A000213 -id:A000213 - cp's OEIS frontend

A077957 Powers of 2 alternating with zeros.

1, 0, 2, 0, 4, 0, 8, 0, 16, 0, 32, 0, 64, 0, 128, 0, 256, 0, 512, 0, 1024, 0, 2048, 0, 4096, 0, 8192, 0, 16384, 0, 32768, 0, 65536, 0, 131072, 0, 262144, 0, 524288, 0, 1048576, 0, 2097152, 0, 4194304, 0, 8388608, 0, 16777216, 0, 33554432, 0, 67108864, 0, 134217728, 0, 268435456

Offset: 0

N. J. A. Sloane, Nov 17 2002

Normally sequences like this are not included, since with the alternating 0's deleted it is already in the database.
Inverse binomial transform of A001333. - Paul Barry, Feb 25 2003
"Sloping binary representation" of powers of 2 (A000079), slope=-1 (see A037095 and A102370). - Philippe Deléham, Jan 04 2008
0,1,0,2,0,4,0,8,0,16,... is the inverse binomial transform of A000129 (Pell numbers). - Philippe Deléham, Oct 28 2008
Number of maximal self-avoiding walks from the NW to SW corners of a 3 X n grid.
Row sums of the triangle in A204293. - Reinhard Zumkeller, Jan 14 2012
Pisano period lengths: 1, 1, 4, 1, 8, 4, 6, 1, 12, 8, 20, 4, 24, 6, 8, 1, 16, 12, 36, 8, ... . - R. J. Mathar, Aug 10 2012
This sequence occurs in the length L(n) = sqrt(2)^n of Lévy's C-curve at the n-th iteration step. Therefore, L(n) is the Q(sqrt(2)) integer a(n) + a(n-1)*sqrt(2), with a(-1) = 0. For a variant of this C-curve see A251732 and A251733. - Wolfdieter Lang, Dec 08 2014
a(n) counts walks (closed) on the graph G(1-vertex,2-loop,2-loop). Equivalently the middle entry (2,2) of A^n where the adjacency matrix of digraph is A=(0,1,0;1,0,1;0,1,0). - David Neil McGrath, Dec 19 2014
a(n-2) is the number of compositions of n into even parts. For example, there are 4 compositions of 6 into even parts: (6), (222), (42), and (24). - David Neil McGrath, Dec 19 2014
Also the number of alternately constant compositions of n + 2, ranked by A351010. The alternately strict version gives A000213. The unordered version is A035363, ranked by A000290, strict A035457. - Gus Wiseman, Feb 19 2022
a(n) counts degree n fixed points of GF(2)[x]'s automorphisms. Proof: given a field k, k[x]'s automorphisms are determined by k's automorphisms and invertible affine maps x -> ax + b. GF(2) is rigid and has only one unit so its only nontrivial automorphism is x -> x + 1. For n = 0 we have 1 fixed point, the constant polynomial 1. (Taking the convention that 0 is not a degree 0 polynomial.) For n = 1 we have 0 fixed points as x -> x + 1 -> x are the only degree 1 polynomials. Note that if f(x) is a fixed point, then f(x) + 1 is also a fixed point. Given f(x) a degree n fixed point, we can assume WLOG x | f(x). Applying the automorphism, we then have x + 1 | f(x). Now note that f(x) / (x^2 + x) must be a fixed point, so any fixed point of degree n must either be of the form g(x) * (x^2 + x) or g(x) * (x^2 + x) + 1 for a unique degree n - 2 fixed point g(x). Therefore we have the recurrence relation a(n) = 2 * a(n - 2) as desired. - Keith J. Bauer, Mar 19 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A000079, A077966.
Column k=3 of A219946. - Alois P. Heinz, Dec 01 2012
Cf. A016116 (powers repeated).

Flat(List([0..30],n->[2^n,0])); # Muniru A Asiru, Aug 05 2018

a077957 = sum . a204293_row  -- Reinhard Zumkeller, Jan 14 2012

&cat [[2^n,0]: n in [0..20]]; // Vincenzo Librandi, Apr 03 2018

seq(op([2^n,0]),n=0..100); # Robert Israel, Dec 23 2014

a077957[n_] := Riffle[Table[2^i, {i, 0, n - 1}], Table[0, {n}]]; a077957[29] (* Michael De Vlieger, Dec 22 2014 *)
CoefficientList[Series[1/(1 - 2*x^2), {x,0,50}], x] (* G. C. Greubel, Apr 12 2017 *)
LinearRecurrence[{0, 2}, {1, 0}, 54] (* Robert G. Wilson v, Jul 23 2018 *)
Riffle[2^Range[0,30],0,{2,-1,2}] (* Harvey P. Dale, Jan 06 2022 *)

PARI
```
a(n)=if(n<0||n%2, 0, 2^(n/2))
```

def A077957():
    x, y = -1, 1
    while True:
        yield -x
        x, y = x + y, x - y
a = A077957(); [next(a) for i in range(40)]  # Peter Luschny, Jul 11 2013

G.f.: 1/(1-2*x^2).
E.g.f.: cosh(x*sqrt(2)).
a(n) = (1 - n mod 2) * 2^floor(n/2).
a(n) = sqrt(2)^n*(1+(-1)^n)/2. - Paul Barry, May 13 2003
a(n) = 2*a(n-2) with a(0)=1, a(1)=0. - Jim Singh, Jul 12 2018

A231575 Indices of primes in A001590.

Original entry on oeis.org

4, 5, 7, 9, 25, 29, 49, 79, 1613, 15205

Offset: 1

Robert Price, Nov 18 2013

a(11) > 2*10^5.

Cf. A001590, A231574, A000213, A056816, A157611.

v=[1,0,1]; for(n=4,1e4, if(ispseudoprime(t=v[1]+v[2]+v[3]), print1(n", ")); v=[v[2],v[3],t]) \\ Charles R Greathouse IV, Nov 18 2013

A001590(a(n)) = A231574(n). - Arthur O'Dwyer, 24 Jul 2024

Name clarified by Arthur O'Dwyer, Jul 24 2024

A214727 a(n) = a(n-1) + a(n-2) + a(n-3) with a(0) = 1, a(1) = a(2) = 2.

Original entry on oeis.org

1, 2, 2, 5, 9, 16, 30, 55, 101, 186, 342, 629, 1157, 2128, 3914, 7199, 13241, 24354, 44794, 82389, 151537, 278720, 512646, 942903, 1734269, 3189818, 5866990, 10791077, 19847885, 36505952, 67144914, 123498751, 227149617, 417793282

Offset: 0

Abel Amene, Jul 27 2012

Part of a group of sequences defined by a(0), a(1)=a(2), a(n) = a(n-1) + a(n-2) + a(n-3) which is a subgroup of sequences with linear recurrences and constant coefficients listed in the index.
Note: A000073 (with offset=1), 1 followed by A000073, A000213, A141523, A214727, A214825 to A214831 completely define possible sequences with a(0)=0,1,2...9 and a(1)=a(2)=0,1,2...9 excluding any multiples of these sequences and the trivial case of a(0)=a(1)=a(2)=0.
Note: allowing a(0)=0 and a(1)=a(2)=1,2,3....9 leads to A000073 (with offset=1) and its multiples.
Note: allowing a(0)=1,2,3....9 a(1)=a(2)=0 leads to 1 followed by A000073 and its multiples.
With offset of 6 this sequence is the 8th row of tribonacci array A136175.

			G.f. = 1 + 2*x + 2*x^2 + 5 x^3 + 9*x^4 + 16*x^5 + 30*x^6 + 55*x^7 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
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.
Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Cf. A000213, A000288, A000322, A000383, A060455, A136175, A141036, A141523, A214825-A214831.

a:=[1,2,2];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 23 2019

a214727 n = a214727_list !! n
a214727_list = 1 : 2 : 2 : zipWith3 (\x y z -> x + y + z)
   a214727_list (tail a214727_list) (drop 2 a214727_list)
-- Reinhard Zumkeller, Jul 31 2012

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x-x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 23 2019

LinearRecurrence[{1,1,1},{1,2,2},40] (* Ray Chandler, Dec 08 2013 *)

a(n)=([0,1,0; 0,0,1; 1,1,1]^n*[1;2;2])[1,1] \\ Charles R Greathouse IV, Mar 22 2016

my(x='x+O('x^40)); Vec((1+x-x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 23 2019

((1+x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019

G.f.: (1+x-x^2)/(1-x-x^2-x^3).
a(n) = K(n) -2*T(n+1) + 3*T(n), where K(n) = A001644(n), T(n) = A000073(n+1). - G. C. Greubel, Apr 23 2019
a(n) = Sum_{r root of x^3-x^2-x-1} r^n/(-r^2+2*r+1). - Fabian Pereyra, Nov 20 2024

A231574 Primes in A001590.

Original entry on oeis.org

2, 3, 11, 37, 634061, 7256527, 1424681173049, 123937002926372177911

Offset: 1

Robert Price, Nov 18 2013

nonn

a(9) contains 427 digits and is too large to include here.

a(10) contains 4024 digits and is too large to include here.

Cf. A000213, A001590, A056816, A157611, A231575.

Name clarified by Arthur O'Dwyer, Jul 24 2024

A214899 a(n) = a(n-1) + a(n-2) + a(n-3) with a(0)=2, a(1)=1, a(2)=2.

Original entry on oeis.org

2, 1, 2, 5, 8, 15, 28, 51, 94, 173, 318, 585, 1076, 1979, 3640, 6695, 12314, 22649, 41658, 76621, 140928, 259207, 476756, 876891, 1612854, 2966501, 5456246, 10035601, 18458348, 33950195, 62444144, 114852687, 211247026, 388543857

Offset: 0

Abel Amene, Jul 29 2012

With offset of 5 this sequence is the 4th row of the tribonacci array A136175.
For n>0, a(n) is the number of ways to tile a strip of length n with squares, dominoes, and trominoes, such that there must be exactly one "special" square (say, of a different color) in the first three cells. - Greg Dresden and Emma Li, Aug 17 2024
From Greg Dresden and Jiarui Zhou, Jun 30 2025: (Start)
For n >= 3, a(n) is the number of ways to tile this shape of length n-1 with squares, dominos, and trominos (of length 3):
  ._
  ||____________
  |||_|||_|||
    |_|
As an example, here is one of the a(9) = 173 ways to tile this shape of length 8:
  ._
  | |___________
  || |__|_|___|
    |_|. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Robert Price)
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.
Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Cf. A000073, A000213, A035513, A136175, A141036, A141523.

a:=[2,1,2];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 23 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (2-x-x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 23 2019

LinearRecurrence[{1,1,1},{2,1,2},34] (* Ray Chandler, Dec 08 2013 *)

a(n)=([0,1,0;0,0,1;1,1,1]^n*[2;1;2])[1,1] \\ Charles R Greathouse IV, Jun 11 2015

my(x='x+O('x^40)); Vec((2-x-x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 23 2019

((2-x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019

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

a(n) = K(n) - T(n+1) + T(n), where K(n) = A001644(n), T(n) = A000073(n+1). - G. C. Greubel, Apr 23 2019

A214825 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 3.

Original entry on oeis.org

1, 3, 3, 7, 13, 23, 43, 79, 145, 267, 491, 903, 1661, 3055, 5619, 10335, 19009, 34963, 64307, 118279, 217549, 400135, 735963, 1353647, 2489745, 4579355, 8422747, 15491847, 28493949, 52408543, 96394339, 177296831, 326099713, 599790883, 1103187427

Offset: 0

Abel Amene, Jul 28 2012

Part of a group of sequences defined by a(0), a(1)=a(2), a(n) = a(n-1) + a(n-2) + a(n-3) which is a subgroup of sequences with linear recurrences and constant coefficients listed in the index. See Comments in A214727.

Indranil Ghosh, Table of n, a(n) for n = 0..3770
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.
Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Cf. A000073, A000213, A000288, A000322, A000383, A001644, A060455, A136175, A141036, A141523, A214825-A214831.

a:=[1,3,3];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 23 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+2*x-x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 23 2019

LinearRecurrence[{1,1,1},{1,3,3},40] (* Harvey P. Dale, Oct 05 2013 *)

a(n)=([0,1,0; 0,0,1; 1,1,1]^n*[1;3;3])[1,1] \\ Charles R Greathouse IV, Mar 22 2016

my(x='x+O('x^40)); Vec((1+2*x-x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 23 2019

((1+2*x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019

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

a(n) = K(n) - 2*T(n+1) + 4*T(n), where K(n) = A001644(n), and T(n) = A000073(n+1). - G. C. Greubel, Apr 23 2019

A060455 7th-order Fibonacci numbers with a(0)=...=a(6)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 7, 13, 25, 49, 97, 193, 385, 769, 1531, 3049, 6073, 12097, 24097, 48001, 95617, 190465, 379399, 755749, 1505425, 2998753, 5973409, 11898817, 23702017, 47213569, 94047739, 187339729, 373174033, 743349313, 1480725217

Offset: 0

Frank Ellermann, Apr 08 2001

a(n) = number of runs in polyphase sort using 8 tapes and n-6 phases.

			General formula for k-th order numbers: f(n,k) = f(n-1,k) + ... + f(n-1-k,k) for n > k, otherwise f(n,k) = 1.

N. Wirth, Algorithmen und Datenstrukturen, 1975 (table 2.15 chapter 2.3.4).

Indranil Ghosh, Table of n, a(n) for n = 0..3339 (terms 0..200 from T. D. Noe)
R. L. Gilstad, Polyphase Merge Sort - Advanced Technique, Proc. AFIPS Eastern Jt. Comp. Conf. 18 (1960) 143-148.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1).

For k=1..5 see A000045, A000213, A000288, A000322, A000383.
Cf. A253333, A253318: primes and indices of primes in this sequence.
Cf. A122189 Heptanacci numbers with a(0),...,a(6) = 0,0,0,0,0,0,1.

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(  (1-x^2-2*x^3-3*x^4-4*x^5-5*x^6)/(1-x-x^2-x^3-x^4-x^5-x^6-x^7) )); // G. C. Greubel, Feb 03 2019

A060455 := proc(n) option remember: if n >=0 and n<=6 then RETURN(1) fi: procname(n-1)+procname(n-2)+procname(n-3)+procname(n-4)+procname(n-5)+procname(n-6)+procname(n-7) end;

LinearRecurrence[{1,1,1,1,1,1,1},{1,1,1,1,1,1,1},40] (* Harvey P. Dale, Mar 17 2012 *)

Vec((1-x^2-2*x^3-3*x^4-4*x^5-5*x^6)/(1-x-x^2-x^3-x^4-x^5-x^6-x^7) +O(x^40)) \\ Charles R Greathouse IV, Feb 03 2014

((1-x^2-2*x^3-3*x^4-4*x^5-5*x^6)/(1-x-x^2-x^3-x^4-x^5-x^6-x^7) ).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 03 2019

a(n) = a(n-1) + a(n-2) + ... + a(n-7) for n > 6, a(0)=a(1)=...=a(6)=1.

G.f.: (-1 + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6)/(-1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7). - R. J. Mathar, Oct 11 2011

More terms from James Sellers, Apr 11 2001

A081172 Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = 1, a(2) = 0.

Original entry on oeis.org

1, 1, 0, 2, 3, 5, 10, 18, 33, 61, 112, 206, 379, 697, 1282, 2358, 4337, 7977, 14672, 26986, 49635, 91293, 167914, 308842, 568049, 1044805, 1921696, 3534550, 6501051, 11957297, 21992898, 40451246, 74401441, 136845585, 251698272, 462945298, 851489155

Offset: 0

Harry J. Smith, Apr 19 2003

The name "tribonacci number" is less well-defined than "Fibonacci number". The sequence A000073 (which begins 0, 0, 1) is probably the most important version, but the name has also been applied to A000213, A001590, and A081172. - N. J. A. Sloane, Jul 25 2024
Completes the set of tribonacci numbers starting with 0's and 1's in the first three terms:
0,0,0 A000004;
0,0,1 A000073;
0,1,0 A001590;
0,1,1 A000073 starting at a(1);
1,0,0 A000073 starting at a(-1);
1,0,1 A001590;
1,1,0 this sequence;
1,1,1 A000213.

Harry J. Smith, Table of n, a(n) for n = 0..500
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.
N. G. Voll, Some identities for four term recurrence relations, Fib. Quart., 51 (2013), 268-273.
Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Cf. A000004, A000073, A000213, A001590, A020992.

a:=[1,1,0];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 23 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-2*x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 23 2019

A081172 := proc(n)
    option remember;
    if n <= 2 then
        op(n+1,[1,1,0]) ;
    else
        add(procname(n-i),i=1..3) ;
    end if;
end proc: # R. J. Mathar, Aug 09 2012

LinearRecurrence[{1,1,1}, {1,1,0}, 40] (* Vladimir Joseph Stephan Orlovsky, Jun 07 2011 *)

{ a1=1; a2=1; a3=0; write("b081172.txt",0," ",a1); write("b081172.txt",1," ",a2); write("b081172.txt",2," ",a3); for(n=3,500, a=a1+a2+a3; a1=a2; a2=a3; a3=a; write("b081172.txt",n," ",a) ) } \\ Harry J. Smith, Mar 20 2009

my(x='x+O('x^40)); Vec((1-2*x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 23 2019

((1-2*x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019

From R. J. Mathar, Mar 27 2009: (Start)
G.f.: (1-2*x^2)/(1 - x - x^2 - x^3).
a(n) = A000073(n+2) - 2*A000073(n). (End)

A008937 a(n) = Sum_{k=0..n} T(k) where T(n) are the tribonacci numbers A000073.

Original entry on oeis.org

0, 1, 2, 4, 8, 15, 28, 52, 96, 177, 326, 600, 1104, 2031, 3736, 6872, 12640, 23249, 42762, 78652, 144664, 266079, 489396, 900140, 1655616, 3045153, 5600910, 10301680, 18947744, 34850335, 64099760, 117897840, 216847936, 398845537, 733591314, 1349284788

Offset: 0

N. J. A. Sloane, Alejandro Teruel (teruel(AT)usb.ve)

a(n+1) is the number of n-bit sequences that avoid 1100. - David Callan, Jul 19 2004 [corrected by Kent E. Morrison, Jan 08 2019]. Also the number of n-bit sequences avoiding one of the patterns 1000, 0011, 1110, ... or any binary string of length 4 without overlap at beginning and end. Strings where it is not true are: 1111, 1010, 1001, ... and their bitwise complements. - Alois P. Heinz, Jan 09 2019
Row sums of Riordan array (1/(1-x), x(1+x+x^2)). - Paul Barry, Feb 16 2005
Diagonal sums of Riordan array (1/(1-x)^2, x(1+x)/(1-x)), A104698.
A shifted version of this sequence can be found in Eqs. (4) and (3) on p. 356 of Dunkel (1925) with r = 3. (Equation (3) follows equation (4) in the paper!) The whole paper is a study of the properties of this and other similar sequences indexed by the parameter r. For r = 2, we get a shifted version of A000071. For r = 4, we get a shifted version of A107066. For r = 5, we get a shifted version of A001949. For r = 6, we get a shifted version of A172316. See also the table in A172119. - Petros Hadjicostas, Jun 14 2019
Officially, to match A000073, this should start with a(0)=a(1)=0, a(2)=1. - N. J. A. Sloane, Sep 12 2020
Numbers with tribonacci representation that is a prefix of 100100100100... . - Jeffrey Shallit, Jul 10 2024

			G.f. = x + 2*x^2 + 4*x^3 + 8*x^4 + 15*x^5 + 28*x^6 + 52*x^7 + 96*x^8 + 177*x^9 + ... [edited by _Petros Hadjicostas_, Jun 12 2019]

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 41.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Isha Agarwal, Matvey Borodin, Aidan Duncan, Kaylee Ji, Tanya Khovanova, Shane Lee, Boyan Litchev, Anshul Rastogi, Garima Rastogi, and Andrew Zhao, From Unequal Chance to a Coin Game Dance: Variants of Penney's Game, arXiv:2006.13002 [math.HO], 2020.
Kassie Archer and Noel Bourne, Pattern avoidance in compositions and powers of permutations, arXiv:2505.05218 [math.CO], 2025. See pp. 6, 9.
Erik Bates, Blan Morrison, Mason Rogers, Arianna Serafini, and Anav Sood, A new combinatorial interpretation of partial sums of m-step Fibonacci numbers, arXiv:2503.11055 [math.CO], 2025. See p. 3.
Otto Dunkel, Solutions of a probability difference equation, Amer. Math. Monthly, 32 (1925), 354-370; see pp. 356 and 369.
Jia Huang, A coin flip game and generalizations of Fibonacci numbers, arXiv:2501.07463 [math.CO], 2025. See p. 7.
Thomas Langley, Jeffrey Liese, and Jeffrey Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order, J. Int. Seq. 14 (2011), Article # 11.4.2.
William Verreault, MacMahon Partition Analysis: a discrete approach to broken stick problems, arXiv:2107.10318 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (2,0,0,-1).

Partial sums of A000073. Cf. A000213, A018921, A027084, A077908, A209972.
Row sums of A055216.
Column k = 1 of A140997 and second main diagonal of A140994.

a:=[0,1,1];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Sep 13 2019

a008937 n = a008937_list !! n
a008937_list = tail $ scanl1 (+) a000073_list
-- Reinhard Zumkeller, Apr 07 2012

[ n eq 1 select 0 else n eq 2 select 1 else n eq 3 select 2 else n eq 4 select 4 else 2*Self(n-1)-Self(n-4): n in [1..40] ]; // Vincenzo Librandi, Aug 21 2011

A008937 := proc(n) option remember; if n <= 3 then 2^n else 2*procname(n-1)-procname(n-4) fi; end;
a:= n-> (Matrix([[1,1,0,0], [1,0,1,0], [1,0,0,0], [1,0,0,1]])^n)[4,1]: seq(a(n), n=0..50); # Alois P. Heinz, Jul 24 2008

CoefficientList[Series[x/(1-2x+x^4), {x, 0, 40}], x]
Accumulate[LinearRecurrence[{1,1,1},{0,1,1},40]] (* Harvey P. Dale, Dec 04 2017 *)
LinearRecurrence[{2, 0, 0, -1},{0, 1, 2, 4},40] (* Ray Chandler, Mar 01 2024 *)

{a(n) = if( n<0, polcoeff( - x^3 / (1 - 2*x^3 + x^4) + x * O(x^-n), -n), polcoeff( x / (1 - 2*x + x^4) + x * O(x^n), n))}; /* Michael Somos, Aug 23 2014 */

a(n) = sum(j=0, n\2, sum(k=0, j, binomial(n-2*j,k+1)*binomial(j,k)*2^k)); \\ Michel Marcus, Sep 08 2017

def A008937_list(prec):
    P = PowerSeriesRing(ZZ, 'x', prec)
    x = P.gen().O(prec)
    return (x/(1-2*x+x^4)).list()
A008937_list(40) # G. C. Greubel, Sep 13 2019

a(n) = A018921(n-2) = A027084(n+1) + 1.
a(n) = (A000073(n+2) + A000073(n+4) - 1)/2.
From Mario Catalani (mario.catalani(AT)unito.it), Aug 09 2002: (Start)
G.f.: x/((1-x)*(1-x-x^2-x^3)).
a(n) = 2*a(n-1) - a(n-4), a(0) = 0, a(1) = 1, a(2) = 2, a(3) = 4. (End)
a(n) = 1 + a(n-1) + a(n-2) + a(n-3). E.g., a(11) = 1 + 600 + 326 + 177 = 1104. - Philippe LALLOUET (philip.lallouet(AT)orange.fr), Oct 29 2007
a(n) = term (4,1) in the 4 X 4 matrix [1,1,0,0; 1,0,1,0; 1,0,0,0; 1,0,0,1]^n. - Alois P. Heinz, Jul 24 2008
a(n) = -A077908(-n-3). - Alois P. Heinz, Jul 24 2008
a(n) = (A000213(n+2) - 1) / 2. - Reinhard Zumkeller, Apr 07 2012
a(n) = Sum_{j=0..floor(n/2)} Sum_{k=0..j} binomial(n-2j,k+1) *binomial(j,k)*2^k. - Tony Foster III, Sep 08 2017
a(n) = Sum_{k=0..floor(n/2)} (n-2*k)*hypergeom([-k,-n+2*k+1], [2], 2). - Peter Luschny, Nov 09 2017
a(n) = 2^(n-1)*hypergeom([1-n/4, 1/4-n/4, 3/4-n/4, 1/2-n/4], [1-n/3, 1/3-n/3, 2/3-n/3], 16/27) for n > 0. - Peter Luschny, Aug 20 2020
a(n-1) = T(n) + T(n-3) + T(n-6) + ... + T(2+((n-2) mod 3)), for n >= 4, where T is A000073(n+1). - Jeffrey Shallit, Dec 24 2020

A027023 Tribonacci array: triangular array T read by rows: T(n,0)=1 for n >= 0, T(n,1) = T(n,2n) = 1 for n >= 1, T(n,2)=1 for n >= 2 and for n >= 3, T(n,k) = T(n-1,k-3) + T(n-1,k-2) + T(n-1,k-1) for 3 <= k <= 2n-1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 5, 5, 1, 1, 1, 1, 3, 5, 9, 13, 11, 1, 1, 1, 1, 3, 5, 9, 17, 27, 33, 25, 1, 1, 1, 1, 3, 5, 9, 17, 31, 53, 77, 85, 59, 1, 1, 1, 1, 3, 5, 9, 17, 31, 57, 101, 161, 215, 221, 145, 1, 1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 189, 319, 477, 597, 581, 367, 1

Offset: 0

Clark Kimberling

The n-th row has 2n+1 terms.

			The array begins:
  1;
  1, 1, 1;
  1, 1, 1, 3, 1;
  1, 1, 1, 3, 5, 5,  1;
  1, 1, 1, 3, 5, 9, 13, 11, 1;

R. J. Mathar, Table of n, a(n) for n = 0..1000, replaces Zumkeller's file for new offset.

Columns are essentially constant with values from A000213 (tribonacci numbers).
Diagonals T(n, n+c) are A027024 (c=2), A027025 (c=3), A027026 (c=4).
Diagonals T(n, 2n-c) are A027050 (c=1), A027051 (c=2), A027027 (c=3), A027028 (c=4), A027029 (c=5), A027030 (c=6), A027031 (c=7), A027032 (c=8), A027033 (c=9), A027034 (c=10).
Many other sequences are derived from this one: see A027035 A027036 A027037 A027038 A027039 A027040 A027041 A027042 A027043 A027044 A027045 and A027046 A027047 A027048 A027049.
Other arrays of this type: A027052, A027082, A027113.
Cf. A027907.

T:= function(n,k)
    if k<3 or k=2*n then return 1;
    else return T(n-1, k-3) + T(n-1, k-2) + T(n-1, k-1);
    fi;
  end;
Flat(List([0..10], n-> List([0..2*n], k-> T(n,k) ))); # G. C. Greubel, Nov 04 2019

a027023 n k = a027023_tabf !! (n-1) !! (k-1)
a027023_row n = a027023_tabf !! (n-1)
a027023_tabf = [1] : iterate f [1, 1, 1] where
   f row = 1 : 1 : 1 :
           zipWith3 (((+) .) . (+)) (drop 2 row) (tail row) row ++ [1]
-- Reinhard Zumkeller, Jul 06 2014

T:= proc(n, k) option remember;
      if k<3 or k=2*n  then 1
    else T(n-1, k-3) + T(n-1, k-2) + T(n-1, k-1)
      fi
end proc:
seq(seq(T(n, k), k=0..2*n), n=0..10); # G. C. Greubel, Nov 04 2019

T[n_, 0] := 1; T[n_, 1] := 1; T[n_, k_]/; (k==2n) := 1 /; n >=1; T[n_, 2] := 1; T[n_, k_]/; (k <= 2n-1) := T[n, k]=T[n-1, k-3]+T[n-1, k-2]+T[n-1, k-1]

{T(n, k) = if( k<0 || k>2*n, 0, if( k<3 || k==2*n, 1, T(n-1, k-3) + T(n-1, k-2) + T(n-1,k-1)))}; /* Michael Somos, Feb 14 2004 */

def T(n, k):
    if (k<3 or k==2*n): return 1
    else: return T(n-1, k-3) + T(n-1, k-2) + T(n-1, k-1)
[[T(n, k) for k in (0..2*n)] for n in (0..10)] # G. C. Greubel, Nov 04 2019

Edited by N. J. A. Sloane and Ralf Stephan, Feb 13 2004

Offset corrected to 0. - R. J. Mathar, Jun 24 2020

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