A000931 -id:A000931 - cp's OEIS frontend

A264569 T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having directed index change 1,0 1,1 0,-1 or -1,1.

1, 1, 1, 1, 2, 1, 2, 4, 4, 2, 2, 8, 10, 8, 2, 4, 24, 44, 31, 16, 3, 4, 64, 143, 192, 79, 32, 4, 7, 160, 633, 1130, 888, 224, 64, 5, 9, 384, 2172, 8356, 7808, 4104, 646, 128, 7, 13, 960, 8409, 47571, 96429, 57265, 18540, 1784, 256, 9, 18, 2432, 32046, 305844, 868613

Offset: 1

R. H. Hardin, Nov 17 2015

Table starts
.1...1.....1.......2.........2...........4.............4...............7
.1...2.....4.......8........24..........64...........160.............384
.1...4....10......44.......143.........633..........2172............8409
.2...8....31.....192......1130........8356.........47571..........305844
.2..16....79.....888......7808.......96429........868613.........8968735
.3..32...224....4104.....57265.....1133040......16284544.......273793368
.4..64...646...18540....403872....13182464.....298587595......8004883334
.5.128..1784...85752...2873739...152082304....5442431797....235814997396
.7.256..5010..389340..20432891..1755041376...99386232806...6882481295560
.9.512.14026.1787832.144677017.20212718576.1803702025944.200087118615516

			Some solutions for n=4 k=4
..1..2..3..4..8....1..5..6..4..8....1..5..3..4..8....1..2..3..4..8
..0..7.11..9.13....0..7..2..9..3....0..7..2..9.13....0.10.11.12.13
..5..6.16.17.18...11.15.16.17.18...11..6.16.17.18....5..6..7.14..9
.10.20.21.12.14...10.20.12.13.14...10.20.21.12.14...16.20.21.19.23
.15.22.23.24.19...21.22.23.24.19...15.22.23.24.19...15.22.17.24.18

R. H. Hardin, Table of n, a(n) for n = 1..179

Column 1 is A000931(n+4).
Column 2 is A000079(n-1).
Row 1 is A253412(n-2).

Empirical for column k:
k=1: a(n) = a(n-2) +a(n-3)
k=2: a(n) = 2*a(n-1)
k=3: [order 15]
k=4: a(n) = 18*a(n-2) +36*a(n-3) -45*a(n-4) -216*a(n-5) -243*a(n-6) for n>7
k=5: [order 84]
k=6: [order 36] for n>40
Empirical for row n:
n=1: a(n) = a(n-2) +a(n-3) +a(n-4) -a(n-6)
n=2: a(n) = 2*a(n-1) +8*a(n-4)
n=3: [order 70]
n=4: [order 56]

A023434 Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-4).

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 27, 36, 48, 64, 85, 113, 150, 199, 264, 350, 464, 615, 815, 1080, 1431, 1896, 2512, 3328, 4409, 5841, 7738, 10251, 13580, 17990, 23832, 31571, 41823, 55404, 73395, 97228, 128800, 170624, 226029, 299425, 396654, 525455

Offset: 0

N. J. A. Sloane

Limit_{n->infinity} a(n)/a(n-1) = positive root of 1+x-x^3 (smallest Pisot-Vijayaraghavan number, A060006). - Gerald McGarvey, Sep 19 2004
a(n) is the number of distinct even run-types taken over nonempty subsets of [n+1]. The run-type of a set of positive integers is the sequence of lengths when the set is decomposed into maximal runs of consecutive integers and it is even if all its entries are even. For example, the set {2,3,5,6,9,10,11} has run-type (2,2,3) and a(6)=6 counts (2),(4),(6),(2,2),(2,4),(4,2). - David Callan, Jul 14 2006
Partial sums of the sequence obtained by deleting the first 2 terms of A000931. Example: 0+1+0+1+1 = 3 = a(4). - David Callan, Jul 14 2006
One less than the sequence obtained by deleting the first 7 terms of A000931. - Ira M. Gessel, May 02 2007
This sequence counts ordered partitions of (n-1) into parts less than or equal to 3, in which the order of 1's are unimportant. Alternately, the order of 2's and 3's are important (see example). - David Neil McGrath, Apr 26 2015
Interleaving of A289692 and A077855. - Bruce J. Nicholson, Apr 09 2018

			G.f. = x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 11*x^8 + ...
a(7)=8, with (n-1)=6. The partially ordered partitions of 6 are (33),(321,312,132=one),(231,213,123=one),(3111,1311,1131,1113=one),(222),(2211,1122,1221,2112,1212,2121=one),(21111,12111,11211,11121,11112=one),(111111). - _David Neil McGrath_, Apr 26 2015

Robert Israel, Table of n, a(n) for n = 0..7360
O. Bouillot, The Algebra of Multitangent Functions, arXiv:1404.0992 [math.NT], 2014.
O. Bouillot, The Algebra of Multitangent Functions, Journal of Algebra, Volume 410, 15 July 2014, Pages 148-238.
J. H. E. Cohn, Letter to the editor, Fib. Quart. 2 (1964), 108.
V. E. Hoggatt, Jr. and D. A. Lind, The dying rabbit problem, Fib. Quart. 7 (1969), 482-487.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1070
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1).

Cf. A000931, A054405, A060006, A077905.

Cf. A289692, A077855.

[0,1] cat [ n le 4 select (n) else Self(n-1)+Self(n-2)-Self(n-4): n in [1..45] ]; // Vincenzo Librandi, Apr 27 2015

f:= gfun:-rectoproc({a(n)=a(n-1)+a(n-2)-a(n-4),seq(a(i)=[0,1,1,2][i+1],i=0..3)},a(n),remember):
seq(f(i),i=0..100); # Robert Israel, May 04 2015

a[ n_] := If[ n < 0, SeriesCoefficient[ -x^3 / (1 - x^2 - x^3 + x^4), {x, 0, -n}], SeriesCoefficient[ x / (1 - x - x^2 + x^4), {x, 0, n}]]; (* Michael Somos, Nov 29 2013 *)
LinearRecurrence[{1, 1, 0, -1}, {0, 1, 1, 2}, 50] (* Vincenzo Librandi, Apr 27 2015 *)

{a(n) = polcoeff( if( n<0, -x^3 / (1 - x^2 - x^3 + x^4), x / (1 - x - x^2 + x^4)) + x * O(x^abs(n)), abs(n))}; /* Michael Somos, Nov 29 2013 */

x='x+O('x^99); concat(0, Vec(x/((1-x)*(1-x^2-x^3)))) \\ Altug Alkan, Apr 09 2018

a(n) = A000931(n+7)-1.
a(0)=0, a(1)=1, a(2)=1 then for n>2 a(n)=ceiling(r*a(n-1)) where r is the positive root of x^3-x-1=0. - Benoit Cloitre, Jun 19 2004
G.f.: x/((1-x)*(1-x^2-x^3)). - Jon Perry, Jul 04 2004
For n>2 a(n) = floor(sqrt(a(n-3)*a(n-2) + a(n-2)*a(n-1) + a(n-1)*a(n-3))) + 1. - Gerald McGarvey, Sep 19 2004
a(n) = Sum_{k=1..floor((n+2)/3)} binomial(floor((n+2-k)/2),k). This formula counts even run-types by length. - David Callan, Jul 14 2006
a(n) = a(n-2) + a(n-3) + 1. - Mark Dols, Feb 01 2010
a(n) + a(n+1) = A054405(n). Partial sums is A054405. - Michael Somos, Dec 01 2013
a(-3-n) = -A077905(n) for all n in Z. - Michael Somos, Sep 25 2014

A053088 a(n) = 3a(n-2) + 2a(n-3) for n > 2, a(0)=1, a(1)=0, a(2)=3.

Original entry on oeis.org

1, 0, 3, 2, 9, 12, 31, 54, 117, 224, 459, 906, 1825, 3636, 7287, 14558, 29133, 58248, 116515, 233010, 466041, 932060, 1864143, 3728262, 7456549, 14913072, 29826171, 59652314, 119304657, 238609284, 477218599, 954437166, 1908874365

Offset: 0

Pauline Gorman (pauline(AT)gorman65.freeserve.co.uk), Feb 26 2000

Growth of happy bug population in GCSE math course work assignment.
The generalized (3,2)-Padovan sequence p(3,2;n). See the W. Lang link under A000931. - Wolfdieter Lang, Jun 25 2010
With offset 1: a(n) = -2^n*Sum_{k=0..n} k^p*q^k for p=1, q=-1/2. See also A232603 (p=2, q=-1/2), A232604 (p=3, q=-1/2). - Stanislav Sykora, Nov 27 2013
From Paul Curtz, Nov 02 2021 (Start)
a(n-2) difference table (from 0, 0, a(n)):
    0    0    1    0    3    2    9    12    31    54  ...
    0    1   -1    3   -1    7    3    19    23    63  ...
    1   -2    4   -4    8   -4   16     4    40    44  ...
   -3    6   -8   12  -12   20  -12    36     4    84  ...
    9  -14   20  -24   32  -32   48   -32    80     0  ...
  -23   34  -44   56  -64   80  -80   112   -80   176  ...
   57  -78  100 -120  144 -160  192  -192   256  -192  ...
   ... .
The signature is valid for every row.
a(n-2) + a(n-1) = A001045(n).
a(n-2) + a(n+1) = A062510(n) = 3*A001045(n).
a(n-2) + a(n+3) = see A144472(n+1).
Second subdiagonal: 1, 6, 20, 56, 144, 352, ... = A014480(n).
First subdiagonal: -A036895(n) = -2*A001787(n).
Main diagonal: A001787(n) = -first and -third upper diagonals.
Second, fourth and fifth upper diagonals: A001792(n), A045891(n+2) and A172160(n+1). (End)

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Iwan Duursma, Xiao Li, and Hsin-Po Wang, Multilinear Algebra for Distributed Storage, arXiv:2006.08911 [cs.IT], 2020.
Index entries for linear recurrences with constant coefficients, signature (0,3,2).

Cf. A232603, A232604.

Cf. A001045, A001787, A001792, A014480, A036895, A062510, A045891, A172160.

CoefficientList[Series[1/(1 - 3 x^2 - 2 x^3), {x, 0, 32}], x] (* Michael De Vlieger, Sep 30 2019 *)

c(n)=(2^(n+1)-(-1)^n*(3*n+2))/9; a(n)=c(n+1); \\ Stanislav Sykora, Nov 27 2013

G.f.: 1 / (1-3*x^2-2*x^3).
With offset 1: a(1)=1; a(n) = 2*a(n-1) - (-1)^n*n; a(n) = (1/9)*(2^(n+1) - (-1)^n*(3*n+2)). - Benoit Cloitre, Nov 02 2002
a(n) = Sum_{k=0..floor(n/2)} A078008(n-2k). - Paul Barry, Nov 24 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(k, n-2k)*3^k*(2/3)^(n-2k). - Paul Barry, Oct 16 2004
a(n) = Sum_{k=0..n} A078008(k)*(1 - (-1)^(n+k-1))/2. - Paul Barry, Apr 16 2005
a(n) = ( 2^(n+2) + (-1)^n*(3*n+5) )/9 (see also the B. Cloitre comment above). From the o.g.f. 1/(1-3*x^2-2*x^3) = 1/((1-2*x)*(1+x)^2) = (3/(1+x)^2 + 2/(1+x) + 4/(1-2*x))/9. - Wolfdieter Lang, Jun 25 2010
From Wolfdieter Lang, Aug 26 2010: (Start)
a(n) = a(n-1) + 2*a(n-2) + (-1)^n for n > 1, a(0)=1, a(1)=0.
Due to the identity for the o.g.f. A(x): A(x) = x*(1+2*x)*A(x) + 1/(1+x).
(This recurrence was observed by Gary Detlefs in a 08/25/10 e-mail to the author.) (End)
G.f.: Sum_{n>=0} binomial(3*n,n)*x^n / (1+x)^(3*n+3). - Paul D. Hanna, Mar 03 2012
E.g.f.: 1 + (1/9)*(exp(-x)*(3*x - 2) + 2*exp(2*x)). - Stefano Spezia, Sep 27 2019

More terms from James Sellers, Feb 28 2000 and Christian G. Bower, Feb 29 2000

A264476 T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having directed index change 0,1 1,0 2,1 or -1,-1.

Original entry on oeis.org

0, 1, 1, 0, 2, 0, 0, 4, 4, 1, 1, 8, 6, 8, 1, 0, 17, 16, 16, 16, 1, 0, 36, 57, 120, 49, 32, 2, 1, 76, 160, 456, 456, 124, 64, 2, 0, 160, 484, 2272, 3540, 2232, 384, 128, 3, 0, 337, 1449, 11044, 28489, 24773, 10116, 1041, 256, 4, 1, 710, 4250, 49200, 215607, 310748

Offset: 1

R. H. Hardin, Nov 14 2015

Table starts
.0...1....0......0........1..........0............0..............1
.1...2....4......8.......17.........36...........76............160
.0...4....6.....16.......57........160..........484...........1449
.1...8...16....120......456.......2272........11044..........49200
.1..16...49....456.....3540......28489.......215607........1711113
.1..32..124...2232....24773.....310748......4039259.......50217832
.2..64..384..10116...174927....3842048.....74367790.....1462247321
.2.128.1041..45792..1262270...43644384...1358980008....44706530288
.3.256.2868.212112..8905776..505769648..25131223920..1299344783466
.4.512.8189.960336.63373156.5849013488.454913826610.38128043682868

			Some solutions for n=4 k=4
..6..0..1..9..3....6..0..1..2..3....6..7..8..9..3....6..7..8..2..3
.11..5..2..7..4...11.12.13.14..4....0..1..2.14..4....0..1.13.14..4
.16.10.18..8.13....5.10..7.19..9...16.10.11.19.13...16.17.11.12..9
.21.22.23.24.14...21.22.16.24..8...21..5.12.24.18...10..5.23.24.18
.15.20.17.12.19...15.20.17.18.23...15.20.17.22.23...15.20.21.22.19

R. H. Hardin, Table of n, a(n) for n = 1..220

Column 1 is A000931(n+1).
Column 2 is A000079(n-1).
Row 2 is A008999(n-1).

Empirical for column k:
k=1: a(n) = a(n-2) +a(n-3)
k=2: a(n) = 2*a(n-1)
k=3: [order 15]
k=4: a(n) = 18*a(n-2) +36*a(n-3) -45*a(n-4) -216*a(n-5) -243*a(n-6) for n>7
k=5: [order 84] for n>86
k=6: [order 36] for n>40
Empirical for row n:
n=1: a(n) = a(n-3)
n=2: a(n) = 2*a(n-1) +a(n-4)
n=3: [order 15]
n=4: [order 10] for n>11
n=5: [order 84]

A052921 Expansion of (1 - x)/(1 - 3x + 2x^2 - x^3).

Original entry on oeis.org

1, 2, 4, 9, 21, 49, 114, 265, 616, 1432, 3329, 7739, 17991, 41824, 97229, 226030, 525456, 1221537, 2839729, 6601569, 15346786, 35676949, 82938844, 192809420, 448227521, 1042002567, 2422362079, 5631308624, 13091204281, 30433357674, 70748973084, 164471408185

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

First differences of A095263. - R. J. Mathar, Nov 23 2011
Partial sums of A034943 starting (1, 1, 2, 5, 12, 28, 65, ...). - Gary W. Adamson, Feb 15 2012
a(n) is the number of n (decimal) digit integers x such that all digits of x are odd and all digits of 6x are even. - Robert Israel, Apr 17 2014
a(n) is the number of words of length n over the alphabet {0,1,2} that do not contain the substrings 01 or 12 and do not end in 0. - Yiseth K. Rodríguez C., Sep 11 2020

			G.f. = 1 + 2*x + 4*x^2 + 9*x^3 + 21*x^4 + 49*x^5 + 114*x^6 + 265*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Sergio Falcón, Binomial Transform of the Generalized k-Fibonacci Numbers, Communications in Mathematics and Applications (2019) Vol. 10, No. 3, 643-651.
I. M. Gessel and Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 905
Index entries for linear recurrences with constant coefficients, signature (3,-2,1).

Cf. A000931, A034943.

Cf. A097550, A137531.

a:=[1,2,4];; for n in [4..40] do a[n]:=3*a[n-1]-2*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Oct 16 2019

I:=[1,2,4]; [n le 3 select I[n] else 3*Self(n-1)-2*Self(n-2) +Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 14 2012

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1-3*x+2*x^2-x^3) )); // Marius A. Burtea, Oct 16 2019

spec := [S,{S=Sequence(Union(Z,Z,Prod(Sequence(Z),Z,Z,Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..29);
A052921 := proc(n): add(binomial(n+k+1, n-2*k),k=0..n+1) end: seq(A052921(n), n=0..29); # Johannes W. Meijer, Aug 16 2011

LinearRecurrence[{3,-2,1},{1,2,4},40] (* Vincenzo Librandi, Feb 14 2012 *)
CoefficientList[Series[(1-x)/(1-3*x+2*x^2-x^3),{x,0,30}],x] (* Harvey P. Dale, Nov 09 2019 *)

my(x='x+O('x^40)); Vec((1-x)/(1 -3*x +2*x^2 -x^3)) \\ G. C. Greubel, Oct 16 2019

def A077952_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x)/(1 -3*x +2*x^2 -x^3)).list()
A077952_list(40) # G. C. Greubel, Oct 16 2019

G.f.: (1 - x)/(1 - 3*x + 2*x^2 - x^3).
a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3), with a(0)=1, a(1)=2, a(2)=4.
a(n) = Sum_{alpha=RootOf(-1 + 3*z - 2*z^2 + z^3)} (1/23)*(8 - 5*alpha + 7*alpha^2)*alpha^(-1-n).
From Paul Barry, Jun 21 2004: (Start)
Binomial transform of the Padovan sequence A000931(n+5).
a(n) = Sum_{k=0..n+1} C(n+k+1, n-2*k). (End)
a(n) = A000931(3*n + 5). - Michael Somos, Sep 18 2012
a(n) = Sum_{i=1..n+1} A000931(3*i). - David Nacin, Nov 03 2019

A103373 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 1 and for n>6: a(n) = a(n-5) + a(n-6).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 5, 7, 8, 8, 8, 9, 12, 15, 16, 16, 17, 21, 27, 31, 32, 33, 38, 48, 58, 63, 65, 71, 86, 106, 121, 128, 136, 157, 192, 227, 249, 264, 293, 349, 419, 476, 513, 557, 642, 768, 895, 989, 1070, 1199, 1410, 1663, 1884, 2059, 2269

Offset: 1

Jonathan Vos Post, Feb 03 2005

k=5 case of the family of sequences whose k=1 case is the Fibonacci sequence A000045, k=2 case is the Padovan sequence A000931 (offset so as to begin 1,1,1), k=3 case is A079398 (offset so as to begin 1,1,1,1) and k=4 case is A103372.
The general case for integer k>1 is defined: a(1) = a(2) = ... = a(k+1) and for n>(k+1) a(n) = a(n-k) + a(n-[k+1]).
For this k=5 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^6 - x - 1 = 0. This is the real constant 1.1347241384015194926054460545064728402796672263828014859251495516682....
The sequence of prime values in this k=5 case is A103383; the sequence of semiprime values in this k=5 case is A103393.

			a(22) = 9 because a(22) = a(22-5) + a(22-6) = a(17) + a(16) = 5 + 4 = 9.

Zanten, A. J. van, "The golden ratio in the arts of painting, building and mathematics", Nieuw Archief voor Wiskunde, 4 (17) (1999) 229-245.

G. C. Greubel, Table of n, a(n) for n = 1..1000
J.-P. Allouche and T. Johnson, Narayana's cows and delayed morphisms, in G. Assayag, M. Chemillier, and C. Eloy, Troisièmes Journées d'Informatique Musicale, JIM '96, Île de Tatihou, France, 1996, pp. 2-7. [The hal link does not always work. - _N. J. A. Sloane_, Feb 19 2025]
J.-P. Allouche and T. Johnson, Narayana's cows and delayed morphisms, in G. Assayag, M. Chemillier, and C. Eloy, Troisièmes Journées d'Informatique Musicale, JIM '96, Île de Tatihou, France, 1996, pp. 2-7. [Local copy with annotations and a correction from _N. J. A. Sloane_, Feb 19 2025]
Richard Padovan, Dom Hans van der Laan and the Plastic Number.
E. S. Selmer, On the irreducibility of certain trinomials, Math. Scand., 4 (1956) 287-302.
J. Shallit, A generalization of automatic sequences, Theoretical Computer Science, 61 (1988) 1-16.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1,1).

Cf. A000045, A000931, A079398, A103383, A103393.

Cf. A103372, A103374, A103375, A103376, A103377, A103378, A103379, A103380

k = 5; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 65]
RecurrenceTable[{a[n] == a[n - 5] + a[n - 6], a[1] == a[2] == a[3] == a[4] == a[5] == a[6] == 1}, a, {n, 65}] (* or *)
Rest@ CoefficientList[Series[-x (1 + x + x^2 + x^3 + x^4)/(-1 + x^5 + x^6), {x, 0, 65}], x] (* Michael De Vlieger, Oct 03 2016 *)
LinearRecurrence[{0,0,0,0,1,1},{1,1,1,1,1,1},70] (* Harvey P. Dale, Jul 20 2019 *)

a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; 1,1,0,0,0,0]^(n-1)*[1;1;1;1;1;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

x='x+O('x^50); Vec(x*(1+x+x^2+x^3+x^4)/(1-x^5-x^6 )) \\ G. C. Greubel, May 01 2017

G.f.: x*(1+x+x^2+x^3+x^4) / (1-x^5-x^6 ). - R. J. Mathar, Aug 26 2011

Edited by Ray Chandler and Robert G. Wilson v, Feb 06 2005

A020720 Pisot sequences E(7,9), P(7,9).

Original entry on oeis.org

7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739, 10252, 13581, 17991, 23833, 31572, 41824, 55405, 73396, 97229, 128801, 170625, 226030, 299426, 396655, 525456, 696081, 922111, 1221537

Offset: 0

David W. Wilson

Colin Barker, Table of n, a(n) for n = 0..1000
S. B. Ekhad, N. J. A. Sloane, and D. Zeilberger, Automated proofs (or disproofs) of linear recurrences satisfied by Pisot Sequences, arXiv:1609.05570 [math.NT], 2016.
Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
Index entries for linear recurrences with constant coefficients, signature (0,1,1).

A subsequence of A000931.
See A008776 for definitions of Pisot sequences.
The following are basically all variants of the same sequence: A000931, A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.

LinearRecurrence[{0, 1, 1}, {7, 9, 12}, 50] (* Jean-François Alcover, Aug 31 2018 *)
CoefficientList[Series[(7 + 9 x + 5 x^2)/(1 - x^2 - x^3), {x, 0, 50}], x] (* Stefano Spezia, Aug 31 2018 *)

a(n) = a(n-2) + a(n-3) for n>=3. (Proved using the PtoRv program of Ekhad-Sloane-Zeilberger.) - N. J. A. Sloane, Sep 09 2016

G.f.: (7+9*x+5*x^2) / (1-x^2-x^3). - Colin Barker, Jun 05 2016

A089068 a(n) = a(n-1)+a(n-2)+a(n-3)+2 with a(0)=0, a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 0, 1, 3, 6, 12, 23, 43, 80, 148, 273, 503, 926, 1704, 3135, 5767, 10608, 19512, 35889, 66011, 121414, 223316, 410743, 755475, 1389536, 2555756, 4700769, 8646063, 15902590, 29249424, 53798079, 98950095, 181997600, 334745776, 615693473

Offset: 0

Roger L. Bagula, Dec 03 2003

The a(n+2) represent the Kn12 and Kn22 sums of the square array of Delannoy numbers A008288. See A180662 for the definition of these knight and other chess sums. - Johannes W. Meijer, Sep 21 2010

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

Cf. A000931, A000073, A077939, A113300, A001057, A006054, A033505.

Cf. A000073 (Kn11 & Kn21), A089068 (Kn12 & Kn22), A180668 (Kn13 & Kn23), A180669 (Kn14 & Kn24), A180670 (Kn15 & Kn25). - Johannes W. Meijer, Sep 21 2010

Join[{a=0,b=0,c=1},Table[d=a+b+c+2;a=b;b=c;c=d,{n,50}]] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)
RecurrenceTable[{a[0]==a[1]==0,a[2]==1,a[n]==a[n-1]+a[n-2]+a[n-3]+2}, a[n],{n,40}] (* or *) LinearRecurrence[{2,0,0,-1},{0,0,1,3},40] (* Harvey P. Dale, Sep 19 2011 *)

a(n) = A008937(n-2)+A008937(n-1). - Johannes W. Meijer, Sep 21 2010
a(n) = A018921(n-5)+A018921(n-4), n>4. - Johannes W. Meijer, Sep 21 2010
a(n) = A000073(n+2)-1. - R. J. Mathar, Sep 22 2010
From Johannes W. Meijer, Sep 22 2010: (Start)
a(n) = a(n-1)+A001590(n+1).
a(n) = Sum_{m=0..n} A040000(m)*A000073(n-m).
a(n+2) = Sum_{k=0..floor(n/2)} A008288(n-k+1,k+1).
G.f. = x^2*(1+x)/((1-x)*(1-x-x^2-x^3)). (End)
a(n) = 2*a(n-1)-a(n-4), a(0)=0, a(1)=0, a(2)=1, a(3)=3. - Bruno Berselli, Sep 23 2010

Corrected and information added by Johannes W. Meijer, Sep 22 2010, Oct 22 2010

Definition based on arbitrarily set floating-point precision removed by R. J. Mathar, Sep 30 2010

A219967 Number A(n,k) of tilings of a k X n rectangle using straight (3 X 1) trominoes and 2 X 2 tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 2, 3, 3, 2, 1, 1, 1, 0, 2, 4, 3, 4, 2, 0, 1, 1, 0, 3, 8, 8, 8, 8, 3, 0, 1, 1, 1, 4, 13, 21, 28, 21, 13, 4, 1, 1, 1, 0, 5, 19, 31, 65, 65, 31, 19, 5, 0, 1, 1, 0, 7, 35, 70, 170, 267, 170, 70, 35, 7, 0, 1

Offset: 0

Alois P. Heinz, Dec 02 2012

			A(4,4) = 3, because there are 3 tilings of a 4 X 4 rectangle using straight (3 X 1) trominoes and 2 X 2 tiles:
  ._._____.  ._____._.  ._._._._.
  | |_____|  |_____| |  | . | . |
  | | . | |  | | . | |  |___|___|
  |_|___| |  | |___|_|  | . | . |
  |_____|_|  |_|_____|  |___|___| .
Square array A(n,k) begins:
  1,  1,  1,  1,  1,   1,    1,     1,     1, ...
  1,  0,  0,  1,  0,   0,    1,     0,     0, ...
  1,  0,  1,  1,  1,   2,    2,     3,     4, ...
  1,  1,  1,  2,  3,   4,    8,    13,    19, ...
  1,  0,  1,  3,  3,   8,   21,    31,    70, ...
  1,  0,  2,  4,  8,  28,   65,   170,   456, ...
  1,  1,  2,  8, 21,  65,  267,   804,  2530, ...
  1,  0,  3, 13, 31, 170,  804,  2744, 12343, ...
  1,  0,  4, 19, 70, 456, 2530, 12343, 66653, ...

Liang Kai, Antidiagonals n = 0..35, flattened (Antidiagonals 0..27 from Alois P. Heinz)
Kai Liang, Solving tiling enumeration problems by tensor network contractions, arXiv:2503.17698 [math.CO], 2025. See p. 25, Table 4.
Wikipedia, Tromino

Columns (or rows) k=0-10 give: A000012, A079978, A000931(n+3), A219968, A202536, A219969, A219970, A219971, A219972, A219973, A219974.

Main diagonal gives: A219975.

b:= proc(n, l) option remember; local k, t;
      if max(l[])>n then 0 elif n=0 or l=[] then 1
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
    else for k do if l[k]=0 then break fi od;
         b(n, subsop(k=3, l))+
         `if`(k `if`(n>=k, b(n, [0$k]), b(k, [0$n])):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, l_] := b[n, l] = Module[{ k, t}, If [Max[l] > n, 0, If[n == 0 || l == {}, 1, If[ Min[l] > 0 ,t = Min[l]; b[n-t, l-t], k = Position[l, 0, 1][[1, 1]]; b[n, ReplacePart[l, k -> 3]] + If[k < Length[l] && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 2, k+1 -> 2}]], 0] + If[k+1 < Length[l] && l[[k+1]] == 0 && l[[k+2]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 1, k+2 -> 1}]], 0] ] ] ] ]; a[n_, k_] := If[n >= k, b[n, Array[0&, k]], b[k, Array[0&, n]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)

A103374 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = 1 and for n>7: a(n) = a(n-6) + a(n-7).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 5, 7, 8, 8, 8, 8, 9, 12, 15, 16, 16, 16, 17, 21, 27, 31, 32, 32, 33, 38, 48, 58, 63, 64, 65, 71, 86, 106, 121, 127, 129, 136, 157, 192, 227, 248, 256, 265, 293, 349, 419, 475, 504, 521, 558, 642, 768, 894, 979, 1025, 1079

Offset: 1

Jonathan Vos Post, Feb 03 2005

k=6 case of the family of sequences whose k=1 case is the Fibonacci sequence A000045, k=2 case is the Padovan sequence A000931 (offset so as to begin 1,1,1), k=3 case is A079398 (offset so as to begin 1,1,1,1), k=4 case is A103372 and k=5 case is A103373.
The general case for integer k>1 is defined: a(1) = a(2) = ... = a(k+1) and for n>(k+1) a(n) = a(n-k) + a(n-[k+1]).
For this k=6 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^7 - x - 1 = 0. This is the real constant 1.1127756842787... (see A230160).
The sequence of prime values in this k=6 case is A103384; the sequence of semiprime values in this k=6 case is A103394.

			a(32) = 17 because a(32) = a(32-6) + a(32-7) = a(26) + a(25) = 9 + 8 = 17.

Zanten, A. J. van, "The golden ratio in the arts of painting, building and mathematics", Nieuw Archief voor Wiskunde, 4 (17) (1999) 229-245.

G. C. Greubel, Table of n, a(n) for n = 1..1000
J.-P. Allouche and T. Johnson, Narayana's cows and delayed morphisms, in G. Assayag, M. Chemillier, and C. Eloy, Troisièmes Journées d'Informatique Musicale, JIM '96, Île de Tatihou, France, 1996, pp. 2-7. [The hal link does not always work. - _N. J. A. Sloane_, Feb 19 2025]
J.-P. Allouche and T. Johnson, Narayana's cows and delayed morphisms, in G. Assayag, M. Chemillier, and C. Eloy, Troisièmes Journées d'Informatique Musicale, JIM '96, Île de Tatihou, France, 1996, pp. 2-7. [Local copy with annotations and a correction from _N. J. A. Sloane_, Feb 19 2025]
Richard Padovan, Dom Hans van der Laan and the Plastic Number.
E. S. Selmer, On the irreducibility of certain trinomials, Math. Scand., 4 (1956) 287-302.
J. Shallit, A generalization of automatic sequences, Theoretical Computer Science, 61 (1988) 1-16.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1,1).

Cf. A000045, A000931, A079398, A103384, A103394, A230160.

Cf. A103372, A103373, A103375, A103376, A103377, A103378, A103379, A103380

k = 6; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 70]
RecurrenceTable[{a[n] == a[n - 6] + a[n - 7], a[1] == a[2] == a[3] == a[4] == a[5] == a[6] == a[7] == 1}, a, {n, 70}] (* or *)
Rest@ CoefficientList[Series[-x (1 + x) (1 + x + x^2) (x^2 - x + 1)/(-1 + x^6 + x^7), {x, 0, 70}], x] (* Michael De Vlieger, Oct 03 2016 *)
LinearRecurrence[{0,0,0,0,0,1,1},{1,1,1,1,1,1,1},80] (* Harvey P. Dale, Sep 02 2024 *)

a(n)=([0,1,0,0,0,0,0; 0,0,1,0,0,0,0; 0,0,0,1,0,0,0; 0,0,0,0,1,0,0; 0,0,0,0,0,1,0; 0,0,0,0,0,0,1; 1,1,0,0,0,0,0]^(n-1)*[1;1;1;1;1;1;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

x='x+O('x^50); Vec(x*(1+x)*(1+x+x^2)*(x^2-x+1)/(1-x^6-x^7)) \\ G. C. Greubel, May 01 2017

G.f.: x*(1+x)*(1+x+x^2)*(x^2-x+1) / ( 1-x^6-x^7 ). - R. J. Mathar, Aug 26 2011

Edited by Ray Chandler and Robert G. Wilson v, Feb 06 2005

cp's OEIS Frontend

A264569 T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having directed index change 1,0 1,1 0,-1 or -1,1.

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A023434 Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-4).

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A053088 a(n) = 3*a(n-2) + 2*a(n-3) for n > 2, a(0)=1, a(1)=0, a(2)=3.

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A264476 T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having directed index change 0,1 1,0 2,1 or -1,-1.

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A052921 Expansion of (1 - x)/(1 - 3*x + 2*x^2 - x^3).

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A103373 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 1 and for n>6: a(n) = a(n-5) + a(n-6).

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A020720 Pisot sequences E(7,9), P(7,9).

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A053088 a(n) = 3a(n-2) + 2a(n-3) for n > 2, a(0)=1, a(1)=0, a(2)=3.

A052921 Expansion of (1 - x)/(1 - 3x + 2x^2 - x^3).