A000213 -id:A000213 - cp's OEIS frontend

A056816 Primes in the tribonacci sequence A000213.

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

Offset: 1

Harvey P. Dale, Sep 01 2000

nonn

Primes within A000213.

Cf. A092836.

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

A274432 Products of distinct tribonacci numbers (A000213).

Original entry on oeis.org

3, 5, 9, 15, 17, 27, 31, 45, 51, 57, 85, 93, 105, 135, 153, 155, 171, 193, 255, 279, 285, 315, 355, 459, 465, 513, 525, 527, 579, 653, 765, 837, 855, 945, 965, 969, 1065, 1201, 1395, 1539, 1575, 1581, 1737, 1767, 1775, 1785, 1959, 2209, 2295, 2565, 2635

Offset: 1

Clark Kimberling, Jun 22 2016

			The tribonacci numbers are 1,1,1,3,5,9,17,31,..., so that the sequence of all products of distinct members, in increasing order, is (3, 5, 9, 15, 17, 27, 31, 45,...).

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A160009, A274280, A274433 (binary products), A274434 (trinary products).

r[1] := 1; r[2] := 1; r[3] = 1; r[n_] := r[n] = r[n - 1] + r[n - 2] + r[n - 3];
s = {1}; z = 60; f = Map[r, Range[z]]; Take[f, 20]
Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}];
Take[s, 2 z]  (*A274432*)
infQ[n_] := MemberQ[f, n];
ans = Table[#[[Flatten[Position[Map[Apply[Times, #] &, #], s[[n]]]][[1]]]] &[
Rest[Subsets[Map[#[[1]] &, Select[Map[{#, infQ[#]} &, Divisors[s[[n]]]], #[[2]] && #[[1]] > 1 &]]]]], {n, 2, 300}];
Map[Apply[Times, #] &, Select[ans, Length[#] == 2 &]] (* A274433 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 3 &]] (* A274434 *)
(* Peter J. C. Moses, Jun 17 2016 *)

A046735 Nontrivial (i.e., having no nontrivial factors with this property) integers which do not divide any terms of A000213.

Original entry on oeis.org

2, 27, 91, 103, 163, 199, 203, 221, 247, 305, 371, 377, 397, 421, 551, 559, 757, 779, 883, 991, 1021, 1079, 1087, 1123, 1189, 1199, 1237, 1351, 1521, 1543, 1567, 1609, 1651, 1753, 1769, 1799, 1807, 1873, 1883, 1919, 2009, 2071, 2261, 2539

Offset: 1

David W. Wilson

nonn

Robert Israel, Table of n, a(n) for n = 1..1275

nd:= proc(p) local a,b,c,r,R;
   a:= 1; b:= 1; c:= 1; R[1,1,1]:= true;
   do
     r:= a+b+c mod p;
     if r = 0 then return false fi;
     a:= b; b:= c; c:= r;
     if assigned(R[a,b,c]) or nops({a,b,c})=1
         then return true
         else R[a,b,c]:= true
       fi;
   od
end proc:
N:= 10^4: # to get all terms <= N
V:= Vector(N): Res:= NULL:
for n from 1 to N do
  if V[n] = 0 then
    if nd(n) then Res:= Res,n; V[[seq(k*n,k=2..floor(N/n))]]:= 1; fi
  fi;
od:
Res; # Robert Israel, Feb 26 2017

nondivisor[n_] := Module[{a = 1, b = 1, c = 1, t}, For[i = 1, i <= n^2, i++, t = Mod[a+b+c, n]; If[t != 0, a = b; b = c; c = t, Return[False]]; If[c == 1 && b == 1 && a == 1, Return[True]]]];
okQ[n_] := Do[If[nondivisor[d], Return[n == d]], {d, Divisors[n]}];
Select[Range[3000], okQ] (* Jean-François Alcover, Mar 05 2019, from PARI *)

nondivisor(n)=my(a=1,b=1,c=1,t);for(i=1,n^2,t=(a+b+c)%n;if(t,a=b;b=c;c=t,return(0));if(c==1&&b==1&&a==1,return(1)))
is(n)=fordiv(n,d,if(nondivisor(d),return(n==d)));0 \\ Charles R Greathouse IV, Aug 29 2012

Definition corrected by Henry Ayoola (henry.ayoola(AT)googlemail.com), Feb 03 2009

a(1) added by Charles R Greathouse IV, Aug 29 2012

A107240 Sum of squares of first n tribonacci numbers (A000213).

Original entry on oeis.org

1, 2, 3, 12, 37, 118, 407, 1368, 4617, 15642, 52891, 178916, 605325, 2047726, 6927407, 23435376, 79281105, 268206130, 907335091, 3069492092, 10384017717, 35128880742, 118840150983, 402033352264, 1360069089113, 4601080768074

Offset: 1

Jonathan Vos Post, May 14 2005

			a(6) = 1^2 + 1^2 + 1^2 + 3^2 + 5^2 + 9^2 = 118.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
M. Feinberg, Fibonacci-Tribonacci, Fib. Quart. 1(3) (1963), 71-74.
Eric Weisstein's World of Mathematics, Tribonacci Number.
Index entries for linear recurrences with constant coefficients, signature (3, 1, 3, -7, 1, -1, 1).

Cf. A000213, A001654, A107239, A107241-A107247.

Accumulate[LinearRecurrence[{1,1,1},{1,1,1},30]^2] (* Harvey P. Dale, Nov 11 2011 *)
LinearRecurrence[{3, 1, 3, -7, 1, -1, 1},{1, 2, 3, 12, 37, 118, 407},26] (* Ray Chandler, Aug 02 2015 *)

a(n) = Sum_{i=1..n} A000213(i)^2.

a(n)= 3*a(n-1) +a(n-2) +3*a(n-3) -7*a(n-4) +a(n-5) -a(n-6) +a(n-7). G.f.: (x^3-x^2+3*x-1)*(1+x)^2/((x-1)*(x^3+x^2+3*x-1)*(x^3-x^2-x-1)). - R. J. Mathar, Aug 11 2009

A157611 Numbers n such that the n-th tribonacci number as defined by A000213 is prime.

Original entry on oeis.org

3, 4, 6, 7, 10, 12, 13, 19, 23, 26, 32, 63, 95, 150, 324, 449, 1969, 2988, 6199, 8093, 23393, 25427, 34685, 35076, 44171, 45624, 83357, 116404

Offset: 1

Dmitry Kamenetsky, Mar 03 2009

nonn

The next term is greater than 10000.

a(29) > 2*10^5. - Robert Price, Nov 15 2013

Cf. A000213, A056816.

a(21)-a(28) from Robert Price, Nov 15 2013

A141583 Squares of tribonacci numbers A000213.

Original entry on oeis.org

1, 1, 1, 9, 25, 81, 289, 961, 3249, 11025, 37249, 126025, 426409, 1442401, 4879681, 16507969, 55845729, 188925025, 639128961, 2162157001, 7314525625, 24744863025, 83711270241, 283193201281, 958035736849, 3241011678961

Offset: 0

R. J. Mathar, Aug 19 2008

Partial sums are in A107240.

a(n) is also the number of total dominating sets in the (n-1)-ladder graph. - Eric W. Weisstein, Apr 10 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Ladder Graph
Eric Weisstein's World of Mathematics, Total Dominating Set
Index entries for linear recurrences with constant coefficients, signature (2,3,6,-1,0,-1).

Cf. A000213, A085697, A107240.

I:=[1,1,1,9,25,81]; [n le 6 select I[n] else 2*Self(n-1) + 3*Self(n-2) + 6*Self(n-3) - Self(n-4) - Self(n-6): n in [1..30]]; // Vincenzo Librandi, Dec 13 2012

CoefficientList[Series[(1+x)^2*(1-3*x+x^2-x^3)/((1+x+x^2-x^3)*(1-3*x-x^2-x^3)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 13 2012 *)
Table[RootSum[-1 - # - #^2 + #^3 &, 2 #^n - 4 #^(n + 1) + 3 #^(n + 2) &]^2/121, {n, 0, 20}] (* Eric W. Weisstein, Apr 10 2018 *)
LinearRecurrence[{2,3,6,-1,0,-1}, {1,1,9,25,81,289}, {0, 20}] (* Eric W. Weisstein, Apr 10 2018 *)
LinearRecurrence[{1,1,1},{1,1,1},40]^2 (* Harvey P. Dale, Aug 01 2021 *)

@CachedFunction
def T(n): # A000213
    if (n<3): return 1
    else: return T(n-1) +T(n-2) +T(n-3)
def A141583(n): return T(n)^2
[A141583(n) for n in (0..40)] # G. C. Greubel, Nov 22 2021

a(n) = (A000213(n))^2.
O.g.f.: (1+x)^2*(1-3*x+x^2-x^3)/((1+x+x^2-x^3)*(1-3*x-x^2-x^3)).
a(n) = 2*a(n-1) + 3*a(n-2) + 6*a(n-3) - a(n-4) - a(n-6).

A186190 First digit of tribonacci sequence A000213.

Original entry on oeis.org

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

Offset: 0

Carmine Suriano, Feb 14 2011

Sequence obeys the Benford law about distribution of first digit d in a sequence: P(d) = log_10(1+1/d).

			a(10) = 1 since A000213(10) = 193.

Index entries for sequences related to Benford's law

Cf. A000030, A000213.

t = {1,1,1}; Do[AppendTo[t, t[[-1]] + t[[-2]] + t[[-3]]], {100}]; Table[IntegerDigits[i][[1]], {i, t}]

Offset changed by Georg Fischer, Jun 26 2024

A046734 Period of A000213 mod n.

Original entry on oeis.org

1, 1, 13, 4, 31, 13, 48, 8, 39, 31, 110, 52, 168, 48, 403, 16, 96, 39, 360, 124, 624, 110, 553, 104, 155, 168, 117, 48, 140, 403, 331, 32, 1430, 96, 1488, 156, 469, 360, 2184, 248, 560, 624, 308, 220, 1209, 553, 46, 208, 336, 155, 1248, 168, 52, 117

Offset: 1

David W. Wilson

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000
Nick Hobson, Python program for this sequence

isper(v,startAt=1)=for(k=startAt,#v-3, for(i=1,3,if(v[i]!=v[k+i],next(2)));return(k)); 0
ap(p)=my(v=vector(99),t); v[1]=v[2]=v[3]=1; for(i=4,#v,v[i]=sum(j=i-3,i-1,v[j])%p); while((t=isper(v,if(#v>99,#v/2-2,1)))==0, v=concat(v, vector(#v)); for(i=#v/2+1,#v, v[i]=sum(j=i-3,i-1,v[j])%p));t
ape(p,e)=if(p==2, return(if(e>1,p^e,1))); if(e==1, return(ap(p))); my(pe=p^e, P=ap(p)*p^(e-1), v=vector(P+3)); v[1]=v[2]=v[3]=1; for(i=4,#v,v[i]=sum(j=i-3,i-1,v[j])%pe); isper(v)
a(n)=my(f=factor(n)); lcm(vector(#f~, i, ape(f[i,1],f[i,2]))) \\ Charles R Greathouse IV, Dec 10 2015

A072874 Numbers k such that A000213(k) == 1 (mod k).

Original entry on oeis.org

1, 2, 4, 8, 16, 24, 28, 32, 36, 47, 53, 56, 64, 72, 80, 94, 103, 106, 128, 163, 192, 199, 206, 256, 257, 269, 311, 326, 336, 384, 397, 398, 401, 419, 421, 456, 499, 512, 514, 536, 538, 587, 599, 617, 622, 683, 700, 757, 768, 773, 794, 802, 838, 842, 863, 883, 907

Offset: 1

Benoit Cloitre, Jul 28 2002

Includes A000079. - Robert Israel, Jul 23 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000213, A023173.

m = 1000; t = LinearRecurrence[{1, 1, 1}, {1, 1, 3}, m]; Position[(t - 1)/Range[m], ?IntegerQ] // Flatten (* _Amiram Eldar, May 05 2022 *)

a(1) = 1 inserted by Amiram Eldar, May 05 2022

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)

cp's OEIS Frontend

A056816 Primes in the tribonacci sequence A000213.

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A274432 Products of distinct tribonacci numbers (A000213).

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Mathematica

A046735 Nontrivial (i.e., having no nontrivial factors with this property) integers which do not divide any terms of A000213.

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Maple

Mathematica

PARI

Extensions

A107240 Sum of squares of first n tribonacci numbers (A000213).

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Mathematica

Formula

A157611 Numbers n such that the n-th tribonacci number as defined by A000213 is prime.

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A141583 Squares of tribonacci numbers A000213.

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Magma

Mathematica

Sage

Formula

A186190 First digit of tribonacci sequence A000213.

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A046734 Period of A000213 mod n.

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PARI

A072874 Numbers k such that A000213(k) == 1 (mod k).

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