A192918 -id:A192918 - cp's OEIS frontend

A317202 Decimal expansion of 3 + (t^2+t^4)/2, where t = 0.543689... (A192918) is the real root of x^3 + x^2 + x = 1.

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

Offset: 1

N. J. A. Sloane, Aug 05 2018

The minimal polynomial of this constant is 2*x^3 - 12*x^2 + 22*x - 13, and it is its unique real root. - Amiram Eldar, May 30 2023

			3.191487883953118747061354268227517293474691...

Bo Tan and Zhi-Ying Wen, Some properties of the Tribonacci sequence, European Journal of Combinatorics, 28 (2007) 1703-1719. See Theorem 4.5.

Cf. A316711, A192918.

RealDigits[x /. FindRoot[2*x^3 - 12*x^2 + 22*x - 13, {x, 3}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, May 30 2023 *)

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

A057597 a(n) = -a(n-1) - a(n-2) + a(n-3), a(0)=0, a(1)=0, a(2)=1.

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, -963, -441, 2452, -2974, 81, 5345, -8400, 3136, 10609, -22145, 14672, 18082, -54899, 51489, 21492, -127880, 157877, -8505, -277252, 443634, -174887, -545999, 1164520, -793408, -917111

Offset: 0

N. J. A. Sloane, Oct 06 2000

Reflected (A074058) tribonacci numbers A000073: A000073(n) = a(1-n).
There is an alternative way to produce this sequence, from A000073, which is 0,0,1,1,2,4,7,13,24,44,... Call this {b(n)}. Taking x1 = (b(2))^2 - b(1)*b(3) = 0; x2 = (b(3))^2 - b(2)*b(4) = 1; x3 = (b(4))^2 - b(3)*b(5) = -1; x4 = 0, x5 = 2, we generate (0),0,1,-1,0,2,-3,1. - John McNamara, Jan 02 2004
Pisano period lengths: 1, 4, 13, 8, 31, 52, 48, 16, 39, 124, 110, 104, 168, 48, 403, 32, 96, 156, 360, 248, ... - R. J. Mathar, Aug 10 2012
The negative powers of the tribonacci constant t = A058265 are t^(-n) = a(n+1)*t^2 + b(n)*t + a(n+2)*1, for n >= 0, with b(n) = A319200(n) = -(a(n+1) - a(n)), for n >= 0. 1/t =  t^2 - t - 1 = A192918. See the example in A319200 for the first powers. - Wolfdieter Lang, Oct 23 2018

Petho Attila, Posting to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Oct 06 2000.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Milan Janjic, Recurrence Relations and Determinants, arXiv preprint arXiv:1112.2466 [math.CO], 2011.
Milan Janjic, Determinants and Recurrence Sequences, Journal of Integer Sequences, 15 (2012), Article 12.3.5. - From _N. J. A. Sloane_, Sep 16 2012.
Ronald C. King, Generating functions for some series of characters of classical Lie groups, arXiv:2303.00576 [math.CO], 2023, p. 9.
A. G. Shannon, H. M. Srivastava, and József Sàndor, Towards a new generalized Simson's identity, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 479-490. See p. 482.
Index entries for linear recurrences with constant coefficients, signature (-1,-1,1).

Cf. A000073, A058265, A319200. First differences of A077908.

a:=[0,0,1];;  for n in [4..55] do a[n]:=-a[n-1]-a[n-2]+a[n-3]; od; a; # Muniru A Asiru, Oct 23 2018

a057597 n = a057597_list !! n
a057597_list = 0 : 0 : 1 : zipWith3 (\x y z -> - x - y + z)
               (drop 2 a057597_list) (tail a057597_list) a057597_list
-- Reinhard Zumkeller, Oct 07 2012

seq(coeff(series(x^2/(1+x+x^2-x^3),x,n+1), x, n), n = 0 .. 50); # Muniru A Asiru, Oct 23 2018

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

{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 */

G.f.: x^2/(1+x+x^2-x^3).
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
G.f. -x*T(1/x), where T is the g.f. of A000073. - Wolfdieter Lang, Oct 26 2018

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

A066447 Number of basis partitions (or basic partitions) of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 6, 8, 10, 13, 16, 20, 26, 32, 40, 50, 61, 74, 90, 108, 130, 156, 186, 222, 264, 313, 370, 436, 512, 600, 702, 818, 952, 1106, 1282, 1484, 1715, 1978, 2278, 2620, 3008, 3448, 3948, 4512, 5150, 5872, 6684, 7600, 8632, 9791, 11094, 12558, 14198, 16036, 18096, 20398

Offset: 0

Herbert S. Wilf, Dec 29 2001

The k-th successive rank of a partition pi = (pi_1, pi_2, ..., pi_s) of the integer n is r_k = pi_k - pi'_k, where pi' denotes the conjugate partition. A partition pi is basic if the number of dots in its Ferrers diagram is the least among all the Ferrers diagrams of partitions with the same rank vector.

The g.f. Sum_{n >= 0} x^(n^2) * Product_{k = 1..n} (1 + x^k)/(1 - x^k) = Sum_{n >= 0} x^(n^2) * Product_{k = 1..n} (1 + x^k)/(1 + x^k - 2*x^k) == Sum_{n >= 0} x^(n^2) (mod 2). It follows that a(n) is odd iff n is a square (Nolan et al., equation 6, p. 282). - Peter Bala, Jan 08 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
George E. Andrews, Partition Identities for Two-Color Partitions, Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2021, Special Commemorative volume in honour of Srinivasa Ramanujan, 2021, 44, pp.74-80. hal-03498190. See p. 79.
J. M. Nolan, C. D. Savage and H. S. Wilf, Basis partitions, Discrete Math. 179 (1998), 277-283.

Row sums of A066448.
Cf. A001130, A001935, A003114, A306734, A333374.
Cf. A192918, A376841.

b := proc(n,d); option remember; if n=0 and d=0 then RETURN(1) elif n<=0 or d<=0 then RETURN(0) else RETURN(b(n-d,d)+b(n-2*d+1,d-1)+b(n-3*d+1,d-1)) fi: end: A066447 := n->add(b(n,d),d=0..n);

nmax = 60; CoefficientList[Series[Sum[x^(n^2)*Product[(1 + x^k)/(1 - x^k), {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 17 2020 *)
nmax = 60; p = 1; s = 1; Do[p = Normal[Series[p*(1 + x^k)/(1 - x^k)*x^(2*k - 1), {x, 0, nmax}]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Mar 17 2020 *)

N=66; x='x+O('x^N); s=sum(n=0,N,x^(n^2)*prod(k=1,n,(1+x^k)/(1-x^k))); Vec(s) /* Joerg Arndt, Apr 07 2011 */

G.f.: Sum_{n >= 0} x^(n^2) * Product_{k = 1..n} (1 + x^k)/(1 - x^k) [Given in Nolan et al. reference]. - Joerg Arndt, Apr 07 2011
Limit_{n->infinity} a(n) / A333374(n) = A058265 = (1 + (19+3*sqrt(33))^(1/3) + (19-3*sqrt(33))^(1/3))/3 = 1.839286755214... - Vaclav Kotesovec, Mar 17 2020
a(n) ~ c * d^sqrt(n) / n^(3/4), where d = A376841 = 7.1578741786143524880205... = exp(2*sqrt(log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))) and c = 0.193340468476900308848561788251945... = (log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))^(1/4) * sqrt(1/24 + cosh(arccosh(53*sqrt(11/2)/64)/3) / (3*sqrt(22))) / sqrt(Pi), where r = A192918 = 0.54368901269207636157... is the real root of the equation r^2*(1+r) = 1-r. - Vaclav Kotesovec, Mar 19 2020, updated Oct 10 2024

A286998 0-limiting word of the morphism 0->10, 1->20, 2->0.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0

Offset: 1

Clark Kimberling, May 22 2017

Starting with 0, the first 5 iterations of the morphism yield words shown here:
  1st: 10
  2nd: 2010
  3rd: 0102010
  4th: 1020100102010
  5th: 201001020101020100102010
The 2-limiting word is the limit of the words for which the number of iterations is congruent to 2 mod 3.
Let U, V, W be the limits of u(n)/n, v(n)/n, w(n)/n, respectively. Then 1/U + 1/V + 1/W = 1, where
  U = 1.8392867552141611325518525646532866..., (A058265)
  V = U^2 = 3.3829757679062374941227085364..., (A276800)
  W = U^3 = 6.2222625231203986266745611011.... (A276801)
If n >=2, then u(n) - u(n-1) is in {1,2}, v(n) - v(n-1) is in {2,3,4}, and w(n) - w(n-1) is in {4,6,7}.
From Jiri Hladky, Aug 29 2021: (Start)
This is also Arnoux-Rauzy word sigma_0 x sigma_1 x sigma_2, where sigmas are defined as:
  sigma_0 : 0 -> 0,  1 -> 10, 2 -> 20;
  sigma_1 : 0 -> 01, 1 -> 1,  2 -> 21;
  sigma_2 : 0 -> 02, 1 -> 12, 2 -> 2.
Fixed point of the morphism 0->0102010, 1->102010, 2->2010, starting from a(1)=0. This definition has the benefit that EACH iteration yields the prefix of the limiting word.
Frequency of letters:
  0: 1/t   ~ 54.368% (A192918)
  1: 1/t^2 ~ 29.559%
  2: 1/t^3 ~ 16.071%
where t is tribonacci constant A058265.
Equals A347290 with a re-mapping of values 1->2, 2->1.
(End)

			3rd iterate: 0102010
6th iterate: 01020101020100102010201001020101020100102010

Jiri Hladky, Table of n, a(n) for n = 1..20000 (terms 1..10000 from Clark Kimberling).
L. Balková, M. Bucci, A. De Luca, J. Hladký, and S. Puzynina: Aperiodic Pseudorandom Number Generators Based on Infinite Words, Theoret. Comput. Sci. 647 (2016), 85-100.
Julien Cassaigne, Sebastien Ferenczi, and Luca Q. Zamboni, Imbalances in Arnoux-Rauzy sequences, Annales de l'institut Fourier, 50 (2000), 1265-1276.
D. Damanik and L. Q. Zamboni, Arnoux-Rauzy subshifts: linear recurrence, powers and palindromes, arXiv:math/0208137 [math.CO], 2002.
J. Patera, GENERATING THE FIBONACCI CHAIN IN O (log n) SPACE AND O (n) TIME (2003)
Gérard Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110.2 (1982): 147-178.
Wikipedia, Rauzy fractal
Index entries for sequences that are fixed points of mappings

Cf. A080843, A286999, A287000, A287001, A287112, A287174, A347290 (values 0,2,1).

s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 0}, 2 -> 0}] &, {0}, 9] (* A286998 *)
Flatten[Position[s, 0]] (* A286999 *)
Flatten[Position[s, 1]] (* A287000 *)
Flatten[Position[s, 2]] (* A287001 *)
(* Using the 0->0102010, 1->102010, 2->2010 rule: *)
Nest[ Flatten[# /. {0 -> {0, 1, 0, 2, 0, 1, 0}, 1 -> {1, 0, 2, 0, 1, 0}, 2 -> {2, 0, 1, 0}}] &, {0}, 3]

A333374 G.f.: Sum_{k>=1} (x^(k(k+1)) Product_{j=1..k} (1 + x^j)/(1 - x^j)).

Original entry on oeis.org

1, 0, 1, 2, 2, 2, 3, 4, 6, 8, 10, 12, 15, 18, 22, 28, 34, 42, 52, 62, 75, 90, 106, 126, 150, 176, 208, 246, 288, 338, 397, 462, 538, 626, 724, 838, 968, 1114, 1282, 1474, 1690, 1936, 2217, 2532, 2890, 3296, 3750, 4264, 4844, 5492, 6222, 7042, 7958, 8986, 10138

Offset: 0

Vaclav Kotesovec, Mar 17 2020

nonn

The g.f. Sum_{k >= 1} x^(k*(k+1)) * Product_{j = 1..k} (1 + x^j)/(1 - x^j) = Sum_{k >= 1} x^(k*(k+1)) * Product_{j = 1..k} (1 + x^j)/(1 - x^j + 2*x^j) == Sum_{k >= 1} x^(k*(k+1)) (mod 2). It follows that a(n) is odd iff n = k*(k + 1) for some nonnegative integer k. - Peter Bala, Jan 04 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A001935, A001936, A003106, A053261, A066447, A003114, A306734, A333179.

Cf. A192918, A376841.

nmax = 100; CoefficientList[Series[Sum[x^(n*(n+1))*Product[(1+x^k)/(1-x^k), {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Normal[Series[p*(1 + x^k)/(1 - x^k)*x^(2*k), {x, 0, nmax}]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Limit_{n->infinity} A066447(n) / a(n) = A058265 = (1 + (19+3*sqrt(33))^(1/3) + (19-3*sqrt(33))^(1/3))/3 = 1.839286755214... (the tribonacci constant).
Compare with: A306734(n) / A333179(n) -> A060006 (the plastic constant) and A003114(n) / A003106(n) -> A001622 (golden ratio).
a(n) ~ c * d^sqrt(n) / n^(3/4), where d = A376841 = 7.1578741786143524880205... = exp(2*sqrt(log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))) and c = 0.10511708841962944170826735560432... = (log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))^(1/4) * sqrt(1/24 - sinh(arcsinh(sqrt(11)/4)/3) / (12*sqrt(11))) / sqrt(Pi), where r = A192918 = 0.54368901269207636157... is the real root of the equation r^2*(1+r) = 1-r. - Vaclav Kotesovec, Mar 17 2020, updated Oct 10 2024

A086253 Decimal expansion of Feller's alpha coin-tossing constant.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jul 13 2003

			1.0873780253841527231417119436....

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.11 Feller's coin tossing constants, p. 339.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Run
Wikipedia, Feller's coin-tossing constants

Cf. A086254, A058265, A192918.

evalf[120](solve(x^3+2*x^2+4*x-8=0,x)[1]); # Muniru A Asiru, Nov 25 2018

alpha = Root[1-x+(x/2)^4, x, 1]; RealDigits[alpha, 10, 102] // First (* Jean-François Alcover, Jun 03 2014 *)

solve(x=1, 3/2, 1-x+(x/2)^4) \\ Michel Marcus, Oct 14 2018

Equals -2/3 - 4/(3*(17 + 3*sqrt(33))^(1/3)) +  2*(17 + 3*sqrt(33))^(1/3)/3. - Vaclav Kotesovec, Oct 14 2018
Positive real root of x^3 + 2*x^2 + 4*x - 8. - Peter Luschny, Oct 14 2018
Equals 2/A058265 = 2*A192918. - Jon Maiga, Nov 24 2018

A199220 Triangle read by rows: T(n,k) = (n-1-k)*abs(s(n,n+1-k)), where s(n,k) are the signed Stirling numbers of the first kind and 1 <= k <= n.

Original entry on oeis.org

-1, 0, -1, 1, 0, -2, 2, 6, 0, -6, 3, 20, 35, 0, -24, 4, 45, 170, 225, 0, -120, 5, 84, 525, 1470, 1624, 0, -720, 6, 140, 1288, 5880, 13538, 13132, 0, -5040, 7, 216, 2730, 18144, 67347, 134568, 118124, 0, -40320, 8, 315, 5220, 47250, 253092, 807975, 1447360, 1172700, 0, -362880, 9, 440, 9240, 108900, 788865, 3608220, 10250790, 16819000, 12753576, 0, -3628800

Offset: 1

Frank M Jackson, Nov 04 2011

Use the T(n,k) as coefficients to generate a polynomial of degree n-1 in d as Sum_{k=1..n} T(n,k)d^(k-1) and let g(n) be the greatest root of this polynomial. Then a polygon of n sides that form a harmonic progression in the ratio 1 : 1/(1+d) : 1/(1+2d) : ... : 1/(1+(n-1)d) can only exist if the common difference d of the denominators is limited to the range f(n) < d < g(n). The lower limit f(n) is the greatest root of another group of polynomials defined by coefficients in the triangle A199221.

			Triangle starts:
  -1;
   0, -1;
   1,  0,  -2;
   2,  6,   0,  -6;
   3, 20,  35,   0, -24;
   4, 45, 170, 225,   0, -120;

Cf. A094638, A192918, A199221.

Flatten[Table[(n-1-k)Abs[StirlingS1[n,n+1-k]],{n,1,20},{k,1,n}]]

T(n,k) = (n-1-k)*abs(stirling(n,n+1-k,1)); \\ Michel Marcus, Sep 30 2018

The triangle of coefficients can be generated by expanding the equation (Sum_{k=1..n} 1/(1+(k-1)*d)) - 2 = 0 into a polynomial of degree n-1 in d.

A199221 Triangle read by rows: T(n,k) = (n+1-k)|s(n,n+1-k)| - 2|s(n-1,n-k)|, where s(n,k) are the signed Stirling numbers of the first kind and 1 <= k <= n.

Original entry on oeis.org

-1, 0, 1, 1, 4, 2, 2, 12, 18, 6, 3, 28, 83, 88, 24, 4, 55, 270, 575, 500, 120, 5, 96, 705, 2490, 4324, 3288, 720, 6, 154, 1582, 8330, 23828, 35868, 24696, 5040, 7, 232, 3178, 23296, 98707, 242872, 328236, 209088, 40320, 8, 333, 5868, 57078, 334740, 1212057, 2658472, 3298932, 1972512, 362880, 9, 460, 10140, 126300, 977865, 4873680, 15637290, 31292600, 36207576, 20531520, 3628800

Offset: 1

Frank M Jackson, Nov 04 2011

Use the T(n,k) as coefficients to generate a polynomial of degree n-1 in d as Sum_{k=1..n} T(n,k)d^(k-1) and let f(n) be the greatest root of this polynomial. Then a polygon of n sides that form a harmonic progression in the ratio 1 : 1/(1+d) : 1/(1+2d) : ... : 1/(1+(n-1)d) can only exist if the common difference d of the denominators is limited to the range f(n) < d < g(n). The higher limit g(n) is the greatest root of another group of polynomials defined by coefficients in the triangle A199220.

			Triangle starts:
  -1;
   0,  1;
   1,  4,   2;
   2, 12,  18,   6;
   3, 28,  83,  88,  24;
   4, 55, 270, 575, 500, 120;

Cf. A094638, A192918, A199220.

Flatten[Table[(n+1-k)Abs[StirlingS1[n,n+1-k]]-2Abs[StirlingS1[n-1,n-k]],{n,1,20},{k,1,n}]]

T(n,k) = (n+1-k)*abs(stirling(n,n+1-k,1)) - 2*abs(stirling(n-1,n-k,1));
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Sep 30 2018

The triangle of coefficients can be generated by expanding the equation (Sum_{k=1..n} 1/(1+(k-1)d)) - 2/(1+(n-1)d) = 0 into a polynomial of degree n-1 in d.

cp's OEIS Frontend

A317202 Decimal expansion of 3 + (t^2+t^4)/2, where t = 0.543689... (A192918) is the real root of x^3 + x^2 + x = 1.

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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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A057597 a(n) = -a(n-1) - a(n-2) + a(n-3), a(0)=0, a(1)=0, a(2)=1.

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A066447 Number of basis partitions (or basic partitions) of n.

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A286998 0-limiting word of the morphism 0->10, 1->20, 2->0.

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A333374 G.f.: Sum_{k>=1} (x^(k(k+1)) Product_{j=1..k} (1 + x^j)/(1 - x^j)).

A199221 Triangle read by rows: T(n,k) = (n+1-k)|s(n,n+1-k)| - 2|s(n-1,n-k)|, where s(n,k) are the signed Stirling numbers of the first kind and 1 <= k <= n.