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.

User: Milan Janjic

Milan Janjic's wiki page.

Milan Janjic has authored 81 sequences. Here are the ten most recent ones:

A255821 Numbers of words on {0,1,...,36} having no isolated zeros.

Original entry on oeis.org

1, 36, 1297, 46729, 1683577, 60656797, 2185374961, 78735837637, 2836736138665, 102203420474269, 3682238546710945, 132665625592223221, 4779746882367738841, 172207232713967895181, 6204372685172893559377, 223534399861459456068709
Offset: 0

Views

Author

Milan Janjic, Mar 07 2015

Keywords

Comments

The number p_n = a(n)/37^n equals the probability that in n trials in single zero (European) Roulette zero will not appear isolated. For example, p_10 is approximately 0.021.

Links

Crossrefs

Cf. A255116, A255118, A254658, A254660, A255633, A255630.

Programs

  • Mathematica
    RecurrenceTable[{a[0] == 1, a[1] == 36,  a[2]== 1297, a[n] == 37 a[n - 1] - 36 a[n - 2] + 36 a[n - 3]}, a[n], {n, 0, 15}]
LinearRecurrence[{37,-36,36}, {1, 36, 1297}, 100] (* G. C. Greubel, Jun 02 2016 *)
  • PARI
    Vec(-(x^2-x+1)/(36*x^3-36*x^2+37*x-1) + O(x^100)) \\ Colin Barker, Mar 09 2015

Formula

G.f.: -(x^2 - x + 1)/(36*x^3 - 36*x^2 + 37*x - 1). - Colin Barker, Mar 09 2015
a(n) = 37*a(n-1) - 36*a(n-2) + 36*a(n-3). - G. C. Greubel, Jun 02 2016

A255814 Numbers of words on {0,1,2,3,4,} having no isolated zeros.

Original entry on oeis.org

1, 4, 17, 73, 313, 1341, 5745, 24613, 105449, 451773, 1935521, 8292309, 35526553, 152205613, 652091089, 2793739205, 11969154121, 51279178141, 219694231041, 941231059125, 4032495084025, 17276328107789, 74016584439345, 317107590101669
Offset: 0

Views

Author

Milan Janjic, Mar 07 2015

Keywords

Links

Crossrefs

Cf. A255116, A255118, A254658, A254660, A255633, A255630.

Programs

  • Magma
    I:=[1,4,17]; [n le 3 select I[n] else 5*Self(n-1)-4*Self(n-2)+4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Feb 26 2018
  • Mathematica
    RecurrenceTable[{a[0] == 1, a[1] == 4,  a[2]== 17, a[n] == 5 a[n - 1] - 4 a[n - 2] + 4 a[n - 3]}, a[n], {n, 0, 23}]
LinearRecurrence[{5, -4, 4}, {1, 4, 17}, 100] (* G. C. Greubel, Jun 02 2016 *)
CoefficientList[Series[(-1 + x - x^2) / (-1 + 5 x -4 x^2 + 4 x^3), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 26 2018 *)

Formula

a(n+3) = 5*a(n+2) - 4*a(n+1)+ 4*a(n) with n>=0, a(0) = 1, a(1) = 4, a(2) = 17.
G.f.: (-1 + x - x^2)/(-1 + 5*x - 4*x^2 + 4*x^3). - R. J. Mathar, Nov 07 2015

A255815 Numbers of words on {0,1,2,3,4,5} having no isolated zeros.

Original entry on oeis.org

1, 5, 26, 136, 711, 3716, 19421, 101501, 530481, 2772486, 14490016, 75730071, 395792776, 2068556381, 10811024761, 56502330541, 295301641346, 1543353319176, 8066123361031, 42156481777036, 220325040452941, 1151498450637621
Offset: 0

Views

Author

Milan Janjic, Mar 07 2015

Keywords

Links

Crossrefs

Cf. A255116, A255118, A254658, A254660, A255633, A255630

Programs

  • Mathematica
    RecurrenceTable[{a[0] == 1, a[1] == 5,  a[2]== 26, a[n] == 6 a[n - 1] - 5 a[n - 2] + 5 a[n - 3]}, a[n], {n, 0, 21}]
LinearRecurrence[{6, -5, 5}, {1, 5, 26}, 100] (* G. C. Greubel, Jun 02 2016 *)

Formula

a(n+3) = 6*a(n+2) - 5*a(n+1)+ 5*a(n) with n>=0, a(0) = 1, a(1) = 5, a(2) = 26.
G.f.: (-1 + x - x^2)/(-1 + 6*x - 5*x^2 + 5*x^3). - R. J. Mathar, Nov 07 2015

A255813 Numbers of words on {0,1,2,3} having no isolated zeros.

Original entry on oeis.org

1, 3, 10, 34, 115, 388, 1309, 4417, 14905, 50296, 169720, 572707, 1932556, 6521263, 22005505, 74255899, 250570870, 845532298, 2853184279, 9627852832, 32488455385, 109629815881, 369937455865, 1248325741972, 4212380047936
Offset: 0

Views

Author

Milan Janjic, Mar 07 2015

Keywords

Links

Crossrefs

Cf. A255116, A255118, A254658, A254660, A255633, A255630

Programs

  • Mathematica
    RecurrenceTable[{a[0] == 1, a[1] == 3,  a[2]== 10, a[n] == 4 a[n - 1] - 3 a[n - 2] + 3 a[n - 3]}, a[n], {n, 0, 25}]
LinearRecurrence[{4, -3, 3}, {1, 3, 10}, 100] (* G. C. Greubel, Jun 02 2016 *)

Formula

a(n+3) = 4*a(n+2) - 3*a(n+1)+ 3*a(n) with n>=0, a(0) = 1, a(1) = 3, a(2) = 10.
G.f.: (-1 + x - x^2)/(-1 + 4*x - 3*x^2 + 3*x^3). - R. J. Mathar, Nov 07 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

A255632 Number of n-length words on {0,1,2,3,4} avoiding runs of zeros of length 1 (mod 3).

Original entry on oeis.org

1, 4, 17, 73, 312, 1333, 5697, 24348, 104057, 444713, 1900592, 8122653, 34714177, 148359668, 634051937, 2709778633, 11580912872, 49493911173, 211524537857, 904002715788, 3863480419017, 16511544365353, 70566191040352
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] == 4,  a[2] == 17,  a[n] == 4*a[n - 1] +  5*a[n - 3]}, a[n], {n, 0, 23}]

Formula

a(n+3) = 4*a(n+2) + 5*a(n) with n > 0, a(0) = 1, a(1) = 4, a(2) = 17.
G.f.: ( -1-x^2 ) / ( -1+4*x+5*x^3 ). - R. J. Mathar, Nov 07 2015

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

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

A255119 Number of n-length words on {0,1,2,3,4,5,6} in which 0 appears only in runs of length 2.

Original entry on oeis.org

1, 6, 37, 228, 1404, 8646, 53244, 327888, 2019204, 12434688, 76575456, 471567960, 2904015888, 17883548064, 110130696144, 678208272192, 4176550921536, 25720089706080, 158389787869632, 975398032747008, 6006708734718528, 36990591135528960
Offset: 0

Views

Author

Milan Janjic, Feb 14 2015

Keywords

Links

Crossrefs

Cf. A000930, A239333, A239340, A254657, A254600, A254664.

Programs

  • Mathematica
    RecurrenceTable[{a[0] == 1, a[1] == 6,  a[2]== 37, a[n] == 6 a[n - 1] + 6 a[n - 3]}, a[n], {n, 0, 20}]
LinearRecurrence[{6,0,6},{1,6,37},30] (* Harvey P. Dale, Nov 06 2017 *)
  • PARI
    Vec(-(x^2+1)/(6*x^3+6*x-1) + O(x^100)) \\ Colin Barker, Feb 15 2015

Formula

a(n+3) = 6*a(n+2) + 6*a(n) with n>1, a(0) = 1, a(1) = 6, a(2) = 37.
G.f.: -(x^2+1) / (6*x^3+6*x-1). - Colin Barker, Feb 15 2015

A255118 Number of n-length words on {0,1,2,3,4,5} in which 0 appears only in runs of length 2.

Original entry on oeis.org

1, 5, 26, 135, 700, 3630, 18825, 97625, 506275, 2625500, 13615625, 70609500, 366175000, 1898953125, 9847813125, 51069940625, 264844468750, 1373461409375, 7122656750000, 36937506093750, 191554837515625, 993387471328125, 5151624887109375, 26715898623125000
Offset: 0

Views

Author

Milan Janjic, Feb 14 2015

Keywords

Links

Crossrefs

Cf. A000930, A239333, A239340, A254657, A254600, A254664.

Programs

  • Mathematica
    RecurrenceTable[{a[0] == 1, a[1] == 5,  a[2]== 26, a[n] == 5 a[n - 1] + 5 a[n - 3]}, a[n], {n, 0, 20}]
  • PARI
    Vec(-(x^2+1)/(5*x^3+5*x-1) + O(x^100)) \\ Colin Barker, Feb 15 2015

Formula

a(n+3) = 5*a(n+2) + 5*a(n) with n>1, a(0) = 1, a(1) = 5, a(2) = 26.
G.f.: -(x^2+1) / (5*x^3+5*x-1). - Colin Barker, Feb 15 2015

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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