A000073 -id:A000073 - cp's OEIS frontend

A356964 Replace 2^k in binary expansion of n with tribonacci(k+3) (where tribonacci corresponds to A000073).

0, 1, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 14, 13, 14, 15, 16, 17, 18, 19, 20, 20, 21, 22, 23, 24, 25, 26, 27, 24, 25, 26, 27, 28, 29, 30, 31, 31, 32, 33, 34, 35, 36, 37, 38, 37, 38, 39, 40, 41, 42, 43, 44, 44, 45, 46, 47, 48, 49, 50, 51, 44, 45, 46, 47

Offset: 0

Rémy Sigrist, Sep 06 2022

This sequence is to tribonacci numbers (A000073) what A022290 is to Fibonacci numbers (A000045).

For any k >= 0, k appears A117546(k) times in this sequence.

			For n = 9:
- 9 = 2^3 + 2^0,
- so a(9) = A000073(3+3) + A000073(0+3) = 7 + 1 = 8.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Index entries for sequences related to Zeckendorf expansion of n

Cf. A000073, A003726, A003796, A022290, A117546.

a(n) = { my (v=0, k); while (n, n-=2^k=valuation(n,2); v+=([0,1,0; 0,0,1; 1,1,1]^(3+k))[2,1]); return (v); }

def A356964(n):
    a, b, c, s = 1,2,4,0
    for i in bin(n)[-1:1:-1]:
        s += int(i)*a
        a, b, c = b, c, a+b+c
    return s # Chai Wah Wu, Sep 10 2022

a(A003726(n+1)) = n.

a(A003796(n+1)) = n.

A359876 a(n) is the smallest tribonacci number (A000073) with exactly n prime factors (counted with multiplicity).

Original entry on oeis.org

1, 2, 4, 44, 24, 5768, 504, 10562230626642, 3136, 7046319384, 615693474, 53798080, 4680045560037375, 35574238430251050319992, 4659412488735286161146176, 23523635785731871586396890786299881280, 79932289960699059086717998848

Offset: 0

Ilya Gutkovskiy, Jan 16 2023

nonn

			a(5) = 5768, because 5768 is a tribonacci number with 5 prime factors (counted with multiplicity) {2, 2, 2, 7, 103} and this is the smallest such number.

Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Tribonacci Number

Cf. A000073, A001222, A072397, A359848, A359877, A359878.

A359878 a(n) is the index of the smallest tribonacci number (A000073) with exactly n prime factors (counted with multiplicity).

Original entry on oeis.org

2, 4, 5, 9, 8, 17, 13, 52, 16, 40, 36, 32, 62, 88, 96, 144, 112, 221, 256, 208, 400, 192

Offset: 0

Ilya Gutkovskiy, Jan 16 2023

Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Tribonacci Number

Cf. A000073, A001222, A072396, A359850, A359876, A359879.

a(17)-a(21) from Daniel Suteu, Jan 18 2023

A386236 Ratio of the period and the reduced period of the Fibonacci 3-step sequence A000073 mod n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 3, 1, 1, 1, 3, 3, 1, 3, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 1, 1, 3, 3, 1, 3, 3, 1, 1, 3, 1, 1, 3, 1, 1, 1, 3, 1, 1, 3, 1, 3, 1, 3, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 3, 1, 3, 3, 1, 1, 3, 3, 3, 3, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 3, 3, 3

Offset: 1

Peter Munn, Jul 16 2025

nonn

The period is A046738(n) and the reduced period is A046737(n).

See also the information in A154754 and A046737.

Peter Munn, Table of n, a(n) for n = 1..1000

Cf. A000073, A046737, A046738.
The equivalent sequence for Fibonacci numbers is A001176.
Cf. A060839 (differs first at n=31), A154754 (restriction to prime indices).

a(n) = A046738(n)/A046737(n).

A074581 a(n) = T(3n+1), where T(n) are tribonacci numbers A000073.

Original entry on oeis.org

0, 2, 13, 81, 504, 3136, 19513, 121415, 755476, 4700770, 29249425, 181997601, 1132436852, 7046319384, 43844049029, 272809183135, 1697490356184, 10562230626642, 65720971788709, 408933139743937, 2544489349890656

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Aug 24 2002

In general, the trisection of a third-order linear recurrence with signature (x,y,z) will result in a third-order recurrence with signature (x^3 + 3*x*y + 3*z, -3*x*y*z + y^3 - 3*z^2, z^3). - Gary Detlefs, May 29 2024

Index entries for linear recurrences with constant coefficients, signature (7,-5,1).

Cf. A000073.

CoefficientList[Series[(2*x-x^2)/(1-7*x+5*x^2-x^3), {x, 0, 40}], x]
LinearRecurrence[{7,-5,1},{0,2,13},30] (* Harvey P. Dale, Jul 22 2021 *)

a(n) = 7*a(n-1) - 5*a(n-2) + a(n-3), a(0)=0, a(1)=2, a(2)=13.

G.f.: (2*x - x^2)/(1 - 7*x + 5*x^2 - x^3). [corrected by Nguyen Tuan Anh, Jan 10 2025]

Definition corrected by David Scambler, Oct 18 2010

A075536 a(n) = ((1+(-1)^n)T(n+1) + (1-(-1)^n)S(n))/2, where T(n) = tribonacci numbers A000073, S(n) = generalized tribonacci numbers A001644.

Original entry on oeis.org

0, 1, 1, 7, 4, 21, 13, 71, 44, 241, 149, 815, 504, 2757, 1705, 9327, 5768, 31553, 19513, 106743, 66012, 361109, 223317, 1221623, 755476, 4132721, 2555757, 13980895, 8646064, 47297029, 29249425, 160004703, 98950096, 541292033, 334745777

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Sep 23 2002

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

Cf. A000073, A001644, A005013, A005247.

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+x+4*x^2+x^3-x^4)/(1-3*x^2-x^4-x^6) )); // G. C. Greubel, Apr 21 2019

A075536 := proc(n)
    if type(n,'even') then
        A000073(n+1) ;
    else
        A001644(n) ;
    end if;
end proc:
seq(A075536(n),n=0..80) ; # R. J. Mathar, Aug 05 2021

CoefficientList[Series[(x+x^2+4x^3+x^4-x^5)/(1-3x^2-x^4-x^6), {x, 0, 40}], x]
LinearRecurrence[{0,3,0,1,0,1},{0,1,1,7,4,21},40] (* Harvey P. Dale, Jul 10 2012 *)

my(x='x+O('x^40)); concat([0], Vec(x*(1+x+4*x^2+x^3-x^4)/(1-3*x^2-x^4-x^6))) \\ G. C. Greubel, Apr 21 2019

(x*(1+x+4*x^2+x^3-x^4)/(1-3*x^2-x^4-x^6)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 21 2019

a(2n) = A073717(n) = A000073(2n+1).
a(2n+1) = A001644(2n+1).
a(n) = 3*a(n-2) + a(n-4) + a(n-6), a(0)=0, a(1)=1, a(2)=1, a(3)=7, a(4)=4, a(5)=21.
O.g.f.: x*(1 + x + 4*x^2 + x^3 - x^4)/(1 - 3*x^2 - x^4 - x^6).

Index in definition corrected. - R. J. Mathar, Aug 05 2021

A112618 Let T(n) = A000073(n+1), n >= 1; a(n) = smallest k such that prime(n) divides T(k).

Original entry on oeis.org

3, 7, 14, 5, 8, 6, 28, 18, 29, 77, 14, 19, 35, 82, 29, 33, 64, 68, 100, 132, 31, 18, 270, 109, 19, 186, 13, 184, 105, 172, 586, 79, 11, 34, 10, 223, 71, 37, 41, 314, 100, 25, 72, 171, 382, 26, 83, 361, 34, 249, 36, 28, 506, 304, 54, 37, 177, 331, 61, 536, 777, 458, 30, 123

Offset: 1

T. D. Noe, Dec 05 2005

nonn

Brenner proves that every prime divides some tribonacci number T(n). For the similar 3-step Lucas sequence A001644, there are primes (A106299) that do not divide any term.

			Sequence T(n) starts 1,1,2,4,7,13,24,44. For the primes 2,3,7,11,13, it is easy to see that a(1)=3, a(2)=7, a(4)=5, a(5)=8, a(6)=6.

T. D. Noe, Table of n, a(n) for n=1..1000
J. L. Brenner, Linear Recurrence Relations, Amer. Math. Monthly, Vol. 61 (1954), 171-173.
Eric Weisstein's World of Mathematics, Tribonacci Number

Equals A112312(n)-1.

a[0]=0; a[1]=a[2]=1; a[n_]:=a[n]=a[n-1]+a[n-2]+a[n-3]; f[n_]:= Module[{k=2, p=Prime[n]}, While[Mod[a[k], p] != 0, k++ ]; k]; Array[f, 64] (* Robert G. Wilson v *)

a(n) = A112305(prime(n)).

A130594 Numbers which are both lucky (A000959) and tribonacci (A000073).

Original entry on oeis.org

1, 7, 13, 927, 1705, 10609

Offset: 1

Jonathan Vos Post, Jun 16 2007

No other terms below 15902591. The next candidate is the odd tribonacci number 15902591. Is this also a lucky number? - Harvey P. Dale, Jul 12 2008
This is to tribonacci as A057589 is to the Fibonacci numbers.
a(7) >= 23837527729. - Kevin P. Thompson, Nov 24 2021

			a(6) = 10609 because it is lucky A000959(1182) and tribonacci A000073(18).

Cf. A000073, A057589, A000959.

A141579 Numbers k such that the arithmetic mean of the first k tribonacci numbers A000073 is an integer.

Original entry on oeis.org

1, 2, 47, 53, 94, 103, 106, 163, 199, 206, 257, 269, 311, 326, 397, 398, 401, 419, 421, 499, 514, 538, 587, 599, 617, 622, 683, 757, 773, 794, 802, 838, 842, 863, 883, 907, 911, 929, 991, 998, 1021, 1087, 1109, 1123, 1174, 1181, 1198, 1210, 1234, 1237, 1291

Offset: 1

R. J. Mathar, Aug 19 2008

nonn

Numbers in this sequence but not in A140973 are 2021 and 2090 (but no others below 8400). - Emeric Deutsch, Aug 19 2008.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1500 from Harvey P. Dale)

Cf. A000073, A008937, A140973.

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: A008937 := proc(n) option remember ; add(A000073(i),i=0..n+1) ; end: isA := proc(n) if n = 1 then RETURN(true) ; fi; if A008937(n-2) mod n = 0 then true; else false ; fi; end: for n from 1 to 2000 do if isA(n) then printf("%d,",n) ; fi; od ;

Module[{nn=1300,tnos},tnos=LinearRecurrence[{1,1,1},{0,0,1},nn];Position[ Table[Mean[Take[tnos,n]],{n,nn}],?(IntegerQ[#]&)]]//Flatten (* _Harvey P. Dale, Oct 05 2020 *)

{k: k | A008937(k-2)}.

A153462 Triangle read by rows, = A000931(n-k+3) * (A000073 * 0^(n-k)).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 0, 4, 2, 1, 1, 2, 0, 7, 2, 2, 1, 2, 4, 0, 13, 3, 2, 2, 2, 4, 7, 0, 24, 4, 3, 2, 4, 4, 7, 13, 0, 44, 5, 4, 3, 4, 8, 7, 13, 24, 0, 81, 7, 5, 4, 6, 8, 14, 13, 24, 44, 0, 149, 9, 7, 5, 8, 12, 14, 26, 24, 44, 81, 0, 274

Offset: 3

Gary W. Adamson, Dec 27 2008

An eigentriangle by rows, the Padovan sequence convolved with the tribonacci numbers.

Sum of n-th row terms = rightmost term of next row. Row sums = the tribonacci numbers, A000073.

			First few rows of the triangle =
   1;
   0, 1;
   1, 0, 1;
   1, 1, 0,  2;
   1, 1, 1,  0,  4;
   2, 1, 1,  2,  0,  7;
   2, 2, 1,  2,  4,  0, 13;
   3, 2, 2,  2,  4,  7,  0, 24;
   4, 3, 2,  4,  4,  7, 13,  0, 44;
   5, 4, 3,  4,  8,  7, 13, 24,  0, 81;
   7, 5, 4,  6,  8, 14, 13, 24, 44,  0, 149;
   9, 7, 5,  8, 12, 14, 26, 24, 44, 81,   0, 274;
  12, 9, 7, 10, 16, 21, 26, 48, 44, 81, 149,   0, 504;
  ...
Row 9 = (2, 2, 1, 2, 4, 0, 13) = termwise products of (1, 1, 1, 2, 4, 7, 13) and (2, 2, 1, 1, 1, 0, 1). Dot product = 24 = A000073(8).

Cf. A000073, A000931.

Triangle read by rows, = A000931(n-k+3) * (A000073 * 0^(n-k)).
Equals infinite lower triangular matrices P*M; where P = a matrix with the Padovan sequence in every column starting with offset 3: (1, 0, 1, 1, 1, 2, 2, 3, 4, 5, ...).
M = an infinite lower triangular matrix with the tribonacci sequence prefaced with a 1 as the main diagonal: (1, 1, 1, 2, 4, 7, 13, ...) and the rest zeros.

cp's OEIS Frontend

A356964 Replace 2^k in binary expansion of n with tribonacci(k+3) (where tribonacci corresponds to A000073).

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A359876 a(n) is the smallest tribonacci number (A000073) with exactly n prime factors (counted with multiplicity).

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A359878 a(n) is the index of the smallest tribonacci number (A000073) with exactly n prime factors (counted with multiplicity).

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A386236 Ratio of the period and the reduced period of the Fibonacci 3-step sequence A000073 mod n.

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A074581 a(n) = T(3n+1), where T(n) are tribonacci numbers A000073.

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A075536 a(n) = ((1+(-1)^n)*T(n+1) + (1-(-1)^n)*S(n))/2, where T(n) = tribonacci numbers A000073, S(n) = generalized tribonacci numbers A001644.

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A112618 Let T(n) = A000073(n+1), n >= 1; a(n) = smallest k such that prime(n) divides T(k).

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A130594 Numbers which are both lucky (A000959) and tribonacci (A000073).

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A141579 Numbers k such that the arithmetic mean of the first k tribonacci numbers A000073 is an integer.

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A075536 a(n) = ((1+(-1)^n)T(n+1) + (1-(-1)^n)S(n))/2, where T(n) = tribonacci numbers A000073, S(n) = generalized tribonacci numbers A001644.