075193) generalized tribonacci numbers A001644.

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A075269 Product of Lucas numbers and inverted Lucas numbers: a(n)=A000032(n)*A075193(n).

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A075270 Sum of Lucas numbers and inverted Lucas numbers: a(n) = A000032(n)*A075193(n).

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

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A186679 First differences of A116697.

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