A000073 -id:A000073 - cp's OEIS frontend

A200541 Product of Fibonacci and tribonacci numbers: a(n) = A000045(n+1)*A000073(n+2).

1, 1, 4, 12, 35, 104, 312, 924, 2754, 8195, 24386, 72576, 215991, 642785, 1912960, 5693016, 16942573, 50421592, 150056090, 446571180, 1329008590, 3955167387, 11770690808, 35029911168, 104250013425, 310251009501, 923315841860, 2747814245904, 8177573467339, 24336691577000

Offset: 0

Paul D. Hanna, Nov 18 2011

nonn

Limit a(n+1)/a(n) = (sqrt(5)+1)/2 * (1+(19+3*sqrt(33))^(1/3)+(19-3*sqrt(33))^(1/3))/3 = 2.9760284849940...

			G.f.: A(x) = 1 + x + 4*x^2 + 12*x^3 + 35*x^4 + 104*x^5 + 312*x^6 + 924*x^7 + 2754*x^8 +...+ A000045(n+1)*A000073(n+2)*x^n +...
where tribonacci numbers (A000073) begin:
[1,1,2,4,7,13,24,44,81,149,274,504,927,1705,3136,5768,10609,...],
and Fibonacci numbers (A000045) begin:
[1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,...].

Paul D. Hanna, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (1,4,5,2,-1,1).

Cf. A001582, A000045, A000073.

Module[{nn=30,fs,ts},fs=Fibonacci[Range[nn]];ts=LinearRecurrence[{1,1,1},{1,1,2},nn];Times@@@Thread[{fs,ts}]] (* or *) LinearRecurrence[ {1,4,5,2,-1,1},{1,1,4,12,35,104},30] (* Harvey P. Dale, Dec 14 2016 *)

{a(n)=polcoeff((1-x^2-x^3)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6 +x*O(x^n)),n)}

{A000073(n)=polcoeff(x^2/(1-x-x^2-x^3+x^3*O(x^n)),n)}
{a(n)=fibonacci(n+1)*A000073(n+2)}

G.f.: (1 - x^2 - x^3) / (1 - x - 4*x^2 - 5*x^3 - 2*x^4 + x^5 - x^6).

A228609 Partial sums of the cubes of the tribonacci sequence A000073.

Original entry on oeis.org

0, 1, 2, 10, 74, 417, 2614, 16438, 101622, 633063, 3941012, 24511836, 152535900, 949133883, 5905611508, 36746590964, 228646935796, 1422699232325, 8852413871022, 55082039340022, 342734883853750, 2132586518002125

Offset: 0

R. J. Mathar, Dec 18 2013

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

Michael De Vlieger, Table of n, a(n) for n = 0..1261
H. Ohtsuka, Advanced Problems and Solutions, Fib. Quart. 51 (2) (2013) 186, H-736.
Index entries for linear recurrences with constant coefficients, signature (5,5,25,-58,26,-42,54,-13,1,-3,1).

Cf. A000073, A107239.

CoefficientList[Series[x (-1 + 3 x + 11 x^3 - 5 x^4 + x^5 - 3 x^6 + x^7 + 5 x^2)/((x^3 - 5 x^2 + 7 x - 1) (x^6 + 4 x^5 + 11 x^4 + 12 x^3 + 11 x^2 + 4 x + 1) (x - 1)^2), {x, 0, 21}], x] (* Michael De Vlieger, Jan 12 2022 *)
Accumulate[LinearRecurrence[{1,1,1},{0,1,1},30]^3]  (* or *) LinearRecurrence[ {5,5,25,-58,26,-42,54,-13,1,-3,1},{0,1,2,10,74,417,2614,16438,101622,633063,3941012},30] (* Harvey P. Dale, Sep 11 2022 *)

T(n)=([0, 1, 0; 0, 0, 1; 1, 1, 1]^n)[1, 3]; \\ A000073
a(n) = sum(k=1, n, T(k)^3); \\ Michel Marcus, Jan 12 2022

a(n) = a(n-1) + (A000073(n))^3.

G.f.: x*(-1+3*x+11*x^3-5*x^4+x^5-3*x^6+x^7+5*x^2) / ( (x^3-5*x^2+7*x-1) *(x^6+4*x^5+11*x^4+12*x^3+11*x^2+4*x+1) *(x-1)^2 )

A274518 Numbers k such that k^2 divides A000073(k).

Original entry on oeis.org

1, 103, 112, 2621, 30576, 77168, 694512, 9919728, 24770928, 55638128, 57268848, 80995824, 1300820976

Offset: 1

Seiichi Manyama, Jun 26 2016

			tribonacci(103) = 331800673921785084815380861 = 103^2 * 31275395788649739354829.

Cf. A000073, A232570.

require 'matrix'
def power(a, n, mod)
  return Matrix.I(a.row_size) if n == 0
  m = power(a, n >> 1, mod)
  m = (m * m).map{|i| i % mod}
  return m if n & 1 == 0
  (m * a).map{|i| i % mod}
end
def f(m, n, mod)
  ary0 = Array.new(m, 0)
  ary0[0] = 1
  v = Vector.elements(ary0)
  ary1 = [Array.new(m, 1)]
  (0..m - 2).each{|i|
    ary2 = Array.new(m, 0)
    ary2[i] = 1
    ary1 << ary2
  }
  a = Matrix[*ary1]
  (power(a, n, mod) * v)[m - 1]
end
p (1..10 ** 6).select{|i| f(3, i, i * i) == 0}

a(9)-a(13) from Chai Wah Wu, Jan 29 2018

A292397 p-INVERT of the tribonacci numbers (A000073(k), k>=2), where p(S) = 1 - S - S^2 - S^3.

Original entry on oeis.org

1, 3, 10, 33, 108, 352, 1144, 3714, 12050, 39084, 126752, 411041, 1332923, 4322363, 14016392, 45451793, 147389276, 477948252, 1549872500, 5025868667, 16297700769, 52849583211, 171378684824, 555740504324, 1802134907175, 5843896942499, 18950374573538

Offset: 0

Clark Kimberling, Sep 18 2017

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4, -1, -3, -7, 2, 6, 7, 3, 1)

Cf. A000073, A292322.

I:=[1,3,10,33,108,352,1144,3714,12050]; [n le 9 select I[n] else 4*Self(n-1)-Self(n-2)-3*Self(n-3)-7*Self(n-4)+2*Self(n-5)+6*Self(n-6)+7*Self(n-7)+3*Self(n-8)+Self(n-9): n in [1..30]]; // Vincenzo Librandi, Oct 13 2017

z = 60; s = x/(1 - x - x^2 - x^3); p = 1 - s - s^2 - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000073 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A292397 *)
LinearRecurrence[{4, -1, -3, -7, 2, 6, 7, 3, 1}, {1, 3, 10, 33, 108, 352, 1144, 3714, 12050}, 30] (* Vincenzo Librandi, Oct 13 2017 *)

x='x+O('x^99); Vec(((1+x+x^2)*(1-2*x+x^3+x^4))/(1-4*x+x^2+3*x^3+7*x^4-2*x^5-6*x^6-7*x^7-3*x^8-x^9)) \\ Altug Alkan, Oct 04 2017

G.f.: -(((1 + x + x^2) (1 - 2 x + x^3 + x^4))/(-1 + 4 x - x^2 - 3 x^3 - 7 x^4 + 2 x^5 + 6 x^6 + 7 x^7 + 3 x^8 + x^9)).

a(n) = 4*a(n-1) - a(n-2) - 3*a(n-3) - 7*a(n-4) + 2*a(n-5) + 6*a(n-6) + 7*a(n-7) + 3*a(n-8) + a(n-9) for n >= 10.

A299156 Numbers k such that k*(k+1) divides tribonacci(k) (A000073(k)).

Original entry on oeis.org

1, 256, 397, 1197, 8053, 8736, 9901, 32173, 33493, 33757, 38461, 48757, 56101, 57073, 64153, 76561, 79693, 87517, 100608, 102217, 105253, 105601, 105913, 105997, 107713, 108553, 110976, 116293, 123121, 131437, 138517, 143137, 147541, 151237, 156601, 171253

Offset: 1

Seiichi Manyama, Feb 04 2018

nonn

A subsequence of A232570.

			tribonacci(256) = 10285895715599251294835119279496333059462348558276025598603904254464 = 256 * 257 * 156339611436029476149609668037091638184921397104146789862048642.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1000 from Alois P. Heinz)

Cf. A000073, A217738, A232570, A274518.

with(LinearAlgebra[Modular]):
T:= (n, m)-> MatrixPower(m, Mod(m, <<0|1|0>,
    <0|0|1>, <1|1|1>>, float[8]), n)[1, 3]:
a:= proc(n) option remember; local i, k, ok;
      if n=1 then 1 else
        for k from 1+a(n-1) do ok:= true;
          for i in ifactors(k*(k+1))[2] while ok do
            ok:= is(T(k, i[1]^i[2])=0)
          od; if ok then break fi
        od; k
      fi
    end:
seq(a(n), n=1..10);  # Alois P. Heinz, Feb 06 2018

a = b = 0; c = d = 1; k = 2; lst = {1}; While[k < 171255, If[ Mod[c, k (k + 1)] == 0, AppendTo[lst, k]]; a = b; b = c; c = d; d = a + b + c; k++] (* Robert G. Wilson v, Feb 07 2018 *)

A308189 Numbers of the form t_n or t_n + t_{n+1} where {t_n} are the tribonacci numbers A000073.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 11, 13, 20, 24, 37, 44, 68, 81, 125, 149, 230, 274, 423, 504, 778, 927, 1431, 1705, 2632, 3136, 4841, 5768, 8904, 10609, 16377, 19513, 30122, 35890, 55403, 66012, 101902, 121415, 187427, 223317, 344732, 410744, 634061, 755476, 1166220, 1389537, 2145013, 2555757, 3945294, 4700770, 7256527

Offset: 1

N. J. A. Sloane, Jun 09 2019

Orders of squares in the ternary tribonacci word A080843.

This is A213816 with duplicates removed.

Colin Barker, Table of n, a(n) for n = 1..1000
Hamoon Mousavi and Jeffrey Shallit, Mechanical Proofs of Properties of the Tribonacci Word, arXiv:1407.5841 [cs.FL], 2014.
H. Mousavi and J. Shallit, Mechanical Proofs of Properties of the Tribonacci Word, In: Manea F., Nowotka D. (eds) Combinatorics on Words. WORDS 2015. Lecture Notes in Computer Science, vol 9304. Springer, 2015, pp. 170-190.
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,1).

Cf. A000073, A001590, A080843, A092782, A213816.

LinearRecurrence[{0,1,0,1,0,1},{0,1,2,3,4,6,7,11},100] (* Paolo Xausa, Nov 14 2023 *)

concat(0, Vec(x^2*(1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + x^5 + x^6) / (1 - x^2 - x^4 - x^6) + O(x^50))) \\ Colin Barker, Jun 11 2019

From Colin Barker, Jun 11 2019: (Start)
G.f.: x^2*(1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + x^5 + x^6) / (1 - x^2 - x^4 - x^6).
a(n) = a(n-2) + a(n-4) + a(n-6) for n>8.
(End)

A337281 a(n) = n*T(n), where T(n) = A000073(n) = n-th tribonacci number.

Original entry on oeis.org

0, 0, 2, 3, 8, 20, 42, 91, 192, 396, 810, 1639, 3288, 6552, 12978, 25575, 50176, 98056, 190962, 370747, 717800, 1386252, 2671130, 5136291, 9857856, 18886900, 36127962, 69005439, 131621560, 250735856, 477077730, 906732175, 1721538560, 3265353168, 6187918434, 11716102195, 22164965064, 41900163524

Offset: 0

N. J. A. Sloane, Sep 12 2020

nonn

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

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

Cf. A000073.

LinearRecurrence[{2,1,0,-3,-2,-1},{0,0,2,3,8,20},40] (* Harvey P. Dale, Dec 19 2023 *)

a(n)= n * ([0, 1, 0; 0, 0, 1; 1, 1, 1]^n)[1, 3] \\ David A. Corneth, Sep 13 2020, after Charles R Greathouse IV

concat([0,0], Vec(x^2*(2 - x + x^3) / (1 - x - x^2 - x^3)^2 + O(x^36))) \\ Colin Barker, Sep 13 2020

From Colin Barker, Sep 13 2020: (Start)
G.f.: x^2*(2 - x + x^3) / (1 - x - x^2 - x^3)^2.
a(n) = 2*a(n-1) + a(n-2) - 3*a(n-4) - 2*a(n-5) - a(n-6) for n>5.
(End)

A341266 a(n) is the n-th term of the n-fold self-convolution of the twice left-shifted tribonacci sequence (A000073).

Original entry on oeis.org

1, 1, 5, 25, 125, 646, 3395, 18054, 96885, 523600, 2845700, 15537457, 85160387, 468279280, 2582140370, 14272523740, 79056303957, 438711518556, 2438587839980, 13574970187300, 75668677723100, 422294150816010, 2359326605275755, 13194525668986350, 73857744668632275

Offset: 0

Alois P. Heinz, Feb 07 2021

nonn

The twice left-shifted tribonacci sequence begins: 1, 1, 2, 4, 7, 13, 24, ... .

Alois P. Heinz, Table of n, a(n) for n = 0..1323

Cf. A000073, A002002, A213684.

a:= n-> coeff(series((1/(1-x-x^2-x^3))^n, x, n+1), x, n):
seq(a(n), n=0..25);
# second Maple program:
g:= proc(n) g(n):= `if`(n<2, (n+1)*(2-n)/2, add(g(n-j), j=1..3)) end:
b:= proc(n, k) option remember; `if`(k<2, g(n),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);

a(n) = [x^n] (1/(1-x-x^2-x^3))^n.

A354784 First differences of A000213, also twice A000073.

Original entry on oeis.org

0, 0, 2, 2, 4, 8, 14, 26, 48, 88, 162, 298, 548, 1008, 1854, 3410, 6272, 11536, 21218, 39026, 71780, 132024, 242830, 446634, 821488, 1510952, 2779074, 5111514, 9401540, 17292128, 31805182, 58498850, 107596160, 197900192, 363995202, 669491554

Offset: 0

N. J. A. Sloane, Jul 12 2022

Number of anti-palindromic compositions of n+1 of even length.

Paolo Xausa, Table of n, a(n) for n = 0..1000
George E. Andrews, Matthew Just, and Greg Simay, Anti-palindromic compositions, arXiv:2102.01613 [math.CO], 2021. Also Fib. Q., 60:2 (2022), 164-176. See Table 1.
Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Cf. A000213, A000073, A135491 (essentially the same sequence).

LinearRecurrence[{1, 1, 1}, {0, 0, 2}, 50] (* Paolo Xausa, May 27 2024 *)

From Chai Wah Wu, Jul 12 2022: (Start)
a(n) = a(n-1) + a(n-2) + a(n-3) for n > 2.
G.f.: -2*x^2/(x^3 + x^2 + x - 1). (End)

A356592 Array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) = Sum_{i, j >= 3} t_i * u_j * T(i+j) where Sum_{i >= 3} t_i * T(i) and Sum_{j >= 3} u_j * T(j) are the greedy tribonacci representations of n and k, respectively, and T = A000073.

Original entry on oeis.org

0, 0, 0, 0, 7, 0, 0, 13, 13, 0, 0, 20, 24, 20, 0, 0, 24, 37, 37, 24, 0, 0, 31, 44, 57, 44, 31, 0, 0, 37, 57, 68, 68, 57, 37, 0, 0, 44, 68, 88, 81, 88, 68, 44, 0, 0, 51, 81, 105, 105, 105, 105, 81, 51, 0, 0, 57, 94, 125, 125, 136, 125, 125, 94, 57, 0

Offset: 0

Rémy Sigrist, Sep 11 2022

This sequence is to tribonacci numbers (A000073) what A135090 is to Fibonacci numbers (A000045).

			Array A(n, k) begins:
  n\k | 0   1    2    3    4    5    6    7    8    9   10
  ----+---------------------------------------------------
    0 | 0   0    0    0    0    0    0    0    0    0    0
    1 | 0   7   13   20   24   31   37   44   51   57   64
    2 | 0  13   24   37   44   57   68   81   94  105  118
    3 | 0  20   37   57   68   88  105  125  145  162  182
    4 | 0  24   44   68   81  105  125  149  173  193  217
    5 | 0  31   57   88  105  136  162  193  224  250  281
    6 | 0  37   68  105  125  162  193  230  267  298  335
    7 | 0  44   81  125  149  193  230  274  318  355  399
    8 | 0  51   94  145  173  224  267  318  369  412  463
    9 | 0  57  105  162  193  250  298  355  412  460  517
   10 | 0  64  118  182  217  281  335  399  463  517  581

A. Messaoudi, Tribonacci multiplication, Appl. Math. Lett. 15 (2002), 981-985.
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

Cf. A000045, A000073, A101330, A135090.

PARI
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See Links section.
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A(n, 0) = A(0, k) = 0.
A(n, k) = A(k, n).
A(m, A(n, k)) = A(A(m, n), k) for m, n, k >= 5.

cp's OEIS Frontend

A200541 Product of Fibonacci and tribonacci numbers: a(n) = A000045(n+1)*A000073(n+2).

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A228609 Partial sums of the cubes of the tribonacci sequence A000073.

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A274518 Numbers k such that k^2 divides A000073(k).

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A292397 p-INVERT of the tribonacci numbers (A000073(k), k>=2), where p(S) = 1 - S - S^2 - S^3.

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A299156 Numbers k such that k*(k+1) divides tribonacci(k) (A000073(k)).

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A308189 Numbers of the form t_n or t_n + t_{n+1} where {t_n} are the tribonacci numbers A000073.

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A337281 a(n) = n*T(n), where T(n) = A000073(n) = n-th tribonacci number.

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