A076264 -id:A076264 - cp's OEIS frontend

A038197 4-wave sequence.

1, 1, 1, 1, 2, 3, 4, 7, 9, 10, 19, 26, 30, 56, 75, 85, 160, 216, 246, 462, 622, 707, 1329, 1791, 2037, 3828, 5157, 5864, 11021, 14849, 16886, 31735, 42756, 48620, 91376, 123111, 139997, 263108, 354484, 403104, 757588, 1020696, 1160693, 2181389

Offset: 0

Floor van Lamoen

This sequence is related to the nonagon or 9-gon.

			The first few rows of the T(n,k) array are, n>=1, 1 <= k <=4:
  0,  0,   0,   1
  1,  1,   1,   1
  1,  2,   3,   4
  4,  7,   9,   10
  10, 19,  26,  30
  30, 56,  75,  85
  85, 160, 216, 246

F. v. Lamoen, Wave sequences
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Eric Weisstein's World of Mathematics, Nonagon.
Index entries for linear recurrences with constant coefficients, signature (1,-1,3,-3,3,0,0,0,-1,1,-1).

The a(3*n) lead to A006357; The T(n,k) lead to A076264 and A091024.
Cf. A038196, A038201.
Cf. A120747 (m = 5: hendecagon or 11-gon)

m:=4: nmax:=15: for k from 1 to m-1 do T(1,k):=0 od: T(1,m):=1: for n from 2 to nmax do for k from 1 to m do T(n,k):= add(T(n-1,k1), k1=m-k+1..m) od: od: for n from 1 to nmax/2 do seq(T(n,k), k=1..m) od; a(0):=1: Tx:=1: for n from 2 to nmax do for k from 2 to m do a(Tx):= T(n,k): Tx:=Tx+1: od: od: seq(a(n), n=0..Tx-1); # Johannes W. Meijer, Aug 03 2011

LinearRecurrence[{1,-1,3,-3,3,0,0,0,-1,1,-1},{1,1,1,1,2,3,4,7,9,10,19},50] (* Harvey P. Dale, Oct 02 2015 *)

a(n) = a(n-1)+a(n-2) if n=3*m+1, a(n) = a(n-1)+a(n-4) if n=3*m+2, a(n) = a(n-1)+a(n-6) if n=3*m. Also: a(n) = 2*a(n-3)+3*a(n-6)-a(n-9)-a(n-12).
G.f.: -(-1-x-x^2+x^3-x^5+x^6)/(1-2*x^3-3*x^6+x^9+x^12)
a(n-1) = sequence(sequence(T(n,k), k=2..4), n>=2) with a(0)=1; T(n,k) = sum(T(n-1,k1), k1 = 5-k..4) with T(1,1) = T(1,2) = T(1,3) = 0 and T(1,4) = 1; n>=1 and 1 <= k <= 4. [Steinbach]

Edited by Floor van Lamoen, Feb 05 2002

Edited and information added by Johannes W. Meijer, Aug 03 2011

A206455 T(n,k) = number of 0..k arrays of length n avoiding the consecutive pattern 0..k.

Original entry on oeis.org

2, 3, 3, 4, 9, 4, 5, 16, 26, 5, 6, 25, 64, 75, 6, 7, 36, 125, 255, 216, 7, 8, 49, 216, 625, 1016, 622, 8, 9, 64, 343, 1296, 3124, 4048, 1791, 9, 10, 81, 512, 2401, 7776, 15615, 16128, 5157, 10, 11, 100, 729, 4096, 16807, 46655, 78050, 64257, 14849, 11, 12, 121, 1000

Offset: 1

R. H. Hardin, Feb 07 2012

			Table starts
  2    3     4      5       6       7        8        9        10        11 ...
  3    9    16     25      36      49       64       81       100       121 ...
  4   26    64    125     216     343      512      729      1000      1331 ...
  5   75   255    625    1296    2401     4096     6561     10000     14641 ...
  6  216  1016   3124    7776   16807    32768    59049    100000    161051 ...
  7  622  4048  15615   46655  117649   262144   531441   1000000   1771561 ...
  8 1791 16128  78050  279924  823542  2097152  4782969  10000000  19487171 ...
  9 5157 64257 390125 1679508 5764787 16777215 43046721 100000000 214358881 ...
  ...

R. H. Hardin, Table of n, a(n) for n = 1..10000
Robert Israel, Proof of recurrence for column k

Columns 2, 3... are A076264, A206450, A206451, A206452.

Subdiagonal 1 is A048861(n+1)

N:= 20: # for the first N antidiagonals
for k from 1 to N-1 do
  F[k]:= gfun:-rectoproc({a(n)=(k+1)*a(n-1) - a(n-k-1), seq(a(j)=(k+1)^j,j=1..k),a(k+1)=(k+1)^(k+1)-1},a(n),remember)
od:
seq(seq(F[m-j](j),j=1..m-1),m=1..N); # Robert Israel, Dec 17 2017

nmax = 20;
col[k_] := col[k] = Module[{a}, a[n_ /; n>2] := a[n] = (k+1)*a[n-1]-a[n-k-1]; a[0]=1; a[1]=k+1; a[2]=(k+1)^2; a[_?Negative]=0; Array[a, nmax]];
T[n_, k_] := If[k == 1, n+1, col[k][[n]]];
Table[T[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jul 22 2022 *)

Empirical: T(n,k) = sum{i=0..floor(n/(k+1))} ( (-1)^i * (k+1)^(n-(k+1)*i) * binomial(n-k*i,i) ) (after A076264)
Empirical for column k: a(n) = (k+1)*a(n-1) - a(n-(k+1)).
Formula for column k verified by Robert Israel, Dec 17 2017 (see link).

A207550 T(n,k) = Number of n X k 0..2 arrays avoiding the pattern z-2 z-1 z in any row, column, nw-to-se diagonal or ne-to-sw antidiagonal.

Original entry on oeis.org

3, 9, 9, 26, 81, 26, 75, 676, 676, 75, 216, 5625, 14974, 5625, 216, 622, 46656, 327518, 327899, 46656, 622, 1791, 386884, 7169168, 18584111, 7104784, 386884, 1791, 5157, 3207681, 156957792, 1058580304, 1033847229, 153718531, 3207681, 5157, 14849

Offset: 1

R. H. Hardin, Feb 18 2012

Table starts
....3........9..........26..............75................216
....9.......81.........676............5625..............46656
...26......676.......14974..........327518............7169168
...75.....5625......327899........18584111.........1058580304
..216....46656.....7104784......1033847229.......151596413811
..622...386884...153718531.....57282563287.....21553876840265
.1791..3207681..3323115736...3168082336959...3054158366460488
.5157.26594649.71828785449.175129179194410.432330439527473859

			Some solutions for n=4, k=3
..2..2..1....0..0..0....2..0..2....1..0..1....2..1..2....1..2..2....2..0..1
..2..0..0....2..0..2....1..1..1....2..0..2....1..2..1....0..0..1....0..0..2
..2..0..0....2..0..1....1..0..0....0..2..2....0..2..1....1..2..0....1..1..2
..2..2..2....0..0..0....1..2..2....2..0..2....1..1..2....1..2..1....0..0..1

R. H. Hardin, Table of n, a(n) for n = 1..127

Column 1 is A076264.
Column 2 is A206694.
Row 1 is A076264.
Row 2 is A206694.

A250351 T(n,k)=Number of length n arrays x(i), i=1..n with x(i) in i..i+k and no value appearing more than 2 times.

Original entry on oeis.org

2, 3, 4, 4, 9, 8, 5, 16, 26, 16, 6, 25, 62, 75, 32, 7, 36, 122, 235, 216, 64, 8, 49, 212, 581, 888, 622, 128, 9, 64, 338, 1221, 2724, 3349, 1791, 256, 10, 81, 506, 2287, 6900, 12734, 12620, 5157, 512, 11, 100, 722, 3935, 15186, 38543, 59406, 47545, 14849, 1024, 12

Offset: 1

R. H. Hardin, Nov 19 2014

Table starts
....2.....3......4.......5........6.........7.........8..........9.........10
....4.....9.....16......25.......36........49........64.........81........100
....8....26.....62.....122......212.......338.......506........722........992
...16....75....235.....581.....1221......2287......3935.......6345.......9721
...32...216....888....2724.....6900.....15186.....30072......54888......93924
...64...622...3349...12734....38543.....99344....226247.....467642.....894599
..128..1791..12620...59406...214716....644040...1681860....3932472....8409252
..256..5157..47545..276816..1193739...4164930..12411486...32743710...78177402
..512.14849.179104.1289208..6628042..26882466..91384716..270990642..720784488
.1024.42756.674666.6002949.36773706.173276640.671639928.2238089580.6611373660

			Some solutions for n=6 k=4
..2....3....2....0....4....2....4....3....4....0....3....1....0....4....4....4
..3....1....2....1....3....2....1....4....1....4....5....4....5....4....5....1
..3....4....4....4....2....3....4....5....5....2....4....5....6....6....3....2
..5....7....7....4....5....6....5....3....4....5....4....6....5....6....6....5
..7....6....7....7....7....6....5....5....7....6....8....5....7....7....7....7
..9....6....9....9....6....5....9....8....7....6....5....6....8....5....7....9

R. H. Hardin, Table of n, a(n) for n = 1..418

Column 1 is A000079
Column 2 is A076264(n)
Row 1 is A000027(n+1)
Row 2 is A000290(n+1)
Cf. A248944

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 3*a(n-1) -a(n-3)
k=3: a(n) = 4*a(n-1) -2*a(n-3) -5*a(n-4) +a(n-6) = A250346(n)
k=4: [order 12] = A250347(n)
k=5: [order 24] = A250348(n)
k=6: [order 48] = A250349(n).
k=7: [order 96] = A250350(n).
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = n^2 + 2*n + 1
n=3: a(n) = n^3 + 3*n^2 + 2*n + 2 = A250352(n).
n=4: a(n) = n^4 + 4*n^3 + 2*n^2 + 9*n + 1 for n>1 = A250353(n).
n=5: a(n) = n^5 + 5*n^4 + 25*n^2 + 5*n for n>2 = A250354(n).
n=6: a(n) = n^6 + 6*n^5 - 5*n^4 + 55*n^3 + 25*n^2 - 61*n + 98 for n>3 = A250355(n).
n=7: a(n) = n^7 + 7*n^6 - 14*n^5 + 105*n^4 + 105*n^3 - 532*n^2 + 1252*n - 744 for n>4 = A250356(n).

A200785 T(n,k) is the number of arrays of n+2 elements from {0,1,...,k} with no two consecutive ascents.

Original entry on oeis.org

8, 26, 16, 60, 75, 32, 115, 225, 216, 64, 196, 530, 840, 622, 128, 308, 1071, 2425, 3136, 1791, 256, 456, 1946, 5796, 11100, 11704, 5157, 512, 645, 3270, 12152, 31395, 50775, 43681, 14849, 1024, 880, 5175, 23136, 75992, 169884, 232275, 163020, 42756, 2048

Offset: 1

R. H. Hardin Nov 22 2011

All the conjectured formulas are true, and follow from the Burstein-Mansour paper. - N. J. A. Sloane, May 21 2013

			Table starts
....8.....26......60.......115.......196........308.........456.........645
...16.....75.....225.......530......1071.......1946........3270........5175
...32....216.....840......2425......5796......12152.......23136.......40905
...64....622....3136.....11100.....31395......75992......164004......324087
..128...1791...11704.....50775....169884.....474566.....1160616.....2562633
..256...5157...43681....232275....919413....2964416.....8216484....20273247
..512..14849..163020...1062500...4975322...18514405....58154912...160338680
.1024..42756..608400...4860250..26924106..115637431...411637168..1268210421
.2048.123111.2270580..22232375.145698840..722234149..2913595712.10030582998
.4096.354484.8473921.101698250.788446400.4510869636.20622837480.79335475611
Some arrays for n=4, k=3:
..0....1....0....0....1....0....3....3....0....1....3....0....2....2....2....2
..3....0....2....2....0....2....0....0....3....1....0....0....0....3....3....3
..2....3....2....2....2....2....3....3....1....0....1....0....2....1....3....3
..1....0....2....1....0....0....2....2....2....2....1....2....2....0....0....2
..0....3....0....0....1....2....1....2....0....0....3....2....0....3....1....3
..3....3....0....3....0....2....3....2....0....3....0....0....2....2....1....3

R. H. Hardin, Table of n, a(n) for n = 1..10010
A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, Annals. Combin., 7 (2003), 1-14. See Th. 3.13.

Column 1 is A000079
Column 2 is A076264
Column 3 is A072335
Row 1 is A002413
Cf. A200781.

T(n-2,k) = \sum_{L=0}^n (-1)^L / L! * \sum_{M=0}^{min(L,[(n-L)/2])} binomial(n-L-M,M) * M! * (k+1)^(n-L-2*M) B_{L,M}(x_1,x_2,...), where B_{L,M}() are Bell polynomials, x_i = binomial(k+1,i+2) * i! * f(i), i=1,2,..., and f(i) has period of length 6: [0,1,1,0,-1,-1] (i.e., f(0)=0, f(1)=1, etc.). This formula implies that for a fixed n, T(n,k) is a polynomial in k, which is easy to compute. - Max Alekseyev, Dec 12 2011
Empirical formulas for columns:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 3*a(n-1) -a(n-3)
k=3: a(n) = 4*a(n-1) -4*a(n-3) +a(n-4)
k=4: a(n) = 5*a(n-1) -10*a(n-3) +5*a(n-4)
k=5: a(n) = 6*a(n-1) -20*a(n-3) +15*a(n-4) -a(n-6)
k=6: a(n) = 7*a(n-1) -35*a(n-3) +35*a(n-4) -7*a(n-6) +a(n-7)
k=7: a(n) = 8*a(n-1) -56*a(n-3) +70*a(n-4) -28*a(n-6) +8*a(n-7)
Empirical recurrence for general column k:
0 = sum{i=0..floor(k/3) (binomial(k+1,3*i+1)*T(n-(3*i+1),k))} - sum{i=0..floor((k+1)/3) (binomial(k+1,3*i)*T(n-3*i,k))}
Formulae for rows:
T(1,k) = (5/6)*k^3 + 3*k^2 + (19/6)*k + 1
T(2,k) = (17/24)*k^4 + (43/12)*k^3 + (151/24)*k^2 + (53/12)*k + 1
T(3,k) = (7/12)*k^5 + (47/12)*k^4 + (39/4)*k^3 + (133/12)*k^2 + (17/3)*k + 1
T(4,k) = (349/720)*k^6 + (321/80)*k^5 + (1883/144)*k^4 + (1013/48)*k^3 + (3139/180)*k^2 + (413/60)*k + 1
T(5,k) = (2017/5040)*k^7 + (1427/360)*k^6 + (5759/360)*k^5 + (607/18)*k^4 + (28459/720)*k^3 + (9113/360)*k^2 + (848/105)*k + 1
T(6,k) = (6679/20160)*k^8 + (4799/1260)*k^7 + (26449/1440)*k^6 + (2162/45)*k^5 + (212153/2880)*k^4 + (6019/90)*k^3 + (174571/5040)*k^2 + (3893/420)*k + 1
T(7,k) = (99377/362880)*k^9 + (48247/13440)*k^8 + (243673/12096)*k^7 + (60529/960)*k^6 + (2076437/17280)*k^5 + (274529/1920)*k^4 + (952027/9072)*k^3 + (152461/3360)*k^2 + (26399/2520)*k + 1

A052536 Number of compositions of n when parts 1 and 2 are of two kinds.

Original entry on oeis.org

1, 2, 6, 17, 49, 141, 406, 1169, 3366, 9692, 27907, 80355, 231373, 666212, 1918281, 5523470, 15904198, 45794313, 131859469, 379674209, 1093228314, 3147825473, 9063802210, 26098178316, 75146709475, 216376326215, 623030800329

Offset: 0

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

The g.f. for compositions of k_1 kinds of 1's, k_2 kinds of 2's, ..., k_j kinds of j's, ... is 1/(1 - Sum_{j>=1} k_j * x^j). - Joerg Arndt, Jul 06 2011

			a(2)=6 because we have (2),(2'),(1,1),(1,1'),(1',1) and (1',1').

Michael De Vlieger, Table of n, a(n) for n = 0..2177
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 467
Index entries for linear recurrences with constant coefficients, signature (3,0,-1).

Row sums of A105478.

Cf. A105478, A076264.

spec := [S,{S=Sequence(Union(Z,Prod(Z,Union(Z,Sequence(Z)))))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

a[0] = 1; a[1] = 2; a[2] = 6; a[n_] := a[n] = 3*a[n-1] - a[n-3]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jun 12 2013, after Emeric Deutsch *)

Vec((1-x)/(1-3*x+x^3)+O(x^99)) \\ Charles R Greathouse IV, Nov 20 2011

G.f.: (1-x)/(1 - 3*x + x^3).
G.f.: 1/(1 - (2*x + 2*x^2 + Sum_{j>=3} x^j)). - Joerg Arndt, Jul 06 2011
a(n) = Sum(-(1/9)*(-2 + r^2 - r)*r^(-1-n)), r = RootOf(1 - 3*x + x^3).
a(0)=1, a(1)=2, a(2)=6, a(n) = 3*a(n-1) - a(n-3) for n >= 3. - Emeric Deutsch, Apr 10 2005
a(n) = left term in M^n * [1 0 0], where M = the 3 X 3 matrix [2 1 1 / 1 1 0 / 1 0 0]. Right term in M^n *[1 0 0] is a(n-1); middle term is A076264(n-1). - Gary W. Adamson, Sep 05 2005
3*a(n) = A123891(n+1). - Jeffrey R. Goodwin, Jul 03 2011

More terms from James Sellers, Jun 06 2000
Edited by Emeric Deutsch, Apr 10 2005
More terms from Gary W. Adamson, Sep 05 2005

A206700 T(n,k)=Number of nXk 0..2 arrays avoiding the pattern z-2 z-1 z in any row or column.

Original entry on oeis.org

3, 9, 9, 26, 81, 26, 75, 676, 676, 75, 216, 5625, 15712, 5625, 216, 622, 46656, 363465, 363465, 46656, 622, 1791, 386884, 8366690, 23335217, 8366690, 386884, 1791, 5157, 3207681, 192513201, 1488199144, 1488199144, 192513201, 3207681, 5157, 14849

Offset: 1

R. H. Hardin Feb 11 2012

Table starts
....3........9...........26..............75.................216
....9.......81..........676............5625...............46656
...26......676........15712..........363465.............8366690
...75.....5625.......363465........23335217..........1488199144
..216....46656......8366690......1488199144........262482902920
..622...386884....192513201.....94853765621......46260684361347
.1791..3207681...4428597297...6043775871265....8149739471342829
.5157.26594649.101873060353.385074186960645.1435662206447144342

			Some solutions for n=4 k=3
..2..0..1....2..0..1....0..0..2....2..0..1....2..1..0....2..2..2....2..1..0
..0..0..2....2..0..2....0..0..2....2..0..2....0..1..1....1..2..1....2..1..2
..1..1..2....1..0..2....0..0..0....2..0..2....1..0..0....0..2..2....2..1..1
..0..0..1....1..2..1....1..2..0....2..0..1....1..2..2....0..0..2....0..1..1

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

Column 1 is A076264

A052541 Expansion of 1/(1-3*x-x^3).

Original entry on oeis.org

1, 3, 9, 28, 87, 270, 838, 2601, 8073, 25057, 77772, 241389, 749224, 2325444, 7217721, 22402387, 69532605, 215815536, 669848995, 2079079590, 6453054306, 20029011913, 62166115329, 192951400293, 598883212792, 1858815753705, 5769398661408, 17907079197016

Offset: 0

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

A transform of A000244 under the mapping mapping g(x)->(1/(1-x^3))g(x/(1-x^3)). - Paul Barry, Oct 20 2004

a(n) equals the number of n-length words on {0,1,2,3} such that 0 appears only in a run which length is a multiple of 3. - Milan Janjic, Feb 17 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 475
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3
José L. Ramírez, Víctor F. Sirvent, A note on the k-Narayana sequence, Annales Mathematicae et Informaticae, 45 (2015) pp. 91-105.
Index entries for linear recurrences with constant coefficients, signature (3,0,1).

Cf. A076264.

a:=[1,3,9];; for n in [4..30] do a[n]:=3*a[n-1]+a[n-3]; od; a; # G. C. Greubel, May 09 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( x/(1-3*x-x^3) )); // G. C. Greubel, May 09 2019

spec := [S,{S=Sequence(Union(Z,Z,Z,Prod(Z,Z,Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..30);

CoefficientList[Series[x/(1-3*x-x^3), {x, 0, 30}], x] (* Zerinvary Lajos, Mar 29 2007 *)
LinearRecurrence[{3,0,1},{1,3,9},30] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)

my(x='x+O('x^30)); Vec(x/(1-3*x-x^3)) \\ G. C. Greubel, May 09 2019

(x/(1-3*x-x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 09 2019

G.f.: 1/(1 - 3*x - x^3).
a(n) = 3*a(n-1) + a(n-3), with a(0)=1, a(1)=3.
a(n) = Sum_{alpha = RootOf(-1+3*x+x^3)} (1/15)*(4 + alpha + 2*alpha^2) * alpha^(-n-1).
a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k, k) * 3^(n-3*k). - Paul Barry, Oct 20 2004

More terms from James Sellers, Jun 06 2000

A207317 T(n,k)=Number of nXk 0..2 arrays avoiding the pattern z-2 z-1 z in any row, column or nw-to-se diagonal.

Original entry on oeis.org

3, 9, 9, 26, 81, 26, 75, 676, 676, 75, 216, 5625, 15390, 5625, 216, 622, 46656, 347502, 347502, 46656, 622, 1791, 386884, 7791488, 21162579, 7791488, 386884, 1791, 5157, 3207681, 174545777, 1274682671, 1274682671, 174545777, 3207681, 5157, 14849

Offset: 1

R. H. Hardin Feb 16 2012

Table starts
....3........9..........26..............75................216
....9.......81.........676............5625..............46656
...26......676.......15390..........347502............7791488
...75.....5625......347502........21162579.........1274682671
..216....46656.....7791488......1274682671.......205235353935
..622...386884...174545777.....76655305645.....32960886054362
.1791..3207681..3908531208...4606553380932...5287481507599689
.5157.26594649.87515884741.276789915709747.847996342709895834

			Some solutions for n=4 k=3
..2..0..1....2..0..2....0..0..2....2..2..2....2..2..1....2..2..0....2..1..2
..2..0..2....1..1..1....0..0..2....1..2..1....2..0..2....0..0..2....1..2..1
..1..0..2....1..0..0....0..0..0....0..2..2....2..2..0....2..0..0....0..2..1
..1..2..1....1..2..2....1..2..0....0..0..2....0..2..1....1..0..1....1..1..2

R. H. Hardin, Table of n, a(n) for n = 1..180

Column 1 is A076264

Column 2 is A206694

A091024 Let v(0) be the column vector (1,0,0,0)'; for n>0, let v(n) = [1 1 1 1 / 1 1 1 0 / 1 1 0 0/ 1 0 0 0] v(n-1). Sequence gives third entry of v(n).

Original entry on oeis.org

0, 1, 2, 7, 19, 56, 160, 462, 1329, 3828, 11021, 31735, 91376, 263108, 757588, 2181389, 6281058, 18085587, 52075371, 149945056, 431749580, 1243173370, 3579575053, 10306975580, 29677753369, 85453685055, 246054079584

Offset: 0

Gary W. Adamson, Dec 14 2003

nonn

First entry of v(n) gives 1,1,4,10,30,85 = A006357 prefixed with an initial 1, the second entry gives 0,1,3,9,26,... = A076264 prefixed with an initial 0.
A sequence derived from 9-gonal diagonal ratios.
a(n)/a(n-1) converges to D = 2.879385... = longest 9-Gon diagonal with edge = 1. E.g., a(7)/a(6) = 707/246 = 2.873983...(a(n)/a(n-1) of all 4 columns converge to 2.8739...). For each row, left to right, terms converge upon 9-Gon ratios: (2.879...):(2.53208...):(1.87938...):(1) Example: row 7 = 707 622 462 246, from A006357, A076264, A091024 and A006357(offset), respectively. The ratios 707/246, 622/246, 462/246 and 246/246 are: (2.8739...):(2.528...):(1.87804...):(1)
From L. Edson Jeffery, Mar 15 2011: (Start)
In fact, the above ratios (2.8739...):(2.528...):(1.87804...):(1) converge to Q_3(w):Q_2(w):Q_1(w):Q_0(w), where the polynomials Q_r(w) are defined by Q_r(w)=w*Q_(r-1)(w)-Q_(r-2)(w) (r>1), Q_0(w)=1, Q_1(w)=w, and w=2*cos(Pi/9).
Moreover, this sequence and a variant of its g.f. are related to rhombus substitution tilings showing 9-fold rotational symmetry (cf. A187503, A187504, A187505, A187506). (End)

			A006357, A076264, a(n) and A006357 (offset) gives the 4 components of v(n) transposed:
1 0 0 0
1 1 1 1
4 3 2 1
10 9 7 4
30 26 19 10
85 75 56 30

Jay Kappraff, "Beyond Measure, A Guided Tour Through Nature, Myth and Number" (p. 497 gives the analogous case for the Heptagon).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-1,-1).

Cf. A006357, A076264.

a[n_] := (MatrixPower[{{1, 1, 1, 1}, {1, 1, 1, 0}, {1, 1, 0, 0}, {1, 0, 0, 0}}, n].{{1}, {0}, {0}, {0}})[[3, 1]]; Table[ a[n], {n, 0, 26}] (* Robert G. Wilson v, Feb 21 2005 *)
LinearRecurrence[{2,3,-1,-1},{0,1,2,7},30] (* Harvey P. Dale, Feb 18 2016 *)

Recurrence: a(n) = 2*a(n-1) + 3*a(n-2) - a(n-3) - a(n-4), with initial conditions {a(k)}={0,1,2,7}, k=0,1,2,3. - L. Edson Jeffery, Mar 15 2011
G.f.: x/(1 - 2*x - 3*x^2 + x^3 + x^4). - L. Edson Jeffery, Mar 15 2011
G.f.: Q(0)*x/(2+2*x) , where Q(k) = 1 + 1/(1 - x*(12*k-3 + x^2)/( x*(12*k+3 + x^2 ) - 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 12 2013

More terms from Robert G. Wilson v, Feb 21 2005

cp's OEIS Frontend

A038197 4-wave sequence.

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A206455 T(n,k) = number of 0..k arrays of length n avoiding the consecutive pattern 0..k.

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A207550 T(n,k) = Number of n X k 0..2 arrays avoiding the pattern z-2 z-1 z in any row, column, nw-to-se diagonal or ne-to-sw antidiagonal.

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A250351 T(n,k)=Number of length n arrays x(i), i=1..n with x(i) in i..i+k and no value appearing more than 2 times.

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A200785 T(n,k) is the number of arrays of n+2 elements from {0,1,...,k} with no two consecutive ascents.

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A052536 Number of compositions of n when parts 1 and 2 are of two kinds.

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A206700 T(n,k)=Number of nXk 0..2 arrays avoiding the pattern z-2 z-1 z in any row or column.

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A052541 Expansion of 1/(1-3*x-x^3).

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