A000073 -id:A000073 - cp's OEIS frontend

A357238 Inverse Moebius transform of tribonacci numbers (A000073).

0, 1, 1, 3, 4, 9, 13, 27, 45, 86, 149, 285, 504, 941, 1710, 3163, 5768, 10662, 19513, 35978, 66026, 121565, 223317, 411053, 755480, 1390042, 2555802, 4701713, 8646064, 15904390, 29249425, 53801243, 98950246, 182003370, 334745794, 615704412, 1132436852, 2082895617, 3831006934

Offset: 1

Ilya Gutkovskiy, Sep 19 2022

nonn

Cf. A000073, A007435, A357239.

nmax = 39; CoefficientList[Series[Sum[x^(2 k)/(1 - x^k - x^(2 k) - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

f(n) = ([0, 1, 0; 0, 0, 1; 1, 1, 1]^n)[1, 3]; \\ A000073
a(n) = sumdiv(n, d, f(d)); \\ Michel Marcus, Sep 20 2022

G.f.: Sum_{k>=1} x^(2*k) / (1 - x^k - x^(2*k) - x^(3*k)).
G.f.: Sum_{k>=1} A000073(k) * x^k / (1 - x^k).
a(n) = Sum_{d|n} A000073(d).

A096233 Number of digits of the 10^n-th tribonacci number (A000073).

Original entry on oeis.org

2, 26, 264, 2646, 26464, 264649, 2646494, 26464944, 264649443, 2646494434, 26464944348, 264649443484, 2646494434842, 26464944348425, 264649443484250, 2646494434842508, 26464944348425087

Offset: 0

Michael Taktikos, Aug 09 2004

a(n)/10^n tends towards log[10](rho) = .26464944348425087191..., where rho is the real root of x^3-x^2-x-1 = 0. - Vladeta Jovovic, Sep 01 2004

Cf. A068070.

rho=1/3*((19+3*Sqrt[33])^(1/3)+(19-3*Sqrt[33])^(1/3)+1); triboappr[n_]:=N[(rho-1)/(4rho-6)*rho^(n-3), 3000]; Table[MantissaExponent[triboappr[10^i]][[2]], {i, 1, 7}]

A106310 Primes p such that p^2 divides some T(k), yet p does not divide any T(j) for any jA000073).

Original entry on oeis.org

47, 617, 2693

Offset: 1

T. D. Noe, May 17 2005

No other p < 10^6. For Fibonacci numbers, A000045, there are no known primes with this property.

			47 is here because the 29th tribonacci number, 15902591, is the first tribonacci number divisible by 47 and 47^2 also divides it. Similarly, 617^2 divides T(409) and 2693^2 divides T(10553).

FibonacciZero[n_, kMax_, m_] := Module[{a, s, k}, a=Join[{1}, Table[0, {n-1}]]; a=Mod[a, m]; k=0; While[k++; s=Mod[Plus@@a, m]; a=RotateLeft[a]; a[[n]]=s; s>0&&k

A115792 a(n) = ceiling(g(A000073(n))) with g(k) = (k-1)^2/(4k).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 6, 11, 20, 37, 69, 126, 232, 426, 784, 1442, 2652, 4878, 8973, 16503, 30354, 55829, 102686, 188869, 347384, 638939, 1175193, 2161516, 3975648, 7312356, 13449520, 24737524, 45499400, 83686444, 153923369, 283109213, 520719026, 957751607

Offset: 2

Roger L. Bagula, Mar 13 2006

Old name: A dihedral D1 elliptical transform on A000073.

A D1 elliptical invariant transform leaves the ratio unchanged.

Cf. A000073, A132151.

g[x_] = (x - 1)^2/(-4*x) M = {{0, 1, 0}, {0, 0, 1}, {1, 1, 1}} w[0] = {0, 1, 1}; w[n_] := w[n] = M.w[n - 1] a0 = Table[ -Floor[g[w[n][[1]]]], {n, 1, 25}] b0 = Table[N[a0[[n + 1]]/a0[[n]]], {n, 2, 24}]

Conjectures from Chai Wah Wu, Dec 21 2023: (Start)
a(n) = a(n-2) + 2*a(n-3) + 2*a(n-4) + 2*a(n-5) + 2*a(n-6) + 2*a(n-7) + 3*a(n-8) + 2*a(n-9) + a(n-10) for n > 11.
G.f.: x^4*(-x^2 - x - 1)/((x + 1)*(x^2 + 1)*(x^4 + 1)*(x^3 + x^2 + x - 1)). (End)
For n >= 5, a(n) = a(n-1) + a(n-2) + a(n-3) - A132151(n+2). - Peter Munn, Jul 17 2025

Edited by Peter Munn, Jul 17 2025

A116573 A Binet type formula from a polynomial whose coefficient expansion gives a tribonacci used as its first derivative InverseZtransform: A000073.

Original entry on oeis.org

1, 0, 4, 17, 1, 82, 324, 49, 961, 5185, 2501, 5776, 57600, 54290, 15625, 497026, 801025, 1, 3437317, 9120400, 1256641, 18714277, 85766122, 38850289, 72999937

Offset: 0

Roger L. Bagula, Mar 19 2006

A polynomial derived in Mathematica by Bob Hanlon that is different from that in A000073; the first derivative sequence is different as well. Bob Hanlon's code: Needs["DiscreteMath`RSolve`"]; eqns={a[n]==a[n-1]+a[n-2]+a[n-3], a[0]==0,a[1]==a[2]==1}; Clear[f0,f1,f2,f3]; f0[0]=0;f0[1]=f0[2]=1; f0[n_Integer?Positive]:= f0[n]=f0[n-1]+f0[n-2]+f0[n-3]; f1[n_Integer]=a[n]/. RSolve[eqns,a[n],n][[1]]// ToRadicals//Simplify; (*Note that f1[n] is not restricted to nonnegative values of n.*) (*RSolve can also provide the generating function*) gf[x_]=GeneratingFunction[ eqns,a[n],n,x][[1,1]] -(x/(x^3 + x^2 + x - 1)) f2[n_Integer?NonNegative]:= SeriesCoefficient[ Series[gf[x],{x,0,n}],n];

Private email from Bob Hanlon (hanlonr(AT)cox.net), Mar 18 2006

Cf. A000073.

g[x_] = -(x/(x^3 + x^2 + x - 1)); dg[x_] = D[g[x], {x, 1}]; w[n_] := InverseZTransform[dg[x], x, n] // ToRadicals; Table[Abs[Floor[N[w[n]]]]^2, {n, 1, 25}]

g[x_] = -(x/(x^3 + x^2 + x - 1)); dg[x_] = D[g[x], {x, 1}]; w[n_] := InverseZTransform[dg[x], x, n] // ToRadicals; a(n) =Abs[w[n]]^2

A116574 A Binet type formula from a polynomial whose coefficient expansion gives a tribonacci used as it first derivative InverseZtransform: A000073.

Original entry on oeis.org

0, 1, 10, 1, 49, 225, 36, 730, 4097, 2025, 4761, 48401, 46225, 13456, 432965, 703922, 1, 3066002, 8185321, 1134225, 16974401, 78145601, 35545444, 67043345, 632572802

Offset: 0

Roger L. Bagula, Mar 19 2006

x^2/(1 - x - x^2 - x^3) is similar to the polynomial: -(x/(x^3 + x^2 + x - 1)) but not the same. As the last is machine derived, it is probably more correct than the one quoted presently in A000073.

Cf. A000073.

(*Source : A000073*) g[x_] = x^2/(1 - x - x^2 - x^3); dg[x_] = D[g[x], {x, 1}]; w[n_] := InverseZTransform[dg[x], x, n] // ToRadicals; Table[Abs[Floor[N[w[n]]]]^2, {n, 1, 25}]

(*Source : A000073*) g[x_] = x^2/(1 - x - x^2 - x^3); dg[x_] = D[g[x], {x, 1}]; w[n_] := InverseZTransform[dg[x], x, n] // ToRadicals; a(n) =Abs[w[n]]^2

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

Artur Jasinski, Oct 28 2006

Also, indices of primes in A282718.

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

A000073(a(n) + 2) = A092836(n).

a(n) = A303263(n) - 2 = A092835(n) - 1.

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

A200543 Product of tribonacci numbers: a(n) = A000073(n+2)*A000213(n).

Original entry on oeis.org

1, 1, 2, 12, 35, 117, 408, 1364, 4617, 15645, 52882, 178920, 605331, 2047705, 6927424, 23435384, 79281057, 268206185, 907335090, 3069491988, 10384017875, 35128880685, 118840150776, 402033352684, 1360069088841, 4601080767717, 15565344749410, 52657184101648, 178137977818211, 602636462317425

Offset: 0

Paul D. Hanna, Nov 19 2011

The g.f. of the tribonacci numbers are as follows: g.f. for A000073 is x^2/(1-x-x^2-x^3), and g.f. for A000213 is (1-x^2)/(1-x-x^2-x^3).

			G.f.: A(x) = 1 + x + 2*x^2 + 12*x^3 + 35*x^4 + 117*x^5 + 408*x^6 +...
where A(x) = 1*1 + 1*1*x + 2*1*x^2 + 4*3*x^3 + 7*5*x^4 + 13*9*x^5 + 24*17*x^6 + 44*31*x^7 + 81*57*x^8 + 149*105*x^9 +...+ A000073(n+2)*A000213(n)*x^n +...

Index entries for linear recurrences with constant coefficients, signature (2,3,6,-1,0,-1).

Cf. A000073, A000213.

{a(n)=polcoeff((1-x-3*x^2-x^3)/((1-3*x-x^2-x^3)*(1+x+x^2-x^3)+x*O(x^n)),n)}

G.f.: (1 - x - 3*x^2 - x^3) / ((1 - 3*x - x^2 - x^3)*(1 + x + x^2 - x^3)).

A240562 Integers whose squares are in A000073 (tribonacci numbers).

Original entry on oeis.org

0, 1, 2, 9, 56, 103

Offset: 1

J. Lowell, Apr 16 2014

a(6) > 10^7. - Tom Edgar, Apr 26 2014
Is this sequence finite?
No more terms < 10^19300. I conjecture that the sequence is finite. - Manfred Scheucher, Aug 17 2015

			9^2 = 81 is in the tribonacci sequence, so 9 is a term.

Cf. A000073 (tribonacci numbers), A128911 (the corresponding squares).

Select[Sqrt[#]&/@LinearRecurrence[{1,1,1},{0,0,1},200],IntegerQ]// Union (* Harvey P. Dale, Aug 16 2021 *)

def tribs():
    a,b,c = 0,0,1
    while True:
        yield a
        a,b,c = b,c,a+b+c
for n in tribs():
    m = sqrt(n)
    if m.is_integer():
        print(m) # Manfred Scheucher, Aug 17 2015

Zero prepended by Harvey P. Dale, Aug 16 2021

A338192 Sum of Fibonacci and tribonacci numbers: a(n) = A000073(n) + A000045(n).

Original entry on oeis.org

0, 1, 2, 3, 5, 9, 15, 26, 45, 78, 136, 238, 418, 737, 1304, 2315, 4123, 7365, 13193, 23694, 42655, 76958, 139126, 251974, 457112, 830501, 1510930, 2752175, 5018581, 9160293, 16734631, 30595694, 55976389, 102474674, 187700488, 343973242, 630623826, 1156594669

Offset: 0

Gary Detlefs, Oct 15 2020

In general, the sum of a second-order sequence with signature (a,b) and a third-order sequence with signature (x,y,z) will be a fifth-order sequence with signature (a+x,-x*a+b+y, -y*a+z-b*x,-a*z-b*y,-b*z). In this instance, a=b=x=y=z=1 resulting in a signature of (2,1,-1,-2,-1).

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

Cf. A000045, A000073.

LinearRecurrence[{2, 1, -1, -2, -1}, {0, 1, 2, 3, 5}, 50] (* Amiram Eldar, Oct 15 2020 *)

a(n) = A000073(n) + A000045(n).
a(n) = 2*a(n-1) + a(n-2) - a(n-3) - 2*a(n-4) - a(n-5) for n > 4 with a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=5.
G.f.: x*(1 - 2*x^2 - 2*x^3)/(1 - 2*x - x^2 + x^3 + 2*x^4 + x^5). - Stefano Spezia, Oct 15 2020

cp's OEIS Frontend

A357238 Inverse Moebius transform of tribonacci numbers (A000073).

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A096233 Number of digits of the 10^n-th tribonacci number (A000073).

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A106310 Primes p such that p^2 divides some T(k), yet p does not divide any T(j) for any jA000073).

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A115792 a(n) = ceiling(g(A000073(n))) with g(k) = (k-1)^2/(4k).

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A116573 A Binet type formula from a polynomial whose coefficient expansion gives a tribonacci used as its first derivative InverseZtransform: A000073.

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A116574 A Binet type formula from a polynomial whose coefficient expansion gives a tribonacci used as it first derivative InverseZtransform: A000073.

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

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A200543 Product of tribonacci numbers: a(n) = A000073(n+2)*A000213(n).

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A240562 Integers whose squares are in A000073 (tribonacci numbers).

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