cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Examples

			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 + ...

References

  • M. Agronomof, Sur une suite récurrente, Mathesis (Series 4), Vol. 4 (1914), pp. 125-126.
  • A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 47, ex. 4.
  • S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.
  • Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
  • J. Riordan, An Introduction to Combinatorial Analysis, Princeton University Press, Princeton, NJ, 1978.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

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).

Programs

  • GAP
    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
  • Haskell
    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
  • Magma
    [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
  • Maple
    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
  • Mathematica
    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 *)
  • Maxima
    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 */
  • PARI
    {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 */
  • PARI
    my(x='x+O('x^99)); concat([0, 0], Vec(x^2/(1-x-x^2-x^3))) \\ Altug Alkan, Apr 04 2016
  • PARI
    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
  • Python
    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

Formula

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

Extensions

Minor edits by M. F. Hasler, Apr 18 2018
Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A056816 Primes in the tribonacci sequence A000213.

Original entry on oeis.org

3, 5, 17, 31, 193, 653, 1201, 46499, 532159, 3311233, 128199521, 20512526282340991, 6036724301884645488192191, 2170304813579195568101904406358277391153
Offset: 1

Views

Author

Harvey P. Dale, Sep 01 2000

Keywords

Comments

Primes within A000213.

Crossrefs

Cf. A092836.

Formula

a(n) = A000213(A157611(n)). - R. J. Mathar, Dec 14 2011

A092835 Indices of prime tribonacci numbers, minus 1.

Original entry on oeis.org

3, 5, 6, 10, 86, 97, 214, 801, 4201, 18698, 96878
Offset: 1

Views

Author

Eric W. Weisstein, Mar 06 2004

Keywords

Comments

The prime for n=96878 was found by Kamil Duszenko. - T. D. Noe, Mar 08 2005
a(12) > 6*10^5. - Robert Price, Nov 21 2013

Links

Crossrefs

Cf. A000073, A092836.

Formula

A000073(a(n)+1) = A092836(n). - Ray Chandler and Jonathan Vos Post, Dec 26 2004

Extensions

2 more terms from T. D. Noe, Mar 08 2005

A303263 Indices of primes in tribonacci sequence A000073.

Original entry on oeis.org

4, 6, 7, 11, 87, 98, 215, 802, 4202, 18699, 96879
Offset: 1

Views

Author

M. F. Hasler, Apr 18 2018

Keywords

Comments

T = A000073 is defined by T(n+1) = T(n) + T(n-1) + T(n-2), T(2) = 1, T(1) = T(0) = 0.
The largest terms correspond to unproven probable primes T(a(n)).

Crossrefs

Cf. A000073 (= T), A092836 (= T(a(n))), A092835 (= a(n) - 1).
Cf. A001605 (indices of primes in Fibonacci numbers A000045).

Programs

  • Mathematica
    -1 + Position[LinearRecurrence[{1, 1, 1}, {0, 0, 1}, 10^4], ?PrimeQ][[All, 1]] (* _Michael De Vlieger, Apr 21 2018 *)
  • PARI
    a(n,N=4,S=vector(N,i,i>N-2))={for(i=N,oo,ispseudoprime(S[i%N+1]=2*S[(i-1)%N+1]-S[i%N+1])&&!n--&&return(i))}

Formula

a(n) = A092835(n) + 1 = index of A092836(n) in A000073.

A105758 Indices of prime hexanacci (or Fibonacci 6-step) numbers A001592 (using offset -4).

Original entry on oeis.org

3, 36, 37, 92, 660, 6091, 8415, 11467, 13686, 38831, 49828, 97148
Offset: 1

Views

Author

T. D. Noe, Apr 22 2005

Keywords

Comments

No other n < 30000.
This sequence uses the convention of the Noe and Post reference. Their indexing scheme differs by 4 from the indices in A001592. Sequence A249635 lists the indices of the same primes (A105759) using the indexing scheme as defined in A001592. - Robert Price, Nov 02 2014 [Edited by M. F. Hasler, Apr 22 2018]
a(13) > 3*10^5. - Robert Price, Nov 02 2014

Links

Crossrefs

Cf. A105759 (prime Fibonacci 6-step numbers), A249635 (= a(n) + 4), A001592.
Cf. A000045, A000073, A000078 (and A001631), A001591, A122189 (or A066178), A079262, A104144, A122265, A168082, A168083 (Fibonacci, tribonacci, tetranacci numbers and other generalizations).
Cf. A005478, A092836, A104535, A105757, A105761, ... (primes in these sequence).
Cf. A001605, A303263, A303264 (and A104534 and A247027), A248757 (and A105756), ... (indices of primes in A000045, A000073, A000078, ...).

Programs

  • Mathematica
    a={1, 0, 0, 0, 0, 0}; lst={}; Do[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s; If[PrimeQ[s], AppendTo[lst, n]], {n, 30000}]; lst

Formula

a(n) = A249635(n) - 4. A105759(n) = A001592(A249635(n)) = A001592(a(n) + 4). - M. F. Hasler, Apr 22 2018

Extensions

a(10)-a(12) from Robert Price, Nov 02 2014
Edited by M. F. Hasler, Apr 22 2018

A100550 a(n) = a(n-1) + 2*a(n-2) + 3*a(n-3), for n>3, otherwise a(n) = n.

Original entry on oeis.org

0, 1, 2, 4, 11, 25, 59, 142, 335, 796, 1892, 4489, 10661, 25315, 60104, 142717, 338870, 804616, 1910507, 4536349, 10771211, 25575430, 60726899, 144191392, 342371480, 812934961, 1930252097, 4583236459, 10882545536, 25839774745, 61354575194
Offset: 0

Views

Author

gamo (gamo(AT)telecable.es), Nov 27 2004

Keywords

Comments

A recursive and iterative algorithm for the computation of a(n) appear as Exercise 1.11 in the book Structure and Interpretation of Computer Programs. - Bas Kok (no(AT)spam.com), Jan 31 2008

References

  • Harold Abelson and Gerald Jay Sussman with Julie Sussman, Structure and Interpretation of Computer Programs, MIT Press, 1996.

Links

Crossrefs

Cf. A092836, A092835, A100477, A101822.

Programs

  • Magma
    [n le 3 select n-1 else Self(n-1) +2*Self(n-2) +3*Self(n-3): n in [1..41]]; // G. C. Greubel, Mar 27 2023
  • Mathematica
    LinearRecurrence[{1,2,3},{0,1,2},40] (* Harvey P. Dale, Mar 19 2023 *)
  • Perl
    perl -e '@a=(0,1,2);for(3..30){$a[$]=$a[$-1]+2*$a[$-2]+3*$a[$-3];} print "@a ";'
  • SageMath
    @CachedFunction
def a(n): # a = A100550
    if (n<3): return n
    else: return a(n-1) + 2*a(n-2) + 3*a(n-3)
[a(n) for n in range(41)] # G. C. Greubel, Mar 27 2023

Formula

From R. J. Mathar, Aug 22 2008: (Start)
O.g.f.: x*(1+x)/(1-x-2*x^2-3*x^3).
a(n) = A101822(n-1) + A101822(n-2). (End)

A303264 Indices of primes in tetranacci sequence A000078.

Original entry on oeis.org

5, 9, 13, 14, 38, 58, 403, 2709, 8419, 14098, 31563, 50698, 53194, 155184
Offset: 1

Views

Author

M. F. Hasler, Apr 18 2018

Keywords

Comments

T = A000078 is defined by T(n) = Sum_{k=1..4} T(n-k), T(3) = 1, T(n) = 0 for n < 3.
The largest terms correspond to unproven probable primes T(a(n)).

Crossrefs

Cf. A000045, A000073, A000078, A001591, A001592, A122189 (or A066178), ... (Fibonacci, tribonacci, tetranacci numbers).
Cf. A005478, A092836, A104535, A105757, A105759, A105761, ... (primes in Fibonacci numbers and above generalizations).
Cf. A001605, A303263, A303264, A248757, A249635, ... (indices of primes in A000045, A000073, A000078, ...).
Cf. A247027: Indices of primes in the tetranacci sequence A001631 (starting 0, 0, 1, 0...), A104534 (a variant: a(n) - 2), A105756 (= A248757 - 3), A105758 (= A249635 - 4).

Programs

  • PARI
    a(n,N=5,S=vector(N,i,i>N-2))={for(i=N,oo,ispseudoprime(S[i%N+1]=2*S[(i-1)%N+1]-S[i%N+1])&&!n--&&return(i))}

Formula

a(n) = A104534(n) + 2.

A131354 Number of primes in the open interval between successive tribonacci numbers.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 3, 5, 8, 12, 23, 38, 61, 109, 179, 312, 537, 920, 1598, 2779, 4835, 8461, 14784, 25984, 45696, 80505, 142165, 251300, 444930, 788828, 1400756, 2489594, 4430712, 7892037, 14073786, 25118167, 44869652, 80223172, 143535369, 257014148, 460524864, 825732764
Offset: 0

Views

Author

Jonathan Vos Post, Oct 21 2007

Keywords

Comments

This is to tribonacci numbers A000073 as A052011 is to Fibonacci numbers and as A052012 is to Lucas numbers A000204. It is mere coincidence that all values until a(12) = 38 are themselves Fibonacci. The formula plus the known asymptotic prime distribution gives the asymptotic approximation of this sequence, which is the same even if we use one of the alternative definitions of tribonacci with different initial values.

Examples

			Between Trib(8)=24 and Trib(9)=44 we find the following primes: 29, 31, 37, 41, 43, hence a(8)=5.

Crossrefs

Cf. A000073, A000720, A092836, A052011, A052012, A056816.

Programs

  • Maple
    A131354 := proc(n)
    a := 0 ;
    for k from 1+A000073(n)  to A000073(n+1)-1 do
        if isprime(k) then
            a := a+1 ;
        end if;
    end do;
    a ;
end proc: # R. J. Mathar, Dec 14 2011
  • Mathematica
    trib[n_] := SeriesCoefficient[x^2/(1 - x - x^2 - x^3), {x, 0, n}];
a[n_] := PrimePi[trib[n + 1] - 1] - PrimePi[trib[n]];
a /@ Range[0, 42] (* Jean-François Alcover, Apr 10 2020 *)
  • PARI
    \\ here b(n) is A000073(n).
b(n)={polcoef(x^2/(1-x-x^2-x^3) + O(x*x^n), n)}
a(n)={primepi(b(n+1)-1) - primepi(b(n))} \\ Andrew Howroyd, Jan 02 2020

Formula

a(n) = A000720(A000073(n+1) - 1) - A000720(A000073(n)) for n >= 3. [formula edited Andrew Howroyd, Jan 02 2020]

Extensions

Terms a(26) and beyond from Andrew Howroyd, Jan 02 2020

A122991 Indices of primes in tribonacci sequence A000073, minus 2.

Original entry on oeis.org

2, 4, 5, 9, 85, 96, 213, 800, 4200, 18697, 96877
Offset: 1

Views

Author

Artur Jasinski, Oct 28 2006

Keywords

Comments

Also, indices of primes in A282718.

Crossrefs

Cf. A000073, A092835, A092836, A282718, A303263.

Formula

A000073(a(n) + 2) = A092836(n).
a(n) = A303263(n) - 2 = A092835(n) - 1.

Extensions

Name corrected and a(10)-a(11) added by Andrew Howroyd, Oct 10 2024
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