cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

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

Views

Author

Milan Janjic, Feb 04 2015

Keywords

Links

Crossrefs

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

Programs

  • Magma
    [n le 1 select 9^n else 8*Self(n)+6*Self(n-1): n in [0..20]];
  • Mathematica
    RecurrenceTable[{a[0] == 1, a[1] == 9, a[n] == 8 a[n - 1] + 6 a[n - 2]}, a[n], {n, 0, 20}]
  • PARI
    Vec((1+x)/(1-8*x-6*x^2) + O(x^30)) \\ Colin Barker, Nov 16 2016

Formula

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

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

Views

Author

Milan Janjic, Feb 04 2015

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Magma
    [n le 1 select 8^n else 7*Self(n)+4*Self(n-1): n in [0..20]];
  • Mathematica
    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 *)
  • PARI
    Vec((1 + x) / (1 - 7*x -4*x^2) + O(x^30)) \\ Colin Barker, Jan 21 2017

Formula

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

Views

Author

Milan Janjic, Feb 04 2015

Keywords

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Milan Janjic, Feb 04 2015

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Magma
    [n le 1 select 8^n else 7*Self(n)+2*Self(n-1): n in [0..20]];
  • Mathematica
    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 *)
  • PARI
    Vec((1 + x)/(1 - 7*x - 2*x^2) + O(x^30)) \\ Colin Barker, Jan 22 2017

Formula

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

Views

Author

Milan Janjic, Feb 04 2015

Keywords

Comments

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

Links

Crossrefs

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

Programs

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

Formula

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]

A255633 Number of n-length words on {0,1,2,3,4,5} avoiding runs of zeros of length 1 (mod 3).

Original entry on oeis.org

1, 5, 26, 136, 710, 3706, 19346, 100990, 527186, 2752006, 14365970, 74992966, 391476866, 2043580150, 10667858546, 55688153926, 290702250530, 1517518403926, 7921720943186, 41352818219110, 215869201519106, 1126876333254646, 5882498575587890, 30707708087054086
Offset: 0

Views

Author

Milan Janjic, Feb 28 2015

Keywords

Links

Crossrefs

Cf. A254598, A254602, A255115, A255117, A254601, A254663.

Programs

  • Mathematica
    RecurrenceTable[{a[0] == 1, a[1] == 5,  a[2] == 26,  a[n] == 5* a[n - 1] +  6*a[n - 3]}, a[n], {n, 0, 20}]
LinearRecurrence[{5,0,6},{1,5,26},30] (* Harvey P. Dale, Aug 11 2023 *)
  • PARI
    Vec((1 + x^2)/(1 - 5*x - 6*x^3) + O(x^30)) \\ Andrew Howroyd, May 01 2020

Formula

a(n+3) = 5*a(n+2) + 6*a(n) with n > 0, a(0) = 1, a(1) = 5, a(2) = 26.
G.f.: (1 + x^2)/(1 - 5*x - 6*x^3). - Andrew Howroyd, May 01 2020

Extensions

Terms a(20) and beyond from Andrew Howroyd, May 01 2020

A255630 Number of n-length ternary words avoiding runs of zeros of length 1 (mod 3).

Original entry on oeis.org

1, 2, 5, 13, 32, 79, 197, 490, 1217, 3025, 7520, 18691, 46457, 115474, 287021, 713413, 1773248, 4407559, 10955357, 27230458, 67683593, 168233257, 418157888, 1039366555, 2583432881, 6421339426, 15960778517, 39671855677, 98607729632
Offset: 0

Views

Author

Milan Janjic, Feb 28 2015

Keywords

Links

Crossrefs

Cf. A254598, A254602, A255115, A255117, A254601, A254663.

Programs

  • Mathematica
    RecurrenceTable[{a[0] == 1, a[2] == 2,  a[3] == 5,  a[n] == 2* a[n - 1] +  3*a[n - 3]}, a[n], {n, 0, 29}]

Formula

a(n+3) = 2*a(n+2) + 3*a(n) with n > 0, a(0) = 1, a(2) = 2, a(3) = 5.
G.f.: ( -1-x^2 ) / ( -1+2*x+3*x^3 ). - R. J. Mathar, Aug 07 2015
a(n) = A099525(n)+A099525(n-2). - R. J. Mathar, Aug 07 2015

A254659 Numbers of words on alphabet {0,1,...,8} with no subwords ii, where i is from {0,1,2,3}.

Original entry on oeis.org

1, 9, 77, 661, 5673, 48689, 417877, 3586461, 30781073, 264180889, 2267352477, 19459724261, 167014556473, 1433415073089, 12302393367077, 105586222302061, 906201745251873, 7777545073525289, 66751369314461677, 572898679883319861, 4916946285638867273
Offset: 0

Views

Author

Milan Janjic, Feb 04 2015

Keywords

Comments

a(n) is the number of nonary sequences of length n such that no two consecutive terms have distance 7. - David Nacin, May 31 2017

Links

Crossrefs

Cf. A015575, A055099, A126473, A126501, A126528, A254598, A254602.

Programs

  • Magma
    [n le 1 select 9^n else 8*Self(n)+5*Self(n-1): n in [0..20]];
  • Mathematica
    RecurrenceTable[{a[0] == 1, a[1] == 9, a[n] == 8 a[n - 1] + 5 a[n - 2]}, a[n], {n, 0, 20}]
  • PARI
    Vec((1 + x)/(1 - 8*x -5*x^2) + O(x^30)) \\ Colin Barker, Jan 22 2017

Formula

G.f.: (1 + x)/(1 - 8*x - 5*x^2).
a(n) = 8*a(n-1) + 5*a(n-2) with n>1, a(0) = 1, a(1) = 9.
a(n) = ((4-r)^n*(-5+r) + (4+r)^n*(5+r)) / (2*r), where r=sqrt(21). - Colin Barker, Jan 22 2017

A254662 Numbers of words on alphabet {0,1,...,7} with no subwords ii, where i is from {0,1,...,4}.

Original entry on oeis.org

1, 8, 59, 437, 3236, 23963, 177449, 1314032, 9730571, 72056093, 533584364, 3951258827, 29259564881, 216670730648, 1604473809179, 11881328856197, 87982723420916, 651523050515003, 4824609523867769, 35726835818619392, 264561679301939051, 1959112262569431533
Offset: 0

Views

Author

Milan Janjic, Feb 04 2015

Keywords

Links

Crossrefs

Cf. A015559, A055099, A126473, A126501, A126528, A254598, A254602.

Programs

  • Magma
    [n le 1 select 8^n else 7*Self(n)+3*Self(n-1): n in [0..20]];
  • Mathematica
    RecurrenceTable[{a[0] == 1, a[1] == 8, a[n] == 7 a[n - 1] + 3 a[n - 2]}, a[n], {n, 0, 20}]
  • PARI
    Vec((1+x)/(1-7*x-3*x^2) + O(x^30)) \\ Colin Barker, Sep 08 2016

Formula

G.f.: (1 + x)/(1 - 7*x -3*x^2).
a(n) = 7*a(n-1) + 3*a(n-2) with n>1, a(0) = 1, a(1) = 8.
a(n) = (2^(-1-n) * ((7-sqrt(61))^n * (-9+sqrt(61)) + (7+sqrt(61))^n * (9+sqrt(61)))) / sqrt(61). - Colin Barker, Sep 08 2016

A255631 Number of n-length words on {0,1,2,3} avoiding runs of zeros of length 1 (mod 3).

Original entry on oeis.org

1, 3, 10, 34, 114, 382, 1282, 4302, 14434, 48430, 162498, 545230, 1829410, 6138222, 20595586, 69104398, 231866082, 777980590, 2610359362, 8758542414, 29387549602, 98604086254, 330846428418, 1110089483662, 3724684796002, 12497440101678, 41932678239682
Offset: 0

Views

Author

Milan Janjic, Feb 28 2015

Keywords

Links

Crossrefs

Cf. A254598, A254602, A255115, A255117, A254601, A254663.

Programs

  • Mathematica
    RecurrenceTable[{a[0] == 1, a[1] == 3,  a[2] == 10,  a[n] == 3* a[n - 1] +  4*a[n - 3]}, a[n], {n, 0, 25}]
LinearRecurrence[{3,0,4},{1,3,10},40] (* Harvey P. Dale, Aug 01 2021 *)

Formula

a(n+3) = 3*a(n+2) + 4*a(n) with n > 0, a(0) = 1, a(1) = 3, a(2) = 10.
G.f.: -(x^2+1) / (4*x^3+3*x-1). - Colin Barker, Mar 20 2015
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