A000073 -id:A000073 - cp's OEIS frontend

A102125 Iccanobirt numbers (15 of 15): a(n) = R(R(a(n-1)) + R(a(n-2)) + R(a(n-3))), where R is the digit reversal function A004086.

0, 0, 1, 1, 2, 4, 7, 31, 42, 44, 18, 941, 472, 405, 729, 5071, 6313, 8675, 90601, 31591, 9853, 11733, 31865, 31149, 736481, 365533, 313416, 3154311, 9834802, 5123383, 7112507, 12796921, 35055832, 19867834, 56610708, 906334841, 561210372

Offset: 0

Jonathan Vos Post and Ray Chandler, Dec 30 2004

Digit reversal variation of tribonacci numbers A000073.

Inspired by Iccanobif numbers: A001129, A014258-A014260.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A102111-A102124.

R:= n-> (s-> parse(cat(s[-i]$i=1..length(s))))(""||n):
a:= proc(n) option remember; `if`(n<3, binomial(n,2),
      R(R(a(n-1)) + R(a(n-2)) + R(a(n-3))))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jun 18 2014

R[n_]:=FromDigits[Reverse[IntegerDigits[n]]];Clear[a];a[0]=0;a[1]=0;a[2]=1;a [n_]:=a[n]=R[R[a[n-1]]+R[a[n-2]]+R[a[n-3]]];Table[a[n], {n, 0, 40}]
rev[n_]:=FromDigits[Reverse[IntegerDigits[n]]]; nxt[{a_, b_, c_}] := {b, c, rev[rev[a] + rev[b] + rev[c]]}; Transpose[NestList[nxt,{0,0,1},40]][[1]] (* Harvey P. Dale, Mar 20 2015 *)
nxt[{a_,b_,c_}]:=With[{ir=IntegerReverse},{b,c,ir[ir[a]+ir[b]+ir[c]]}]; NestList[nxt,{0,0,1},40][[;;,1]] (* Harvey P. Dale, Jul 22 2025 *)

a(n) = A004086(A102117(n)).

A104144 a(n) = Sum_{k=1..9} a(n-k); a(8) = 1, a(n) = 0 for n < 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272, 32512, 64960, 129792, 259328, 518145, 1035269, 2068498, 4132920, 8257696, 16499120, 32965728, 65866496, 131603200, 262947072, 525375999, 1049716729, 2097364960

Offset: 0

Jean Lefort (jlefort.apmep(AT)wanadoo.fr), Mar 07 2005

Sometimes called the Fibonacci 9-step numbers.
For n >= 8, this gives the number of integers written without 0 in base ten, the sum of digits of which is equal to n-7. E.g., a(11) = 8 because we have the 8 numbers: 4, 13, 22, 31, 112, 121, 211, 1111.
The offset for this sequence is fairly arbitrary. - N. J. A. Sloane, Feb 27 2009

T. D. Noe, Table of n, a(n) for n = 0..208
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Taras Goy and Mark Shattuck, Some Toeplitz-Hessenberg Determinant Identities for the Tetranacci Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.6.8.
Kai Wang, Identities for generalized enneanacci numbers, Generalized Fibonacci Sequences (2020).
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1).

Cf. A000045, A000073, A000078, A001591, A001592, A066178, A079262 (Fibonacci n-step numbers).

Cf. A255529 (Indices of primes in this sequence).

for n from 0 to 50 do k(n):=sum((-1)^i*binomial(n-8-9*i,i)*2^(n-8-10*i),i=0..floor((n-8)/10))-sum((-1)^i*binomial(n-9-9*i,i)*2^(n-9-10*i),i=0..floor((n-9)/10)):od:seq(k(n),n=0..50);a:=taylor((z^8-z^9)/(1-2*z+z^(10)),z=0,51);for p from 0 to 50 do j(p):=coeff(a,z,p):od :seq(j(p),p=0..50); # Richard Choulet, Feb 22 2010

a={1, 0, 0, 0, 0, 0, 0, 0, 0}; Table[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s, {n, 50}]
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 1}, 50] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
With[{nn=9},LinearRecurrence[Table[1,{nn}],Join[Table[0,{nn-1}],{1}],50]] (* Harvey P. Dale, Aug 17 2013 *)

a(n)=([0,1,0,0,0,0,0,0,0; 0,0,1,0,0,0,0,0,0; 0,0,0,1,0,0,0,0,0; 0,0,0,0,1,0,0,0,0; 0,0,0,0,0,1,0,0,0; 0,0,0,0,0,0,1,0,0; 0,0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,0,1; 1,1,1,1,1,1,1,1,1]^n*[0;0;0;0;0;0;0;0;1])[1,1] \\ Charles R Greathouse IV, Jun 16 2015

A104144(n,m=9)=(matrix(m,m,i,j,j==i+1||i==m)^n)[1,m] \\ M. F. Hasler, Apr 22 2018

a(n) = Sum_{k=1..9} a(n-k) for n > 8, a(8) = 1, a(n) = 0 for n=0..7.
G.f.: x^8/(1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9). - N. J. A. Sloane, Dec 04 2011
Another form of the g.f. f: f(z) = (z^8-z^9)/(1-2*z+z^(10)), then a(n) = Sum_((-1)^i*binomial(n-8-9*i,i)*2^(n-8-10*i), i=0..floor((n-8)/10))-Sum_((-1)^i*binomial(n-9-9*i,i)*2^(n-9-10*i), i=0..floor((n-9)/10)) with Sum_(alpha(i), i=m..n)=0 for m>n. - Richard Choulet, Feb 22 2010
From N. J. A. Sloane, Dec 04 2011: (Start)
Let b be the smallest root (in magnitude) of g(x) := 1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9, b = 0.50049311828655225605926845999420216157202861343888...
Let c = -b^8/g'(b) = 0.00099310812055463178382193226558248643030626601288701...
Then a(n) is the nearest integer to c/b^n. (End)

Edited by N. J. A. Sloane, Aug 15 2006 and Nov 11 2006
Incorrect formula deleted by N. J. A. Sloane, Dec 04 2011
Name edited by M. F. Hasler, Apr 22 2018

A352103 a(n) is the maximal (or lazy) tribonacci representation of n using a binary system of vectors not containing three consecutive 0's.

Original entry on oeis.org

0, 1, 10, 11, 100, 101, 110, 111, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10010, 10011, 10100, 10101, 10110, 10111, 11001, 11010, 11011, 11100, 11101, 11110, 11111, 100100, 100101, 100110, 100111, 101001, 101010, 101011, 101100, 101101, 101110, 101111, 110010

Offset: 0

Amiram Eldar, Mar 05 2022

Each nonnegative integer has 2 unique representations as sums of distinct positive tribonacci numbers (A000073): 1, 2, 4, 7, 13, 24, ...: the minimal (or greedy, A278038) representation in which there are no 3 consecutive 1's (i.e., no 3 consecutive tribonacci numbers appear in the sum), and the maximal (or lazy) representation of n in which no 3 consecutive 0's appear.

			a(5) = 101 = 4 + 1.
a(6) = 110 = 4 + 2.
a(7) = 111 = 4 + 2 + 1.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000073, A007088, A003796, A278038.

Similar sequences: A104326 (Fibonacci), A130311 (Lucas).

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; trib[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; IntegerDigits[Total[2^(s - 1)], 2]]; a[n_] := Module[{v = trib[n]}, nv = Length[v]; i = 1; While[i <= nv - 3, If[v[[i ;; i + 3]] == {1, 0, 0, 0}, v[[i ;; i + 3]] = {0, 1, 1, 1}; If[i > 3, i -= 4]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 0, FromDigits[v[[i[[1, 1]] ;; -1]]]]]; Array[a, 100, 0]

a(n) = A007088(A003796(n+1)).

A104621 Heptanacci-Lucas numbers.

Original entry on oeis.org

7, 1, 3, 7, 15, 31, 63, 127, 247, 493, 983, 1959, 3903, 7775, 15487, 30847, 61447, 122401, 243819, 485679, 967455, 1927135, 3838783, 7646719, 15231991, 30341581, 60439343, 120393007, 239818559, 477709983, 951581183, 1895515647, 3775799303, 7521257025

Offset: 0

Jonathan Vos Post, Mar 17 2005

This 7th-order linear recurrence is a generalization of the Lucas sequence A000032. Mario Catalani would refer to this is a generalized heptanacci sequence, had he not stopped his series of sequences after A001644 "generalized tribonacci", A073817 "generalized tetranacci", A074048 "generalized pentanacci", A074584 "generalized hexanacci." T. D. Noe and I have noted that each of these has many more primes than the corresponding tribonacci A000073 (see A104576), tetranacci A000288 (see A104577), pentanacci, hexanacci and heptanacci (see A104414). For primes in Heptanacci-Lucas numbers, see A104622. For semiprimes in Heptanacci-Lucas numbers, see A104623.

G. C. Greubel, Table of n, a(n) for n = 0..3300 (terms 0..200 from T. D. Noe)
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Mario Catalani, Polymatrix and Generalized Polynacci Numbers, arXiv:math/0210201 [math.CO], 2002.
C. A. Charalambides, Lucas numbers and polynomials of order k and the length of the longest circular success run, The Fibonacci Quarterly, 29 (1991), 290-297.
Spiros D. Dafnis, Andreas N. Philippou, Ioannis E. Livieris, An Alternating Sum of Fibonacci and Lucas Numbers of Order k, Mathematics (2020) Vol. 9, 1487.
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1).

Cf. A000032, A001644, A073817, A074048, A074584, A104414, A104576, A104577.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (-7+6*x+ 5*x^2+4*x^3+3*x^4+2*x^5+x^6)/(-1+x +x^2+x^3+x^4+x^5+x^6+x^7) )); // G. C. Greubel, Apr 22 2019

A104621 := proc(n)
    option remember;
    if n <=6 then
        op(n+1,[7, 1, 3, 7, 15, 31, 63])
    else
        add(procname(n-i),i=1..7) ;
    end if;
end proc: # R. J. Mathar, Mar 26 2015

a[0]=7; a[1]=1; a[2]=3; a[3]=7; a[4]=15; a[5]=31; a[6]=63; a[n_]:= a[n]= a[n-1]+a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]+a[n-7]; Table[a[n], {n,0,40}] (* Robert G. Wilson v, Mar 17 2005 *)
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1}, {7, 1, 3, 7, 15, 31, 63}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)

my(x='x+O('x^40)); Vec((-7+6*x+5*x^2+4*x^3+3*x^4+2*x^5+x^6)/(-1+x +x^2+x^3+x^4+x^5+x^6+x^7)) \\ G. C. Greubel, Dec 18 2017

polsym(polrecip(1-x-x^2-x^3-x^4-x^5-x^6-x^7), 40) \\ G. C. Greubel, Apr 22 2019

((-7+6*x+5*x^2+4*x^3+3*x^4+2*x^5+x^6)/(-1+x +x^2+x^3+x^4+x^5+x^6 +x^7)).series(x, 41).coefficients(x, sparse=False) # G. C. Greubel, Apr 22 2019

a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) + a(n-6) + a(n-7); a(0) = 7, a(1) = 1, a(2) = 3, a(3) = 7, a(4) = 15, a(5) = 31, a(6) = 63.
From R. J. Mathar, Nov 16 2007: (Start)
G.f.: (7 - 6*x - 5*x^2 - 4*x^3 - 3*x^4 - 2*x^5 - x^6)/(1 - x - x^2 - x^3 - x^4 - x^5 - x^6 - x^7).
a(n) = 7*A066178(n) - 6*A066178(n-1) - 5*A066178(n-2) - ... - 2*A066178(n-5) - A066178(n-6) if n >= 6. (End)

A113153 Sum of the first n nonzero tribonacci numbers, in ascending order, as bases, with the same, in descending order, as exponents.

Original entry on oeis.org

1, 2, 4, 8, 17, 54, 472, 27216, 84738887, 299164114847940, 311903053042108587337426568, 5846720173185251353387753850814872871131756204168

Offset: 1

Jonathan Vos Post, Jan 04 2006

			For the tribonacci sequence starting t(1)=t(2)=1, t(3)=2, that is, the nonzero terms of A000073:
a(1) = t(1)^t(1) = 1^1 = 1.
a(2) = t(1)^t(2) + t(2)^t(1) = 1^1 + 1^1 = 2.
a(3) = t(1)^t(3) + t(2)^t(2) + t(3)^t(1) = 1^2 + 1^1 + 2^1 = 4.
a(4) = t(1)^t(4) + t(2)^t(3) + t(3)^t(2) + t(4)^t(1) = 1^4 + 1^2 + 2^1 + 4^1 = 8.
a(5) = 1^7 + 1^4 + 2^2 + 4^1 + 7^1 = 17.
a(6) = 1^13 + 1^7 + 2^4 + 4^2 + 7^1 + 13^1 = 54.

Cf. A000073.

a[0] = a[1] = 0 ; a[2] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2] + a[n - 3]; Table[Sum[a[k + 2]^(a[n - k + 1]), {k, 1, n}], {n, 1, 10}] (* G. C. Greubel, May 18 2017 *)

a(n) = Sum_{i=1..n} A000073(i+1)^A000073(n-i+2).

Name clarified by Arthur O'Dwyer, Jul 24 2024

A135491 Number of ways to toss a coin n times and not get a run of four.

Original entry on oeis.org

1, 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, 1231386948, 2264873704

Offset: 0

James R FitzSimons (cherry(AT)getnet.net), Feb 07 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Elena Barcucci, Antonio Bernini, Stefano Bilotta, Renzo Pinzani, Non-overlapping matrices, arXiv:1601.07723 [cs.DM], 2016. See column 2 of Table 2 p. 11.
Elena Barcucci, Antonio Bernini, Stefano Bilotta and Renzo Pinzani, Non-overlapping matrices, Theoretical Computer Science, Vol. 658, Part A (2017), 36-45.
B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, On Curling Numbers of Integer Sequences, arXiv:1212.6102 [math.CO], 2012-2013.
B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, On Curling Numbers of Integer Sequences, Journal of Integer Sequences, Vol. 16 (2013), Article 13.4.3.
Emrah Kılıç, Talha Arıkan, Evaluation of Hessenberg determinants with recursive entries: generating function approach, Filomat (2017) Vol. 31, Issue 15, pp. 4945-4962.
A. V. Zharkova, Inaccesible States in Dynamic Systems Associated with Paths and Cycles, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 11 (2011), 116-122.
Index entries for linear recurrences with constant coefficients, signature (1, 1, 1).

Cf. A000073. Column 2 of A265624. Cf. A135492, A135493, A000213, A058265.

LinearRecurrence[{1, 1, 1}, {1, 2, 4, 8}, 36] (* Vladimir Joseph Stephan Orlovsky, Jul 23 2011; first term 1 added by Georg Fischer, Apr 02 2019 *)

Vec(1-2*x*(1+x+x^2)/(-1+x+x^2+x^3) + O(x^100)) \\ Altug Alkan, Dec 10 2015

a(n) = 2*A000073(n+2) for n > 0.
a(n) = a(n-1) + a(n-2) + a(n-3) for n > 3.
G.f.: -(x+1)*(x^2+1)/(x^3+x^2+x-1).
a(n) = nearest integer to b*c^n, where b = 1.2368... and c = 1.839286755... is the real root of x^3-x^2-x-1 = 0. See A058265. - N. J. A. Sloane, Jan 06 2010
G.f.: (1-x^4)/(1-2*x+x^4) and generally to "not get a run of k" (1-x^k)/(1-2*x+x^k). - Geoffrey Critzer, Feb 01 2012
G.f.: Q(0)/x^2 - 2/x- 1/x^2, where Q(k) = 1 + (1+x)*x^2 + (2*k+3)*x - x*(2*k+1 +x+x^2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 04 2013
a(n) = A000213(n+3) - A000213(n+2), n>=1. - Peter M. Chema, Jan 11 2017.

More terms from Robert G. Wilson v, Feb 10 2008

a(0)=1 prepended by Alois P. Heinz, Dec 10 2015

A159741 a(n) = 8*(2^n - 1).

Original entry on oeis.org

8, 24, 56, 120, 248, 504, 1016, 2040, 4088, 8184, 16376, 32760, 65528, 131064, 262136, 524280, 1048568, 2097144, 4194296, 8388600, 16777208, 33554424, 67108856, 134217720, 268435448, 536870904, 1073741816, 2147483640, 4294967288, 8589934584, 17179869176, 34359738360

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Apr 20 2009

Fifth diagonal of the array which contains m-acci numbers in the m-th row.
The base array is constructed from m-acci numbers starting each with 1, 1, and 2 and filling one row of the table (see the examples).
The main and the upper diagonals of the table are the powers of 2, A000079.
The first subdiagonal is essentially A000225, followed by essentially A036563.
The next subdiagonal is this sequence here, followed by A159742, A159743, A159744, A159746, A159747, A159748.
a(n) written in base 2: 1000, 11000, 111000, 1111000, ..., i.e., n times 1 and 3 times 0 (A161770). - Jaroslav Krizek, Jun 18 2009
Also numbers for which n^8/(n+8) is an integer. - Vicente Izquierdo Gomez, Jan 03 2013

			From _R. J. Mathar_, Apr 22 2009: (Start)
The base table is
.1..1....1....1....1....1....1....1....1....1....1....1....1....1
.1..1....1....1....1....1....1....1....1....1....1....1....1....1
.2..2....2....2....2....2....2....2....2....2....2....2....2....2
.0..2....3....4....4....4....4....4....4....4....4....4....4....4
.0..2....5....7....8....8....8....8....8....8....8....8....8....8
.0..2....8...13...15...16...16...16...16...16...16...16...16...16
.0..2...13...24...29...31...32...32...32...32...32...32...32...32
.0..2...21...44...56...61...63...64...64...64...64...64...64...64
.0..2...34...81..108..120..125..127..128..128..128..128..128..128
.0..2...55..149..208..236..248..253..255..256..256..256..256..256
.0..2...89..274..401..464..492..504..509..511..512..512..512..512
.0..2..144..504..773..912..976.1004.1016.1021.1023.1024.1024.1024
.0..2..233..927.1490.1793.1936.2000.2028.2040.2045.2047.2048.2048
.0..2..377.1705.2872.3525.3840.3984.4048.4076.4088.4093.4095.4096
Columns: A000045, A000073, A000078, A001591, A001592 etc. (End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000079, A000225, A036563, A159742, A159743, A159744, A159746, A159747, A159748.

Cf. A000918, A028399, A161770, A173787, A175161, A175166.

[8*(2^n -1): n in [1..50]]; // G. C. Greubel, May 22 2018

T := proc(n,m) option remember ; if n < 0 then 0; elif n <= 1 then 1; elif n = 2 then 2; else add(procname(n-i,m),i=1..m) ; fi: end: A159741 := proc(n) T(n+4,n+1) ; end: seq(A159741(n),n=1..40) ; # R. J. Mathar, Apr 22 2009

Table[8(2^n-1),{n,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{3,-2},{8,24},30] (* Harvey P. Dale, Jan 01 2019 *)

a(n)=8*(2^n-1) \\ Charles R Greathouse IV, Sep 24 2015

From R. J. Mathar, Apr 22 2009: (Start)
a(n) = 3*a(n-1) - 2*a(n-2).
a(n) = 8*(2^n-1).
G.f.: 8*x/((2*x-1)*(x-1)). (End)
From Jaroslav Krizek, Jun 18 2009: (Start)
a(n) = Sum_{i=3..(n+2)} 2^i.
a(n) = Sum_{i=1..n} 2^(i+2).
a(n) = a(n-1) + 2^(n+2) for n >= 2. (End)
a(n) = A173787(n+3,3) = A175166(2*n)/A175161(n). - Reinhard Zumkeller, Feb 28 2010
From Elmo R. Oliveira, Jun 15 2025: (Start)
E.g.f.: 8*exp(x)*(exp(x) - 1).
a(n) = 8*A000225(n) = 4*A000918(n+1) = 2*A028399(n+2). (End)

More terms from R. J. Mathar, Apr 22 2009
Edited by Al Hakanson (hawkuu(AT)gmail.com), May 11 2009
Comments claiming negative entries deleted by R. J. Mathar, Aug 24 2009

A050231 a(n) is the number of n-tosses having a run of 3 or more heads for a fair coin (i.e., probability is a(n)/2^n).

Original entry on oeis.org

0, 0, 1, 3, 8, 20, 47, 107, 238, 520, 1121, 2391, 5056, 10616, 22159, 46023, 95182, 196132, 402873, 825259, 1686408, 3438828, 6999071, 14221459, 28853662, 58462800, 118315137, 239186031, 483072832, 974791728, 1965486047, 3960221519, 7974241118, 16047432332, 32276862265

Offset: 1

Eric W. Weisstein

a(n-1) is the number of compositions of n with at least one part >= 4. - Joerg Arndt, Aug 06 2012

W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed. New York: Wiley, p. 300, 1968.

T. D. Noe, Table of n, a(n) for n = 1..300
David Broadhurst, Multiple Landen values and the tribonacci numbers, arXiv:1504.05303 [hep-th], 2015.
Simon Cowell, A Formula for the Reliability of a d-dimensional Consecutive-k-out-of-n:F System, arXiv preprint arXiv:1506.03580 [math.CO], 2015.
Erich Friedman, Illustration of initial terms
T. Langley, J. Liese, and J. Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order, J. Int. Seq. 14 (2011) # 11.4.2.
Eric Weisstein's World of Mathematics, Run
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1,-2).

Cf. A000073, A008466, A050232, A050233.

Column 4 of A109435.

R:= PowerSeriesRing(Integers(), 60);
[0,0] cat Coefficients(R!( x^3/((1-2*x)*(1-x-x^2-x^3)) )); // G. C. Greubel, Jun 01 2025

LinearRecurrence[{3, -1, -1, -2}, {0, 0, 1, 3}, 50] (* David Nacin, Mar 07 2012 *)

concat([0,0], Vec(1/(1-2*x)/(1-x-x^2-x^3)+O(x^99))) \\ Charles R Greathouse IV, Feb 03 2015

def a(n, adict={0:0, 1:0, 2:1, 3:3}):
    if n in adict:
        return adict[n]
    adict[n]=3*a(n-1)-a(n-2)-a(n-3)-2*a(n-4)
    return adict[n] # David Nacin, Mar 07 2012

def A050231_list(prec):
    P.= PowerSeriesRing(QQ, prec)
    return P( x^3/((1-2*x)*(1-x-x^2-x^3)) ).list()
a=A050231_list(41); a[1:] # G. C. Greubel, Jun 01 2025

a(n) = 2^n - tribonacci(n+3), see A000073. - Vladeta Jovovic, Feb 23 2003
G.f.: x^3/((1-2*x)*(1-x-x^2-x^3)). - Geoffrey Critzer, Jan 29 2009
a(n) = 2 * a(n-1) + 2^(n-4) - a(n-4) since we can add T or H to a sequence of n-1 flips which has HHH, and H to one which ends in THH and does not have HHH among the first (n-4) flips. - Toby Gottfried, Nov 20 2010
a(n) = 3*a(n-1) - a(n-2) - a(n-3) - 2*a(n-4), a(0)=0, a(1)=0, a(2)=1, a(3)=3. - David Nacin, Mar 07 2012

A057597 a(n) = -a(n-1) - a(n-2) + a(n-3), a(0)=0, a(1)=0, a(2)=1.

Original entry on oeis.org

0, 0, 1, -1, 0, 2, -3, 1, 4, -8, 5, 7, -20, 18, 9, -47, 56, 0, -103, 159, -56, -206, 421, -271, -356, 1048, -963, -441, 2452, -2974, 81, 5345, -8400, 3136, 10609, -22145, 14672, 18082, -54899, 51489, 21492, -127880, 157877, -8505, -277252, 443634, -174887, -545999, 1164520, -793408, -917111

Offset: 0

N. J. A. Sloane, Oct 06 2000

Reflected (A074058) tribonacci numbers A000073: A000073(n) = a(1-n).
There is an alternative way to produce this sequence, from A000073, which is 0,0,1,1,2,4,7,13,24,44,... Call this {b(n)}. Taking x1 = (b(2))^2 - b(1)*b(3) = 0; x2 = (b(3))^2 - b(2)*b(4) = 1; x3 = (b(4))^2 - b(3)*b(5) = -1; x4 = 0, x5 = 2, we generate (0),0,1,-1,0,2,-3,1. - John McNamara, Jan 02 2004
Pisano period lengths: 1, 4, 13, 8, 31, 52, 48, 16, 39, 124, 110, 104, 168, 48, 403, 32, 96, 156, 360, 248, ... - R. J. Mathar, Aug 10 2012
The negative powers of the tribonacci constant t = A058265 are t^(-n) = a(n+1)*t^2 + b(n)*t + a(n+2)*1, for n >= 0, with b(n) = A319200(n) = -(a(n+1) - a(n)), for n >= 0. 1/t =  t^2 - t - 1 = A192918. See the example in A319200 for the first powers. - Wolfdieter Lang, Oct 23 2018

Petho Attila, Posting to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Oct 06 2000.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Milan Janjic, Recurrence Relations and Determinants, arXiv preprint arXiv:1112.2466 [math.CO], 2011.
Milan Janjic, Determinants and Recurrence Sequences, Journal of Integer Sequences, 15 (2012), Article 12.3.5. - From _N. J. A. Sloane_, Sep 16 2012.
Ronald C. King, Generating functions for some series of characters of classical Lie groups, arXiv:2303.00576 [math.CO], 2023, p. 9.
A. G. Shannon, H. M. Srivastava, and József Sàndor, Towards a new generalized Simson's identity, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 479-490. See p. 482.
Index entries for linear recurrences with constant coefficients, signature (-1,-1,1).

Cf. A000073, A058265, A319200. First differences of A077908.

a:=[0,0,1];;  for n in [4..55] do a[n]:=-a[n-1]-a[n-2]+a[n-3]; od; a; # Muniru A Asiru, Oct 23 2018

a057597 n = a057597_list !! n
a057597_list = 0 : 0 : 1 : zipWith3 (\x y z -> - x - y + z)
               (drop 2 a057597_list) (tail a057597_list) a057597_list
-- Reinhard Zumkeller, Oct 07 2012

seq(coeff(series(x^2/(1+x+x^2-x^3),x,n+1), x, n), n = 0 .. 50); # Muniru A Asiru, Oct 23 2018

CoefficientList[Series[x^2/(1+x+x^2-x^3), {x, 0, 50}], x]

{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 */

G.f.: x^2/(1+x+x^2-x^3).
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
G.f. -x*T(1/x), where T is the g.f. of A000073. - Wolfdieter Lang, Oct 26 2018

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A048888 a(n) = Sum_{m=1..n} T(m,n+1-m), array T as in A048887.

Original entry on oeis.org

0, 1, 2, 4, 7, 13, 23, 42, 76, 139, 255, 471, 873, 1627, 3044, 5718, 10779, 20387, 38673, 73561, 140267, 268065, 513349, 984910, 1892874, 3643569, 7023561, 13557019, 26200181, 50691977, 98182665, 190353369, 369393465, 717457655

Offset: 0

Clark Kimberling

nonn

From Marc LeBrun, Dec 12 2001: (Start)
Define a "numbral arithmetic" by replacing addition with binary bitwise inclusive-OR (so that [3] + [5] = [7] etc.) and multiplication becomes shift-&-OR instead of shift-&-add (so that [3] * [3] = [7] etc.). [d] divides [n] means there exists an [e] with [d] * [e] = [n]. For example the six divisors of [14] are [1], [2], [3], [6], [7] and [14]. Then it appears that this sequence gives the number of proper divisors of [2^n-1]. Conjecture confirmed by Richard C. Schroeppel, Dec 14 2001. (End)
The number of "prime endofunctions" on n points, meaning the cardinality of the subset of the A001372(n) mappings (or mapping patterns) up to isomorphism from n (unlabeled) points to themselves (endofunctions) which are neither the sum of prime endofunctions (i.e., whose disjoint connected components are prime endofunctions) nor the categorical product of prime endofunctions. The n for which a(n) is prime (n such that the number of prime endofunctions on n points is itself prime) are 2, 4, 5, 6, 9, 13, 19, ... - Jonathan Vos Post, Nov 19 2006
For n>=1, compositions p(1)+p(2)+...+p(m)=n such that p(k)<=p(1)+1, see example. - Joerg Arndt, Dec 28 2012

			From _Joerg Arndt_, Dec 28 2012: (Start)
There are a(6)=23 compositions p(1)+p(2)+...+p(m)=6 such that p(k)<=p(1)+1:
[ 1]  [ 1 1 1 1 1 1 ]
[ 2]  [ 1 1 1 1 2 ]
[ 3]  [ 1 1 1 2 1 ]
[ 4]  [ 1 1 2 1 1 ]
[ 5]  [ 1 1 2 2 ]
[ 6]  [ 1 2 1 1 1 ]
[ 7]  [ 1 2 1 2 ]
[ 8]  [ 1 2 2 1 ]
[ 9]  [ 2 1 1 1 1 ]
[10]  [ 2 1 1 2 ]
[11]  [ 2 1 2 1 ]
[12]  [ 2 1 3 ]
[13]  [ 2 2 1 1 ]
[14]  [ 2 2 2 ]
[15]  [ 2 3 1 ]
[16]  [ 3 1 1 1 ]
[17]  [ 3 1 2 ]
[18]  [ 3 2 1 ]
[19]  [ 3 3 ]
[20]  [ 4 1 1 ]
[21]  [ 4 2 ]
[22]  [ 5 1 ]
[23]  [ 6 ]
(End)

D. Applegate, M. LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial Interpretations, J. Integer Seq., Vol. 9 (2006), Article 06.3.1.

Cf. A007059.

Cf. A000312, A001372, A002861, A006961, A001373, A054050, A054745, A125024.

N = 66;  x = 'x + O('x^N);
gf = sum(n=0,N,  (1-x^n)*x^n/(1-2*x+x^(n+1)) ) + 'c0;
v = Vec(gf);  v[1]-='c0;  v
/* Joerg Arndt, Apr 14 2013 */

G.f.: Sum_{k>0} x^k*(1-x^k)/(1-2*x+x^(k+1)). - Vladeta Jovovic, Feb 25 2003

a(m) = Sum_{ n=2..m+1 } Fn(m) where Fn is a Fibonacci n-step number (Fibonacci, tetranacci, etc.) indexed as in A000045, A000073, A000078. - Gerald McGarvey, Sep 25 2004

cp's OEIS Frontend

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A104144 a(n) = Sum_{k=1..9} a(n-k); a(8) = 1, a(n) = 0 for n < 8.

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A352103 a(n) is the maximal (or lazy) tribonacci representation of n using a binary system of vectors not containing three consecutive 0's.

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A104621 Heptanacci-Lucas numbers.

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A113153 Sum of the first n nonzero tribonacci numbers, in ascending order, as bases, with the same, in descending order, as exponents.

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A135491 Number of ways to toss a coin n times and not get a run of four.

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A159741 a(n) = 8*(2^n - 1).

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