A057597 -id:A057597 - cp's OEIS frontend

A319200 a(n) = -(A(n) - A(n-1)) where A(n) = A057597(n+1), for n >= 0.

0, -1, 2, -1, -2, 5, -4, -3, 12, -13, -2, 27, -38, 9, 56, -103, 56, 103, -262, 215, 150, -627, 692, 85, -1404, 2011, -522, -2893, 5426, -3055, -5264, 13745, -11536, -7473, 32754, -36817, -3410, 72981, -106388, 29997, 149372, -285757, 166382, 268747, -720886, 618521, 371112, -1710519, 1957928, 123703, -3792150

Offset: 0

Wolfdieter Lang, Oct 23 2018

This sequence appears in the reduction formula for negative powers of the tribonacci constant t = A058265: t^(-n) = A(n)*t^2 + a(n)*t + A(n+1)*1, with A(n) = A057597(n+1), for n >= 0. This follows from t^3 = t^2 + t + 1, or 1/t = t^2 - t - 1 = A192918, leading to the recurrence: A(n) = -A(n) - A(n-1) + A(n-2), with inputs A(-3) = 1, A(-2) = 1 and A(-1) = 0 and a(n) = -(A(n) - A(n-1)). See the example below.

			The coefficients of t^2, t, 1 for t^(-n) begin, for n >= -3:
n     t^2  t   1
-----------------
-3     1   1   1
-2     1   0   0
-1     0   1   0
----------------
+0     0   0   1
+1     1  -1  -1
+2    -1   2   0
+3     0  -1   2
+4     2  -2  -3
+5    -3   5   1
+6     1  -4   4
+7     4  -3  -8
+8    -8  12   5
+9     5 -13   7
10     7  -2 -20
...

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

Cf. A057597, A058265, A078016(n+1) (different signs), A192918.

LinearRecurrence[{-1,-1,1},{0,-1,2},60] (* Harvey P. Dale, Jul 20 2025 *)

a057597(n) = polcoeff( if( n<0, x / ( 1 - x - x^2 - x^3), x^2 / ( 1 + x + x^2 - x^3) ) + x*O(x^abs(n)), abs(n)) \\ after Michael Somos in A057597
a(n) = -(a057597(n+1)-a057597(n)) \\ Felix Fröhlich, Oct 23 2018

a(n) = -(A057597(n+1) - A057597(n)), for n >= 0.
Recurrence  a(n) = -a(n-1) - a(n-2) + a(n-3), for n >=0, with a(-3) = 1, a(-2) = 0 and a(-1) = 1.
G.f.: (1 + 1/x)/(1 + x + x^2 - x^3).

A075297 Duplicate of A057597.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 3, 1, 4, 8, 5, 7, 20, 18, 9, 47, 56, 0, 103, 159, 56, 206, 421, 271, 356, 1048

Offset: 0

dead

A000073 Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) for n >= 3 with a(0) = a(1) = 0 and a(2) = 1.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, 121415, 223317, 410744, 755476, 1389537, 2555757, 4700770, 8646064, 15902591, 29249425, 53798080, 98950096, 181997601, 334745777, 615693474, 1132436852

Offset: 0

N. J. A. Sloane

The name "tribonacci number" is less well-defined than "Fibonacci number". The sequence A000073 (which begins 0, 0, 1) is probably the most important version, but the name has also been applied to A000213, A001590, and A081172. - N. J. A. Sloane, Jul 25 2024
Also (for n > 1) number of ordered trees with n+1 edges and having all leaves at level three. Example: a(4)=2 because we have two ordered trees with 5 edges and having all leaves at level three: (i) one edge emanating from the root, at the end of which two paths of length two are hanging and (ii) one path of length two emanating from the root, at the end of which three edges are hanging. - Emeric Deutsch, Jan 03 2004
a(n) is the number of compositions of n-2 with no part greater than 3. Example: a(5)=4 because we have 1+1+1 = 1+2 = 2+1 = 3. - Emeric Deutsch, Mar 10 2004
Let A denote the 3 X 3 matrix [0,0,1;1,1,1;0,1,0]. a(n) corresponds to both the (1,2) and (3,1) entries in A^n. - Paul Barry, Oct 15 2004
Number of permutations satisfying -k <= p(i)-i <= r, i=1..n-2, with k=1, r=2. - Vladimir Baltic, Jan 17 2005
Number of binary sequences of length n-3 that have no three consecutive 0's. Example: a(7)=13 because among the 16 binary sequences of length 4 only 0000, 0001 and 1000 have 3 consecutive 0's. - Emeric Deutsch, Apr 27 2006
Therefore, the complementary sequence to A050231 (n coin tosses with a run of three heads). a(n) = 2^(n-3) - A050231(n-3) - Toby Gottfried, Nov 21 2010
Convolved with the Padovan sequence = row sums of triangle A153462. - Gary W. Adamson, Dec 27 2008
For n > 1: row sums of the triangle in A157897. - Reinhard Zumkeller, Jun 25 2009
a(n+2) is the top left entry of the n-th power of any of the 3 X 3 matrices [1, 1, 1; 0, 0, 1; 1, 0, 0] or [1, 1, 0; 1, 0, 1; 1, 0, 0] or [1, 1, 1; 1, 0, 0; 0, 1, 0] or [1, 0, 1; 1, 0, 0; 1, 1, 0]. - R. J. Mathar, Feb 03 2014
a(n-1) is the top left entry of the n-th power of any of the 3 X 3 matrices [0, 0, 1; 1, 1, 1; 0, 1, 0], [0, 1, 0; 0, 1, 1; 1, 1, 0], [0, 0, 1; 1, 0, 1; 0, 1, 1] or [0, 1, 0; 0, 0, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
Also row sums of A082601 and of A082870. - Reinhard Zumkeller, Apr 13 2014
Least significant bits are given in A021913 (a(n) mod 2 = A021913(n)). - Andres Cicuttin, Apr 04 2016
The nonnegative powers of the tribonacci constant t = A058265 are t^n = a(n)*t^2 + (a(n-1) + a(n-2))*t + a(n-1)*1, for  n >= 0, with  a(-1) = 1 and a(-2) = -1. This follows from the recurrences derived from t^3 = t^2 + t + 1. See the example in A058265 for the first nonnegative powers. For the negative powers see A319200. - Wolfdieter Lang, Oct 23 2018
The term "tribonacci number" was coined by Mark Feinberg (1963), a 14-year-old student in the 9th grade of the Susquehanna Township Junior High School in Pennsylvania. He died in 1967 in a motorcycle accident. - Amiram Eldar, Apr 16 2021
Andrews, Just, and Simay (2021, 2022) remark that it has been suggested that this sequence is mentioned in Charles Darwin's Origin of Species as bearing the same relation to elephant populations as the Fibonacci numbers do to rabbit populations. - N. J. A. Sloane, Jul 12 2022

			G.f. = x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 13*x^7 + 24*x^8 + 44*x^9 + 81*x^10 + ...

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

Cf. A000045, A000078, A000213, A000931, A001590 (first differences, also a(n)+a(n+1)), A001644, A008288 (tribonacci triangle), A008937 (partial sums), A021913, A027024, A027083, A027084, A046738 (Pisano periods), A050231, A054668, A062544, A063401, A077902, A081172, A089068, A118390, A145027, A153462, A230216.
A057597 is this sequence run backwards: A057597(n) = a(1-n).
Row 3 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).
Partitions: A240844 and A117546.
Cf. also A092836 (subsequence of primes), A299399 = A092835 + 1 (indices of primes).

a:=[0,0,1];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # Muniru A Asiru, Oct 24 2018

a000073 n = a000073_list !! n
a000073_list = 0 : 0 : 1 : zipWith (+) a000073_list (tail
                          (zipWith (+) a000073_list $ tail a000073_list))
-- Reinhard Zumkeller, Dec 12 2011

[n le 3 select Floor(n/3) else Self(n-1)+Self(n-2)+Self(n-3): n in [1..70]]; // Vincenzo Librandi, Jan 29 2016

a:= n-> (<<0|1|0>, <0|0|1>, <1|1|1>>^n)[1,3]:
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 19 2016
# second Maple program:
A000073:=proc(n) option remember; if n <= 1 then 0 elif n=2 then 1 else procname(n-1)+procname(n-2)+procname(n-3); fi; end; # N. J. A. Sloane, Aug 06 2018

CoefficientList[Series[x^2/(1 - x - x^2 - x^3), {x, 0, 50}], x]
a[0] = a[1] = 0; a[2] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2] + a[n - 3]; Array[a, 36, 0] (* Robert G. Wilson v, Nov 07 2010 *)
LinearRecurrence[{1, 1, 1}, {0, 0, 1}, 60] (* Vladimir Joseph Stephan Orlovsky, May 24 2011 *)
a[n_] := SeriesCoefficient[If[ n < 0, x/(1 + x + x^2 - x^3), x^2/(1 - x - x^2 - x^3)], {x, 0, Abs @ n}] (* Michael Somos, Jun 01 2013 *)
Table[-RootSum[-1 - # - #^2 + #^3 &, -#^n - 9 #^(n + 1) + 4 #^(n + 2) &]/22, {n, 0, 20}] (* Eric W. Weisstein, Nov 09 2017 *)

A000073[0]:0$
A000073[1]:0$
A000073[2]:1$
A000073[n]:=A000073[n-1]+A000073[n-2]+A000073[n-3]$
  makelist(A000073[n], n, 0, 40);  /* Emanuele Munarini, Mar 01 2011 */

{a(n) = polcoeff( if( n<0, x / ( 1 + x + x^2 - x^3), x^2 / ( 1 - x - x^2 - x^3) ) + x * O(x^abs(n)), abs(n))}; /* Michael Somos, Sep 03 2007 */

my(x='x+O('x^99)); concat([0, 0], Vec(x^2/(1-x-x^2-x^3))) \\ Altug Alkan, Apr 04 2016

a(n)=([0,1,0;0,0,1;1,1,1]^n)[1,3] \\ Charles R Greathouse IV, Apr 18 2016, simplified by M. F. Hasler, Apr 18 2018

def a(n, adict={0:0, 1:0, 2:1}):
    if n in adict:
        return adict[n]
    adict[n]=a(n-1)+a(n-2)+a(n-3)
    return adict[n] # David Nacin, Mar 07 2012
from functools import cache
@cache
def A000073(n: int) -> int:
    if n <= 1: return 0
    if n == 2: return 1
    return A000073(n-1) + A000073(n-2) + A000073(n-3) # Peter Luschny, Nov 21 2022

G.f.: x^2/(1 - x - x^2 - x^3).
G.f.: x^2 / (1 - x / (1 - x / (1 + x^2 / (1 + x)))). - Michael Somos, May 12 2012
G.f.: Sum_{n >= 0} x^(n+2) *[ Product_{k = 1..n} (k + k*x + x^2)/(1 + k*x + k*x^2) ] = x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 13*x^7 + ... may be proved by the method of telescoping sums. - Peter Bala, Jan 04 2015
a(n+1)/a(n) -> A058265.  a(n-1)/a(n) -> A192918.
a(n) = central term in M^n * [1 0 0] where M = the 3 X 3 matrix [0 1 0 / 0 0 1 / 1 1 1]. (M^n * [1 0 0] = [a(n-1) a(n) a(n+1)].) a(n)/a(n-1) tends to the tribonacci constant, 1.839286755... = A058265, an eigenvalue of M and a root of x^3 - x^2 - x - 1 = 0. - Gary W. Adamson, Dec 17 2004
a(n+2) = Sum_{k=0..n} T(n-k, k), where T(n, k) = trinomial coefficients (A027907). - Paul Barry, Feb 15 2005
A001590(n) = a(n+1) - a(n); A001590(n) = a(n-1) + a(n-2) for n > 1; a(n) = (A000213(n+1) - A000213(n))/2; A000213(n-1) = a(n+2) - a(n) for n > 0. - Reinhard Zumkeller, May 22 2006
Let C = the tribonacci constant, 1.83928675...; then C^n = a(n)*(1/C) + a(n+1)*(1/C + 1/C^2) + a(n+2)*(1/C + 1/C^2 + 1/C^3). Example: C^4 = 11.444...= 2*(1/C) + 4*(1/C + 1/C^2) + 7*(1/C + 1/C^2 + 1/C^3). - Gary W. Adamson, Nov 05 2006
a(n) = j*C^n + k*r1^n + L*r2^n where C is the tribonacci constant (C = 1.8392867552...), real root of x^3-x^2-x-1=0, and r1 and r2 are the two other roots (which are complex), r1 = m+p*i and r2 = m-p*i, where i = sqrt(-1), m = (1-C)/2 (m = -0.4196433776...) and p = ((3*C-5)*(C+1)/4)^(1/2) = 0.6062907292..., and where j = 1/((C-m)^2 + p^2) = 0.1828035330..., k = a+b*i, and L = a-b*i, where a = -j/2 = -0.0914017665... and b = (C-m)/(2*p*((C-m)^2 + p^2)) = 0.3405465308... . - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 23 2007
a(n+1) = 3*c*((1/3)*(a+b+1))^n/(c^2-2*c+4) where a=(19+3*sqrt(33))^(1/3), b=(19-3*sqrt(33))^(1/3), c=(586+102*sqrt(33))^(1/3). Round to the nearest integer. - Al Hakanson (hawkuu(AT)gmail.com), Feb 02 2009
a(n) = round(3*((a+b+1)/3)^n/(a^2+b^2+4)) where a=(19+3*sqrt(33))^(1/3), b=(19-3*sqrt(33))^(1/3).. - Anton Nikonov
Another form of the g.f.: f(z) = (z^2-z^3)/(1-2*z+z^4). Then we obtain a(n) as a sum: a(n) = Sum_{i=0..floor((n-2)/4)} ((-1)^i*binomial(n-2-3*i,i)*2^(n-2-4*i)) - Sum_{i=0..floor((n-3)/4)} ((-1)^i*binomial(n-3-3*i,i)*2^(n-3-4*i)) with natural convention: Sum_{i=m..n} alpha(i) = 0 for m > n. - Richard Choulet, Feb 22 2010
a(n+2) = Sum_{k=0..n} Sum_{i=k..n, mod(4*k-i,3)=0} binomial(k,(4*k-i)/3)*(-1)^((i-k)/3)*binomial(n-i+k-1,k-1). - Vladimir Kruchinin, Aug 18 2010
a(n) = 2*a(n-2) + 2*a(n-3) + a(n-4). - Gary Detlefs, Sep 13 2010
Sum_{k=0..2*n} a(k+b)*A027907(n,k) = a(3*n+b), b >= 0 (see A099464, A074581).
a(n) = 2*a(n-1) - a(n-4), with a(0)=a(1)=0, a(2)=a(3)=1. - Vincenzo Librandi, Dec 20 2010
Starting (1, 2, 4, 7, ...) is the INVERT transform of (1, 1, 1, 0, 0, 0, ...). - Gary W. Adamson, May 13 2013
G.f.: Q(0)*x^2/2, where Q(k) = 1 + 1/(1 - x*(4*k+1 + x + x^2)/( x*(4*k+3 + x + x^2) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 09 2013
a(n+2) = Sum_{j=0..floor(n/2)} Sum_{k=0..j} binomial(n-2*j,k)*binomial(j,k)*2^k. - Tony Foster III, Sep 08 2017
Sum_{k=0..n} (n-k)*a(k) = (a(n+2) + a(n+1) - n - 1)/2. See A062544. - Yichen Wang, Aug 20 2020
a(n) = A008937(n-1) - A008937(n-2) for n >= 2. - Peter Luschny, Aug 20 2020
From Yichen Wang, Aug 27 2020: (Start)
Sum_{k=0..n} a(k) = (a(n+2) + a(n) - 1)/2. See A008937.
Sum_{k=0..n} k*a(k) = ((n-1)*a(n+2) - a(n+1) + n*a(n) + 1)/2. See A337282. (End)
For n > 1, a(n) = b(n) where b(1) = 1 and then b(n) = Sum_{k=1..n-1} b(n-k)*A000931(k+2). - J. Conrad, Nov 24 2022
Conjecture: the congruence a(n*p^(k+1)) + a(n*p^k) + a(n*p^(k-1)) == 0 (mod p^k) holds for positive integers k and n and for all the primes p listed in A106282. - Peter Bala, Dec 28 2022
Sum_{k=0..n} k^2*a(k) = ((n^2-4*n+6)*a(n+1) - (2*n^2-2*n+5)*a(n) + (n^2-2*n+3)*a(n-1) - 3)/2. - Prabha Sivaramannair, Feb 10 2024
a(n) = Sum_{r root of x^3-x^2-x-1} r^n/(3*r^2-2*r-1). - Fabian Pereyra, Nov 23 2024

Minor edits by M. F. Hasler, Apr 18 2018

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A058265 Decimal expansion of the tribonacci constant t, the real root of x^3 - x^2 - x - 1.

Original entry on oeis.org

1, 8, 3, 9, 2, 8, 6, 7, 5, 5, 2, 1, 4, 1, 6, 1, 1, 3, 2, 5, 5, 1, 8, 5, 2, 5, 6, 4, 6, 5, 3, 2, 8, 6, 6, 0, 0, 4, 2, 4, 1, 7, 8, 7, 4, 6, 0, 9, 7, 5, 9, 2, 2, 4, 6, 7, 7, 8, 7, 5, 8, 6, 3, 9, 4, 0, 4, 2, 0, 3, 2, 2, 2, 0, 8, 1, 9, 6, 6, 4, 2, 5, 7, 3, 8, 4, 3, 5, 4, 1, 9, 4, 2, 8, 3, 0, 7, 0, 1, 4

Offset: 1

Robert G. Wilson v, Dec 07 2000

"The tribonacci constant, the only real solution to the equation x^3 - x^2 - x - 1 = 0, which is related to tribonacci sequences (in which U_n = U_n-1 + U_n-2 + U_n-3) as the Golden Ratio is related to the Fibonacci sequence and its generalizations. This ratio also appears when a snub cube is inscribed in an octahedron or a cube, by analogy once again with the appearance of the Golden Ratio when an icosahedron is inscribed in an octahedron. [John Sharp, 1997]"
The tribonacci constant corresponds to the Golden Section in a tripartite division 1 = u_1 + u_2 + u_3 of a unit line segment; i.e., if 1/u_1 = u_1/u_2 = u_2/u_3 = c, c is the tribonacci constant. - Seppo Mustonen, Apr 19 2005
The other two polynomial roots are the complex-conjugated pair -0.4196433776070805662759262... +- i* 0.60629072920719936925934... - R. J. Mathar, Oct 25 2008
For n >= 3, round(q^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. - Vladimir Shevelev, Mar 21 2014
Concerning orthogonal projections, the tribonacci constant is the ratio of the diagonal of a square to the width of a rhombus projected by rotating a square along its diagonal in 3D until the angle of rotation equals the apparent apex angle at approximately 57.065 degrees (also the corresponding angle in the formula generating A256099). See illustration in the links. - Peter M. Chema, Jan 02 2017
From Wolfdieter Lang, Aug 10 2018: (Start)
Real eigenvalue t of the tribonacci Q-matrix <<1, 1, 1>,<1, 0, 0>,<0, 1, 0>>.
Limit_{n -> oo} T(n+1)/T(n) = t (from the T recurrence), where T = {A000073(n+2)}_{n >= 0}. (End)
The nonnegative powers of t are t^n = T(n)*t^2 + (T(n-1) + T(n-2))*t + T(n-1)*1, for n >= 0, with T(n) = A000073(n), with T(-1) = 1 and T(-2) = -1, This follows from the recurrences derived from t^3 = t^2 + t + 1. See the examples below. For the negative powers see A319200. - Wolfdieter Lang, Oct 23 2018
Note that we have: t + t^(-3) = 2, and the k-nacci constant approaches 2 when k approaches infinity (Martin Gardner). - Bernard Schott, May 16 2022
The roots of this cubic are found from those of y^3 - (4/3)*y - 38/27, after adding 1/3. - Wolfdieter Lang, Aug 24 2022
The algebraic number t - 1 has minimal polynomial x^3 + 2*x^2 - 2 over Q. The roots coincide with those of y^3 - (4/3)*y - 38/27, after subtracting 2/3. - Wolfdieter Lang, Sep 20 2022
The value of the ratio R/r of the radius R of a uniform ball to the radius r of a spherical hole in it with a common point of contact, such that the center of gravity of the object lies on the surface of the spherical hole (Schmidt, 2002). - Amiram Eldar, May 20 2023

			1.8392867552141611325518525646532866004241787460975922467787586394042032220\
    81966425738435419428307014141979826859240974164178450746507436943831545\
    820499513796249655539644613666121540277972678118941041...
From _Wolfdieter Lang_, Oct 23 2018: (Start)
The coefficients of t^2, t, 1 for t^n begin, for n >= 0:
    n     t^2    t    1
    -------------------
    0      0     0    1
    1      0     1    0
    2      1     0    0
    1      1     1    1
    4      2     2    1
    5      4     3    2
    6      7     6    4
    7     13    11    7
    8     24    20   13
    9     44    37   24
   10     81    68   44
...  (End)

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.
Martin Gardner, The Second Scientific American Book of Mathematical Puzzles and Diversions, "Phi: The Golden Ratio", Chapter 8, p. 101, Simon & Schuster, NY, 1961.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, page 23.

Harry J. Smith, Table of n, a(n) for n = 1..20000
A. Beha et al., The convergence of diffy boxes, American Mathematical Monthly, Vol. 112 (2005), pp. 426-439.
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), Article P1.52.
O. Deveci, Y. Akuzum, E. Karaduman, and O. Erdag, The Cyclic Groups via Bezout Matrices, Journal of Mathematics Research, Vol. 7, No. 2 (2015), pp. 34-41.
Ömür Deveci, Zafer Adıgüzel, and Taha Doğan, On the Generalized Fibonacci-circulant-Hurwitz numbers, Notes on Number Theory and Discrete Mathematics, Vol. 26, No. 1 (2020), 179-190.
Peter M. Chema, Tribonacci constant as ratio of square to rhombus projection.
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
S. Litsyn and Vladimir Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, Vol. 1, No. 4 (2005), 499-512.
Xerardo Neira, A geometric construction of the tribonacci constant with marked ruler and compass.
Tito Piezas III, Tribonacci constant and Pi.
Simon Plouffe, Tribonacci constant to 2000 digits.
Simon Plouffe, The Tribonacci constant(to 1000 digits).
Herbert C. H. Schmidt, Problem 2670, Crux Mathematicorum, Vol. 28, No. 7 (2002), pp. 464-465.
Vladimir Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014).
Nikita Sidorov, Expansions in non-integer bases: Lower, middle and top orders, Journal of Number Theory, Volume 129, Issue 4, April 2009, Pages 741-754. See Lemma 4.1 p. 750.
Kees van Prooijen, The Odd Golden Section.
Kees van Prooijen, Tribonacci Box (analog of Golden Rectangle).
Eric Weisstein's World of Mathematics, Tribonacci Number.
Eric Weisstein's World of Mathematics, Tribonacci Constant.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
Index entries for algebraic numbers, degree 3

Cf. A000073, A019712 (continued fraction), A133400, A254231, A158919 (spectrum = floor(n*t)), A357101 (x^3-2*x^2-2).
Cf. A192918 (reciprocal), A276800 (square), A276801 (cube), A319200.
k-nacci constants: A001622 (Fibonacci), this sequence (tribonacci), A086088 (tetranacci), A103814 (pentanacci), A118427 (hexanacci), A118428 (heptanacci).

Digits:=200; fsolve(x^3=x^2+x+1); # N. J. A. Sloane, Mar 16 2019

RealDigits[ N[ 1/3 + 1/3*(19 - 3*Sqrt[33])^(1/3) + 1/3*(19 + 3*Sqrt[33])^(1/3), 100]] [[1]]
RealDigits[Root[x^3-x^2-x-1,1],10,120][[1]] (* Harvey P. Dale, Mar 23 2019 *)

set_display(none)$ fpprec:100$ bfloat(rhs(solve(t^3-t^2-t-1,t)[3])); /* Dimitri Papadopoulos, Nov 09 2023 */

default(realprecision, 20080); x=solve(x=1, 2, x^3 - x^2 - x - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b058265.txt", n, " ", d));  \\ Harry J. Smith, May 30 2009

q=(1+sqrtn(19+3*sqrt(33),3)+sqrtn(19-3*sqrt(33),3))/3 \\ Use \p# to set 'realprecision'. - M. F. Hasler, Mar 23 2014

t = (1/3)*(1+(19+3*sqrt(33))^(1/3)+(19-3*sqrt(33))^(1/3)). - Zak Seidov, Jun 08 2005
t = 1 - Sum_{k>=1} A057597(k+2)/(T_k*T_(k+1)), where T_n = A000073(n+1). - Vladimir Shevelev, Mar 02 2013
1/t + 1/t^2 + 1/t^3 = 1/A058265 + 1/A276800 + 1/A276801 = 1. - N. J. A. Sloane, Oct 28 2016
t = (4/3)*cosh((1/3)*arccosh(19/8)) + 1/3. - Wolfdieter Lang, Aug 24 2022
t = 2 - Sum_{k>=0} binomial(4*k + 2, k)/((k + 1)*2^(4*k + 3)). - Antonio Graciá Llorente, Oct 28 2024

A085697 a(n) = T(n)^2, where T(n) = A000073(n) is the n-th tribonacci number.

Original entry on oeis.org

0, 0, 1, 1, 4, 16, 49, 169, 576, 1936, 6561, 22201, 75076, 254016, 859329, 2907025, 9834496, 33269824, 112550881, 380757169, 1288092100, 4357584144, 14741602225, 49870482489, 168710633536, 570743986576, 1930813074369, 6531893843049

Offset: 0

Emanuele Munarini, Jul 18 2003

In general, squaring the terms of a third-order linear recurrence with signature (x,y,z) will result in a sixth-order recurrence with signature (x^2 + y, x^2*y + z*x + y^2, x^3*z + 4*x*y*z - y^3 + 2*z^2, x^2*z^2 - x*y^2*z - z^2*y, z^2*y^2 - z^3*x, -z^4). - Gary Detlefs, Jan 10 2023

R. Schumacher, Explicit formulas for sums involving the squares of the first n Tribonacci numbers, Fib. Q., 58:3 (2020), 194-202.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,6,-1,0,-1).

Cf. A000073, A057597, A099463, A107239, A113300.

R:=PowerSeriesRing(Integers(), 40); [0,0] cat Coefficients(R!( x^2*(1-x-x^2-x^3)/((1-3*x-x^2-x^3)*(1+x+x^2-x^3)) )); // G. C. Greubel, Nov 20 2021

LinearRecurrence[{2,3,6,-1,0,-1},{0,0,1,1,4,16},30] (* Harvey P. Dale, Oct 26 2020 *)

t[0]:0$  t[1]:0$  t[2]:1$
t[n]:=t[n-1]+t[n-2]+t[n-3]$
makelist(t[n]^2,n,0,40); /* Emanuele Munarini, Mar 01 2011 */

@CachedFunction
def T(n): # A000073
    if (n<2): return 0
    elif (n==2): return 1
    else: return T(n-1) +T(n-2) +T(n-3)
def A085697(n): return T(n)^2
[A085697(n) for n in (0..40)] # G. C. Greubel, Nov 20 2021

G.f.: x^2*( 1-x-x^2-x^3 )/( (1-3*x-x^2-x^3)*(1+x+x^2-x^3) ).
a(n+6) = 2*a(n+5) + 3*a(n+4) + 6*a(n+3) - a(n+2) - a(n).
a(n) = (-A057597(n-2) + 3*A057597(n-1) + 6*A057597(n) + 5*A113300(n-1) - A099463(n-2))/11. - R. J. Mathar, Aug 19 2008

Offset corrected to match A000073 by N. J. A. Sloane, Sep 12 2020

Name corrected to match corrected offset by Michael A. Allen, Jun 10 2021

A073145 a(n) = -a(n-1) - a(n-2) + a(n-3), a(0)=3, a(1)=-1, a(2)=-1.

Original entry on oeis.org

3, -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

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jul 17 2002

Previous name was: Sum of the determinants of the principal minors of 2nd order of n-th power of Tribomatrix: first row (1, 1, 0); second row (1, 0, 1); third row (1, 0, 0).
a(n) is related to the generalized Lucas numbers S(n). For instance, 2S(n) = a(n)^2 - a(2n).
a(n) is also the reflected (A074058) sequence of the generalized tribonacci sequence (A001644).

			G.f. = 3 - x - x^2 + 5*x^3 - 5*x^4 - x^5 + 11*x^6 - 15*x^7 + 3*x^8 + 23*x^9 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Mario Catalani, Polymatrix and Generalized Polynacci Numbers, arXiv:math/0210201 [math.CO], 2002.
Curtis Cooper, S. Miller, Peter J. C. Moses, M. Sahin, and T. Thanatipanonda, On Identities of Ruggles, Horadam, Howard, and Young, Preprint 2016.
Kai Wang, Identities for generalized enneanacci numbers, Generalized Fibonacci Sequences (2020).
Index entries for linear recurrences with constant coefficients, signature (-1,-1,1).

Cf. A000073, A001644, A057597.

I:=[3,-1,-1]; [n le 3 select I[n] else -Self(n-1)-Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Aug 17 2013

A = Table[0, {3}, {3}]; A[[1, 1]] = 1; A[[1, 2]] = 1; A[[2, 1]] = 1; A[[2, 3]] = 1; A[[3, 1]] = 1; For[i = 1; t = IdentityMatrix[3], i < 50, i++, t = t.A; Print[t[[2, 2]]*t[[3, 3]] - t[[2, 3]]*t[[3, 2]] + t[[1, 1]]*t[[3, 3]] - t[[1, 3]]*t[[3, 1]] + t[[1, 1]]*t[[2, 2]] - t[[1, 2]]*t[[2, 1]]]]
LinearRecurrence[{-1, -1, 1}, {3, -1, -1}, 50] (* Vincenzo Librandi, Aug 17 2013 *)
nxt[{a_,b_,c_}]:={b,c,a-b-c}; NestList[nxt,{3,-1,-1},50][[;;,1]] (* Harvey P. Dale, Jun 16 2024 *)

{a(n) = if( n<0, polsym(1 + x+ x^2 - x^3, -n)[-n+1], polsym(1 - x - x^2 - x^3, n)[n+1])}; /* Michael Somos, Dec 17 2016 */

a(n)=([0,1,0; 0,0,1; 1,-1,-1]^n*[3;-1;-1])[1,1] \\ Charles R Greathouse IV, Feb 07 2017

((3+2*x+x^2)/(1+x+x^2-x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Apr 22 2019

a(n) = -a(n-1) - a(n-2) + a(n-3), a(0)=3, a(1)=-1, a(2)=-1.
O.g.f.: (3 + 2*x + x^2)/(1 + x + x^2 - x^3).
a(n) = -T(n)^2 + 2*T(n-1)^2 + 3*T(n-2)^2 - 2*T(n)*T(n-1) + 2*T(n)*T(n-2) + 4*T(n-1)*T(n-2), where T(n) are tribonacci numbers (A000073).
a(n) = 3*A057597(n+2) + 2*A057597(n+1) + A057597(n). - R. J. Mathar, Jun 06 2011
From Peter Bala, Jun 29 2015: (Start)
a(n) = alpha^n + beta^n + gamma^n, where alpha, beta and gamma are the roots of 1 - x - x^2 - x^3 = 0.
x^2*exp( Sum_{n >= 1} a(n)*x^n/n ) = x^2 - x^3 + 2*x*5 - 3*x^6 + x^7 + ... is the o.g.f. for A057597. (End)
a(n) = A001644(-n) for all n in Z. - Michael Somos, Dec 17 2016

Better name by Joerg Arndt, Aug 17 2013
More terms from Vincenzo Librandi, Aug 17 2013
Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A084610 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+x-x^2)^n.

Original entry on oeis.org

1, 1, 1, -1, 1, 2, -1, -2, 1, 1, 3, 0, -5, 0, 3, -1, 1, 4, 2, -8, -5, 8, 2, -4, 1, 1, 5, 5, -10, -15, 11, 15, -10, -5, 5, -1, 1, 6, 9, -10, -30, 6, 41, -6, -30, 10, 9, -6, 1, 1, 7, 14, -7, -49, -14, 77, 29, -77, -14, 49, -7, -14, 7, -1, 1, 8, 20, 0, -70, -56, 112, 120, -125, -120, 112, 56, -70, 0, 20, -8, 1

Offset: 0

Paul D. Hanna, Jun 01 2003

			Rows:
  1;
  1, 1, -1;
  1, 2, -1,  -2,   1;
  1, 3,  0,  -5,   0,   3, -1;
  1, 4,  2,  -8,  -5,   8,  2,  -4,   1;
  1, 5,  5, -10, -15,  11, 15, -10,  -5,   5, -1;
  1, 6,  9, -10, -30,   6, 41,  -6, -30,  10,  9, -6,   1;
  1, 7, 14,  -7, -49, -14, 77,  29, -77, -14, 49, -7, -14, 7, -1;

Paul D. Hanna, Rows n=0..34 of triangle, flattened
Yahia Djemmada, Abdelghani Mehdaoui, László Németh, and László Szalay, The Fibonacci-Fubini and Lucas-Fubini numbers, arXiv:2407.04409 [math.CO], 2024. See p. 13.

Cf. A002426, A057597, A084600 - A084609, A084611 - A084615.

A084610:= func< n,k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*(-1)^j: j in [0..k]]) >;
[A084610(n,k): k in [0..2*n], n in [0..13]]; // G. C. Greubel, Mar 26 2023

T[n_, k_]:= Sum[Binomial[n,k-j]*Binomial[k-j,j]*(-1)^j, {j,0,k}];
Table[T[n, k], {n,0,12}, {k,0,2*n}]//Flatten (* G. C. Greubel, Mar 26 2023 *)

for(n=0,12, for(k=0,2*n,t=polcoeff((1+x-x^2)^n,k,x); print1(t",")); print(" "))

def A084610(n,k): return sum(binomial(n,k-j)*binomial(k-j,j)*(-1)^j for j in range(k+1))
flatten([[A084610(n,k) for k in range(2*n+1)] for n in range(14)]) # G. C. Greubel, Mar 26 2023

G.f.: G(0)/2 , where G(k)= 1 + 1/( 1 - (1+x-x^2)*x^(2*k+1)/((1+x-x^2)*x^(2*k+1) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 06 2013
From G. C. Greubel, Mar 26 2023: (Start)
T(n, k) = Sum_{j=0..k} binomial(n, k-j)*binomial(k-j, j)*(-1)^j.
T(n, 2*n) = (-1)^n.
T(n, 2*n-1) = (-1)^(n-1)*n, n >= 1.
Sum_{k=0..2*n} T(n, k) = 1.
Sum_{k=0..2*n} (-1)^k*T(n, k) = (-1)^n.
Sum_{k=0..n} T(n-k, k) = floor((n+2)/2).
Sum_{k=0..n} (-1)^k*T(n-k, k) = (-1)^n*A057597(n+2). (End)

A104769 Expansion of g.f. -x/(1+x-x^3).

Original entry on oeis.org

0, -1, 1, -1, 0, 1, -2, 2, -1, -1, 3, -4, 3, 0, -4, 7, -7, 3, 4, -11, 14, -10, -1, 15, -25, 24, -9, -16, 40, -49, 33, 7, -56, 89, -82, 26, 63, -145, 171, -108, -37, 208, -316, 279, -71, -245, 524, -595, 350, 174, -769, 1119, -945, 176, 943, -1888, 2064, -1121, -767, 2831, -3952

Offset: 0

Creighton Dement, Mar 24 2005

Generating floretion is "jesright".

Pisano period lengths: 1, 7, 13, 14, 24, 91, 48, 28, 39, 168, 120, 182, 183, 336, 312, 56, 288, 273, 180, 168,.. (which differs from A104217 for example at index 23). - R. J. Mathar, Aug 10 2012

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (-1,0,1).

Apart from signs, essentially the same as A050935 and A078013.
Cf. A247917 (negative).
Cf. A000931, A057597.

LinearRecurrence[{-1, 0, 1}, {0, -1, 1}, 61] (* or *)
CoefficientList[Series[-x/(1 + x - x^3), {x, 0, 60}], x] (* Michael De Vlieger, Jul 02 2021 *)

a(n)=([0,1,0;0,0,1;1,0,-1]^n*[0;-1;1])[1,1] \\ Charles R Greathouse IV, Jun 11 2015

a(n) = -A247917(n-1).
Recurrence: a(n+3) = a(n) - a(n+2); a(0) = 0, a(1) = -1, a(2) = 1.
a(n+1) - a(n) = ((-1)^(n+1))*a(n+5).
a(n) = ((-1)^n)*A050935(n+1) = ((-1)^n)*A078013(n+2).
a(n) = A104771(n) - A104770(n).

Edited by Ralf Stephan, Apr 05 2009

A078046 Expansion of (1-x)/(1 + x + x^2 - x^3).

Original entry on oeis.org

1, -2, 1, 2, -5, 4, 3, -12, 13, 2, -27, 38, -9, -56, 103, -56, -103, 262, -215, -150, 627, -692, -85, 1404, -2011, 522, 2893, -5426, 3055, 5264, -13745, 11536, 7473, -32754, 36817, 3410, -72981, 106388, -29997, -149372, 285757, -166382, -268747, 720886, -618521, -371112, 1710519, -1957928

Offset: 0

N. J. A. Sloane, Nov 17 2002

The root of the denominator [1 + x + x^2 - x^3] is the tribonacci constant.

This is the negative of the tribonacci numbers, signature (0, 1, 0), in reverse order, starting from A001590(-1), going backwards A001590(-2), A001590(-3), ... - Peter M. Chema, Dec 31 2016

			G.f. = 1 - 2*x + x^2 + 2*x^3 - 5*x^4 + 4*x^5 + 3*x^6 - 12*x^7 + 13*x^8 + ...

Benjamin Braun, W. K. Hough, Matching and Independence Complexes Related to Small Grids, arXiv preprint arXiv:1606.01204 [math.CO], 2016.
Wesley K. Hough, On Independence, Matching, and Homomorphism Complexes, (2017), Theses and Dissertations--Mathematics, 42.
Index entries for linear recurrences with constant coefficients, signature (-1,-1,1).

First differences of A057597.

Cf. A001590.

a[ n_] := If[ n >= 0, SeriesCoefficient[ (1 - x) / (1 + x + x^2 - x^3), {x, 0, n}], SeriesCoefficient [ -x^2 (1 - x) / (1 - x - x^2 - x^3),{x, 0, -n}]]; (* Michael Somos, Jun 01 2014 *)

Vec((1-x)/(1+x+x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

{a(n) = if( n>=0, polcoeff( (1 - x) / (1 + x + x^2 - x^3) + x * O(x^n), n), polcoeff( -x^2 * (1 - x) / (1 - x - x^2 - x^3) + x * O(x^-n), -n))}; /* Michael Somos, Jun 01 2014 */

G.f.: (1-x)/(1+x+x^2-x^3).
Recurrence: a(n) = a(n-3) - a(n-2) - a(n-1) for n > 2.
a(-1 - n) = - A001590(n). - Michael Somos, Jun 01 2014

A077908 Expansion of (1-x)^(-1)/(1+x+x^2-x^3).

Original entry on oeis.org

1, 0, 0, 2, -1, 0, 4, -4, 1, 8, -12, 6, 15, -32, 24, 24, -79, 80, 24, -182, 239, -32, -388, 660, -303, -744, 1708, -1266, -1185, 4160, -4240, -1104, 9505, -12640, 2032, 20114, -34785, 16704, 38196, -89684, 68193, 59688, -217564, 226070, 51183, -494816, 669704, -123704, -1040815, 1834224

Offset: 0

N. J. A. Sloane, Nov 17 2002

			G.f. = 1 + 2*x^3 - x^4 + 4*x^6 - 4*x^7 + x^8 + 8*x^9 - 12*x^10 + 6*x^11 + ...

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

Cf. A008937.

a:= n-> -(<<0|0|1|0>, <1|0|-1|0>, <0|1|-1|0>, <0|0|-1|1>>^(n+3))[4, 1]:
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 24 2008

a[n_] := -MatrixPower[{{0, 0, 1, 0}, {1, 0, -1, 0}, {0, 1, -1, 0}, {0, 0, -1, 1}}, n+3][[4, 1]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 14 2016, after Alois P. Heinz *)
LinearRecurrence[{0,0,2,-1},{1,0,0,2},50] (* Harvey P. Dale, Oct 25 2020 *)

Vec((1-x)^(-1)/(1+x+x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

{a(n) = if( n<0, polcoeff( - x^4 / (1 - 2*x + x^4) + x * O(x^-n), -n), polcoeff( 1 / (1 - 2*x^3 + x^4) + x * O(x^n), n))}; /* Michael Somos, Aug 19 2014 */

a(n) = -1 * term (4,1) in the 4x4 matrix [0,0,1,0; 1,0,-1,0; 0,1,-1,0; 0,0,-1,1]^(n+3) - Alois P. Heinz, Jul 24 2008
a(n) = -A008937(-n-3). - Alois P. Heinz, Jul 24 2008
G.f.: 1 / (1 - 2*x^3 + x^4). - Michael Somos, Aug 19 2014
a(n) = -a(n-1) - a(n-2) + a(n-3) + 1 = 2*a(n-3) - a(n-4) for all n in Z. - Michael Somos, Aug 19 2014
a(n) - a(n-1) = A057597(n+2). (first differences). - R. J. Mathar, Oct 16 2017

cp's OEIS Frontend

A319200 a(n) = -(A(n) - A(n-1)) where A(n) = A057597(n+1), for n >= 0.

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A075297 Duplicate of A057597.

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A000073 Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) for n >= 3 with a(0) = a(1) = 0 and a(2) = 1.

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A058265 Decimal expansion of the tribonacci constant t, the real root of x^3 - x^2 - x - 1.

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A085697 a(n) = T(n)^2, where T(n) = A000073(n) is the n-th tribonacci number.

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A073145 a(n) = -a(n-1) - a(n-2) + a(n-3), a(0)=3, a(1)=-1, a(2)=-1.

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