A126528 -id:A126528 - cp's OEIS frontend

A054458 Convolution triangle based on A001333(n), n >= 1.

1, 3, 1, 7, 6, 1, 17, 23, 9, 1, 41, 76, 48, 12, 1, 99, 233, 204, 82, 15, 1, 239, 682, 765, 428, 125, 18, 1, 577, 1935, 2649, 1907, 775, 177, 21, 1, 1393, 5368, 8680, 7656, 4010, 1272, 238, 24, 1, 3363, 14641, 27312, 28548, 18358, 7506, 1946, 308, 27, 1

Offset: 0

Wolfdieter Lang, Apr 26 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.
The G.f. for the row polynomials p(n,x) (increasing powers of x) is LPell(z)/(1-x*z*LPell(z)) with LPell(z) given in 'Formula'.
Column sequences are A001333(n+1), A054459(n), A054460(n) for m=0..2.
Mirror image of triangle in A209696. - Philippe Deléham, Mar 24 2012
Subtriangle of the triangle given by (0, 3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 25 2012
Riordan array ((1+x)/(1-2*x-x^2), (x+x^2)/(1-2*x-x^2)). - Philippe Deléham, Mar 25 2012

			Fourth row polynomial (n=3): p(3,x)= 17+23*x+9*x^2+x^3.
Triangle begins :
  1
  3, 1
  7, 6, 1
  17, 23, 9, 1
  41, 76, 48, 12, 1
  99, 233, 204, 82, 15, 1
  239, 682, 765, 428, 125, 18, 1. - _Philippe Deléham_, Mar 25 2012
(0, 3, -2/3, -1/3, 0, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins :
  1
  0, 1
  0, 3, 1
  0, 7, 6, 1
  0, 17, 23, 9, 1
  0, 41, 76, 48, 12, 1
  0, 99, 233, 204, 82, 15, 1
  0, 239, 682, 765, 428, 125, 15, 1. - _Philippe Deléham_, Mar 25 2012

Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Cf. A002203(n+1)/2. Row sums: A055099(n).
Cf. A001333, A054459, A054460.
Cf. A040000, A001333, A055099, A126473, A126501, A126528.

a(n, m) := ((n-m+1)*a(n, m-1) + (2n-m)*a(n-1, m-1) + (n-1)*a(n-2, m-1))/(4*m), n >= m >= 1; a(n, 0)= A001333(n+1); a(n, m) := 0 if n

G.f. for column m: LPell(x)*(x*LPell(x))^m, m >= 0, with LPell(x)= (1+x)/(1-2*x-x^2) = g.f. for A001333(n+1).

G.f.: (1+x)/(1-2*x-y*x-x^2-y*x^2). - Philippe Deléham, Mar 25 2012

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = 3 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 25 2012

Sum_{k=0..n} T(n,k)*x^k = A040000(n), A001333(n+1), A055099(n), A126473(n), A126501(n), A126528(n) for x = -1, 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Mar 25 2012

A212835 T(n,k)=Number of 0..k arrays of length n+1 with 0 never adjacent to k.

Original entry on oeis.org

2, 7, 2, 14, 17, 2, 23, 50, 41, 2, 34, 107, 178, 99, 2, 47, 194, 497, 634, 239, 2, 62, 317, 1106, 2309, 2258, 577, 2, 79, 482, 2137, 6306, 10727, 8042, 1393, 2, 98, 695, 3746, 14407, 35954, 49835, 28642, 3363, 2, 119, 962, 6113, 29114, 97127, 204994, 231521

Offset: 1

R. H. Hardin May 28 2012

Table starts
.2.....7......14.......23........34.........47.........62..........79
.2....17......50......107.......194........317........482.........695
.2....41.....178......497......1106.......2137.......3746........6113
.2....99.....634.....2309......6306......14407......29114.......53769
.2...239....2258....10727.....35954......97127.....226274......472943
.2...577....8042....49835....204994.....654797....1758602.....4159927
.2..1393...28642...231521...1168786....4414417...13667858....36590017
.2..3363..102010..1075589...6663906...29760487..106226618...321839625
.2..8119..363314..4996919..37994674..200635007..825593474..2830847119
.2.19601.1293962.23214443.216628994.1352612477.6416514026.24899654327

			Some solutions for n=5 k=4
..1....4....1....1....1....3....2....1....1....4....3....1....0....2....2....3
..1....3....0....4....2....4....1....1....2....2....3....4....1....3....3....1
..1....3....3....1....3....3....2....2....2....4....3....3....1....1....0....3
..1....3....0....4....2....3....3....3....4....2....2....0....4....4....2....1
..1....4....0....3....2....2....3....3....4....4....0....1....4....3....4....2
..1....2....0....4....3....1....0....2....4....2....3....2....1....4....3....4

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 2 is A001333(n+2)
Column 3 is A055099(n+1)
Column 4 is A126473(n+1)
Column 5 is A126501(n+1)
Column 6 is A126528(n+1)
Row 1 is A008865(n+1)

Empirical for column k: a(n) = k*a(n-1) +(k-1)*a(n-2)
Empirical for rows:
n=1: a(k) = k^2 + 2*k - 1
n=2: a(k) = k^3 + 3*k^2 - k - 1
n=3: a(k) = k^4 + 4*k^3 - 4*k + 1
n=4: a(k) = k^5 + 5*k^4 + 2*k^3 - 8*k^2 + k + 1
n=5: a(k) = k^6 + 6*k^5 + 5*k^4 - 12*k^3 - 3*k^2 + 6*k - 1
n=6: a(k) = k^7 + 7*k^6 + 9*k^5 - 15*k^4 - 13*k^3 + 15*k^2 - k - 1
n=7: a(k) = k^8 + 8*k^7 + 14*k^6 - 16*k^5 - 30*k^4 + 24*k^3 + 8*k^2 - 8*k + 1

A254600 Numbers of words on alphabet {0,1,...,10} with no subwords ii, for i from {0,1}.

Original entry on oeis.org

1, 11, 119, 1289, 13961, 151211, 1637759, 17738489, 192124721, 2080893611, 22538058599, 244108628489, 2643928812281, 28636265779211, 310158017102639, 3359306563039289, 36384487784316641, 394078636910520011, 4268246759164049879, 46229175323835178889

Offset: 0

Milan Janjic, Feb 02 2015

a(n) equals the number of sequences over the alphabet {0,1,...,9,10} such that no two consecutive terms have distance 10. - David Nacin, Jun 02 2017

Colin Barker, Table of n, a(n) for n = 0..950
Index entries for linear recurrences with constant coefficients, signature (10,9).

Cf. A015591, A055099, A126473, A126501, A126528.

[n le 1 select 11^n else 10*Self(n)+9*Self(n-1): n in [0..20]]; // Bruno Berselli, Feb 03 2015

RecurrenceTable[{a[0]==1, a[1]==11, a[n]== 10a[n-1] +9a[n-2]}, a[n], {n, 0, 25}]
Table[(-3 I)^(n-1)*(ChebyshevU[n-1, 5*I/3] - 3*I*ChebyshevU[n, 5*I/3]), {n,0,25}] (* G. C. Greubel, Feb 13 2021 *)

Vec((x+1) / (1-10*x-9*x^2) + O(x^30)) \\ Colin Barker, Jan 21 2017

[(-3*i)^(n-1)*( chebyshev_U(n-1, 5*i/3) -3*i*chebyshev_U(n, 5*i/3) ) for n in (0..30)] # G. C. Greubel, Feb 13 2021

G.f.: (1+x)/(1-10*x-9*x^2).
a(n) = 10*a(n-1) + 9*a(n-2) with n>1, a(0) = 1, a(1) = 11.
a(n) = ((5-sqrt(34))^n*(-6+sqrt(34)) + (5+sqrt(34))^n*(6+sqrt(34))) / (2*sqrt(34)). - Colin Barker, Jan 21 2017
a(n) = (-3*i)^(n-1) * (ChebyshevU(n-1, 5*i/3) - 3*i*ChebyshevU(n, 5*i/3)). - G. C. Greubel, Feb 13 2021

A254657 Numbers of words on alphabet {0,1,...,8} with no subwords ii, where i is from {0,1,2}.

Original entry on oeis.org

1, 9, 78, 678, 5892, 51204, 444984, 3867096, 33606672, 292055952, 2538087648, 22057036896, 191684821056, 1665820789824, 14476675244928, 125808326698368, 1093326665056512, 9501463280642304, 82571666235477504, 717582109567673856, 6236086873954255872

Offset: 0

Milan Janjic, Feb 04 2015

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,6).

Cf. A055099, A126473, A126501, A126528, A190560, A254598, A254602.

[n le 1 select 9^n else 8*Self(n)+6*Self(n-1): n in [0..20]];

RecurrenceTable[{a[0] == 1, a[1] == 9, a[n] == 8 a[n - 1] + 6 a[n - 2]}, a[n], {n, 0, 20}]

Vec((1+x)/(1-8*x-6*x^2) + O(x^30)) \\ Colin Barker, Nov 16 2016

G.f.: (1 + x)/(1 - 8*x - 6*x^2).
a(n) = 8*a(n-1) + 6*a(n-2) with n>1, a(0) = 1, a(1) = 9.
a(n) = (((4-sqrt(22))^n*(-5+sqrt(22)) + (4+sqrt(22))^n*(5+sqrt(22))))/(2*sqrt(22)). - Colin Barker, Nov 16 2016

A254601 Numbers of n-length words on alphabet {0,1,...,6} with no subwords ii, where i is from {0,1,2}.

Original entry on oeis.org

1, 7, 46, 304, 2008, 13264, 87616, 578752, 3822976, 25252864, 166809088, 1101865984, 7278432256, 48078057472, 317582073856, 2097804673024, 13857156333568, 91534156693504, 604633565495296, 3993938019745792, 26382162380455936, 174268726361718784

Offset: 0

Milan Janjic, Feb 02 2015

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,4).

Cf. A055099, A126473, A126501, A126528, A135032, A190976 (shifted bin. trans).

[n le 1 select 7^n else 6*Self(n)+4*Self(n-1): n in [0..25]]; // Bruno Berselli, Feb 03 2015

RecurrenceTable[{a[0] == 1, a[1] == 7, a[n] == 6 a[n - 1] + 4 a[n - 2]}, a[n], {n, 0, 25}]
LinearRecurrence[{6,4},{1,7},30] (* Harvey P. Dale, Oct 10 2017 *)

Vec((1 + x)/(1 - 6*x - 4*x^2) + O(x^30)) \\ Colin Barker, Jan 22 2017

G.f.: (1 + x)/(1 - 6*x - 4*x^2).
a(n) =  6*a(n-1) + 4*a(n-2) with n>1, a(0) = 1, a(1) = 7.
a(n) = ((3-r)^n*(-4+r) + (3+r)^n*(4+r)) / (2*r), where r=sqrt(13). - Colin Barker, Jan 22 2017
a(n) = A135032(n-1)+A135032(n). - R. J. Mathar, Apr 07 2022

A254658 Numbers of words on alphabet {0,1,...,7} with no subwords ii, where i is from {0,1,2,3}.

Original entry on oeis.org

1, 8, 60, 452, 3404, 25636, 193068, 1454020, 10950412, 82468964, 621084396, 4677466628, 35226603980, 265296094372, 1997979076524, 15047037913156, 113321181698188, 853436423539940, 6427339691572332, 48405123535166084, 364545223512451916, 2745437058727827748

Offset: 0

Milan Janjic, Feb 04 2015

a(n) equals the number of octonary sequences of length n such that no two consecutive terms differ by 6. - David Nacin, May 31 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,4).

Cf. A015561, A055099, A126473, A126501, A126528, A254598, A254602.

[n le 1 select 8^n else 7*Self(n)+4*Self(n-1): n in [0..20]];

RecurrenceTable[{a[0] == 1, a[1] == 8, a[n] == 7 a[n - 1] + 4 a[n - 2]}, a[n], {n, 0, 20}]
LinearRecurrence[{7,4},{1,8},30] (* Harvey P. Dale, Jan 21 2023 *)

Vec((1 + x) / (1 - 7*x -4*x^2) + O(x^30)) \\ Colin Barker, Jan 21 2017

G.f.: (1 + x)/(1 - 7*x -4*x^2).
a(n) = 7*a(n-1) + 4*a(n-2) with n>1, a(0) = 1, a(1) = 8.
a(n) = (2^(-1-n)*((7-sqrt(65))^n*(-9+sqrt(65)) + (7+sqrt(65))^n*(9+sqrt(65)))) / sqrt(65). - Colin Barker, Jan 21 2017

A254660 Numbers of words on alphabet {0,1,...,6} with no subwords ii, where i is from {0,1,...,4}.

Original entry on oeis.org

1, 7, 44, 278, 1756, 11092, 70064, 442568, 2795536, 17658352, 111541184, 704563808, 4450465216, 28111918912, 177572443904, 1121658501248, 7085095895296, 44753892374272, 282693546036224, 1785669060965888, 11279401457867776, 71247746869138432

Offset: 0

Milan Janjic, Feb 04 2015

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,2).

Cf. A135030, A126473, A126501, A126528, A254598, A254602.

RecurrenceTable[{a[0] == 1, a[1] == 7, a[n] == 6 a[n - 1] + 2 a[n - 2]}, a[n], {n, 0, 20}]
LinearRecurrence[{6,2},{1,7},30] (* Harvey P. Dale, Sep 11 2024 *)

Vec((1 + x) / (1 - 6*x -2*x^2) + O(x^30)) \\ Colin Barker, Jan 21 2017

G.f.: (1 + x)/(1 - 6*x -2*x^2).
a(n) = 6*a(n-1) + 2*a(n-2) with n>1, a(0) = 1, a(1) = 7.
a(n) = ((3-sqrt(11))^n*(-4+sqrt(11)) + (3+sqrt(11))^n*(4+sqrt(11))) / (2*sqrt(11)). - Colin Barker, Jan 21 2017

A254663 Numbers of n-length words on alphabet {0,1,...,7} with no subwords ii, where i is from {0,1,...,5}.

Original entry on oeis.org

1, 8, 58, 422, 3070, 22334, 162478, 1182014, 8599054, 62557406, 455099950, 3310814462, 24085901134, 175222936862, 1274732360302, 9273572395838, 67464471491470, 490798445231966, 3570518059606702, 25975223307710846, 188967599273189326, 1374723641527746974

Offset: 0

Milan Janjic, Feb 04 2015

a(n) equals the number of octonary sequences of length n such that no two consecutive terms differ by 5. - David Nacin, May 31 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,2).

Cf. A015555, A055099, A126473, A126501, A126528, A254598, A254602.

[n le 1 select 8^n else 7*Self(n)+2*Self(n-1): n in [0..20]];

RecurrenceTable[{a[0] == 1, a[1] == 8, a[n] == 7 a[n - 1] + 2 a[n - 2]}, a[n], {n, 0, 20}]
LinearRecurrence[{7,2},{1,8},30] (* Harvey P. Dale, Nov 28 2023 *)

Vec((1 + x)/(1 - 7*x - 2*x^2) + O(x^30)) \\ Colin Barker, Jan 22 2017

G.f.: (1 + x)/(1 - 7*x - 2*x^2).
a(n) = 7*a(n-1) + 2*a(n-2) with n>1, a(0) = 1, a(1) = 8.
a(n) = 2^(-1-n)*((7-r)^n*(-9+r) + (7+r)^n*(9+r)) / r, where r=sqrt(57). - Colin Barker, Jan 22 2017

A254664 Numbers of words on alphabet {0,1,...,8} with no subwords ii, where i is from {0,1,...,5}.

Original entry on oeis.org

1, 9, 75, 627, 5241, 43809, 366195, 3060987, 25586481, 213874809, 1787757915, 14943687747, 124912775721, 1044133269009, 8727804479235, 72954835640907, 609822098564961, 5097441295442409, 42608996659234155, 356164297160200467

Offset: 0

Milan Janjic, Feb 04 2015

a(n) is the number of nonary sequences of length n such that no two consecutive terms have distance 6. - David Nacin, May 31 2017

Index entries for linear recurrences with constant coefficients, signature (8,3).

Cf. A015574, A055099, A126473, A126501, A126528, A254598, A254602.

[n le 1 select 9^n else 8*Self(n)+3*Self(n-1): n in [0..20]];

RecurrenceTable[{a[0] == 1, a[1] == 9, a[n] == 8 a[n - 1] + 3 a[n - 2]}, a[n], {n, 0, 20}]
LinearRecurrence[{8,3},{1,9},20] (* Harvey P. Dale, Feb 16 2024 *)

G.f.: (1 + x)/(1 - 8*x -3*x^2).
a(n) = 8*a(n-1) + 3*a(n-2) with n>1, a(0) = 1, a(1) = 9.
a(n) = ((1+t)*(4-t)^(n+1)+(-1+t)*(4+t)^(n+1))/(6*t), where t=sqrt(19). [Bruno Berselli, Feb 04 2015]

Showing 1-10 of 13 results. Next

cp's OEIS Frontend

A254602 Numbers of n-length words on alphabet {0..7} with no subwords ii, where i is from {0..2}.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A054458 Convolution triangle based on A001333(n), n >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A212835 T(n,k)=Number of 0..k arrays of length n+1 with 0 never adjacent to k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A254600 Numbers of words on alphabet {0,1,...,10} with no subwords ii, for i from {0,1}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A254657 Numbers of words on alphabet {0,1,...,8} with no subwords ii, where i is from {0,1,2}.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A254601 Numbers of n-length words on alphabet {0,1,...,6} with no subwords ii, where i is from {0,1,2}.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A254658 Numbers of words on alphabet {0,1,...,7} with no subwords ii, where i is from {0,1,2,3}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A254660 Numbers of words on alphabet {0,1,...,6} with no subwords ii, where i is from {0,1,...,4}.

Views