A038718 -id:A038718 - cp's OEIS frontend

A171861 Expansion of x(1+x+x^2) / ( (x-1)(x^3+x^2-1) ).

1, 2, 4, 6, 9, 13, 18, 25, 34, 46, 62, 83, 111, 148, 197, 262, 348, 462, 613, 813, 1078, 1429, 1894, 2510, 3326, 4407, 5839, 7736, 10249, 13578, 17988, 23830, 31569, 41821, 55402, 73393, 97226, 128798, 170622, 226027, 299423, 396652, 525453, 696078, 922108

Offset: 1

Ed Pegg Jr, Oct 16 2010

Number of wins in Penney's game if the two players start HHT and TTT and HHT beats TTT.

HHT beats TTT 70% of the time. - Geoffrey Critzer, Mar 01 2014

			a(n) enumerates length n+2 sequences on {H,T} that end in HHT but do not contain the contiguous subsequence TTT.
a(3)=4 because we have: TTHHT, THHHT, HTHHT, HHHHT.
a(4)=6 because we have: TTHHHT, THTHHT, THHHHT, HTTHHT, HTHHHT, HHHHHT. - _Geoffrey Critzer_, Mar 01 2014

G. C. Greubel, Table of n, a(n) for n = 1..1000
Wikipedia, Penney's game
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1).

Related sequences are A000045 (HHH beats HHT, HTT beats TTH), A006498 (HHH beats HTH), A023434 (HHH beats HTT), A000930 (HHH beats THT, HTH beats HHT), A000931 (HHH beats TTH), A077868 (HHT beats HTH), A002620 (HHT beats HTT), A000012 (HHT beats THH), A004277 (HHT beats THT), A070550 (HTH beats HHH), A000027 (HTH beats HTT), A097333 (HTH beats THH), A040000 (HTH beats TTH), A068921 (HTH beats TTT), A054405 (HTT beats HHH), A008619 (HTT beats HHT), A038718 (HTT beats THT), A128588 (HTT beats TTT).

Cf. A164315 (essentially the same sequence).

A171861 := proc(n) option remember; if n <=4 then op(n,[1,2,4,6]); else procname(n-1)+procname(n-2)-procname(n-4) ; end if; end proc:

nn=44;CoefficientList[Series[x(1+x+x^2)/(1-x-x^2+x^4),{x,0,nn}],x] (* Geoffrey Critzer, Mar 01 2014 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,0,1,1]^(n-1)*[1;2;4;6])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

a(n) = a(n-1) +a(n-2) -a(n-4) = A000931(n+10)-3 = A134816(n+6)-3 = A078027(n+12)-3.

a(n) = A164315(n-1). - Alois P. Heinz, Oct 12 2017

A003410 Expansion of (1+x)(1+x^2)/(1-x-x^3).

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 15, 22, 32, 47, 69, 101, 148, 217, 318, 466, 683, 1001, 1467, 2150, 3151, 4618, 6768, 9919, 14537, 21305, 31224, 45761, 67066, 98290, 144051, 211117, 309407, 453458, 664575, 973982, 1427440, 2092015, 3065997, 4493437, 6585452, 9651449

Offset: 0

N. J. A. Sloane

From Emeric Deutsch, Feb 15 2010: (Start)
a(n) is the number of binary words of length n that have no pair of adjacent 1's and have no 0000 subwords. Example: a(4)=7 because we have 0101, 1010, 0010, 1001, 0100, 0001, and 1000.
a(n) = A171855(n,0). (End)
a(n) is the number of solus bitstrings of length n with no runs of 4 zeros. - Steven Finch, Mar 25 2020

R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Steven Finch, Cantor-solus and Cantor-multus distributions, arXiv:2003.09458 [math.CO], 2020.
R. K. Guy, Letter to N. J. A. Sloane, Apr 1975
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (1,0,1).

Essentially the same as A058278 and A097333, partial sums and first differences of A058278, first and second differences of itself and A038718. Equals A038718(n+1) + 1, n>0.

Cf. A171855. - Emeric Deutsch, Feb 15 2010

G:=series((1+x)*(1+x^2)/(1-x-x^3),x=0,42): 1,seq(coeff(G,x^n),n=1..38);
A003410:=-(1+z)*(1+z**2)/(-1+z+z**3); # Simon Plouffe in his 1992 dissertation

Join[{1}, LinearRecurrence[{1, 0, 1}, {2, 3, 5}, 80]] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)
CoefficientList[Series[((1+x)(1+x^2))/(1-x-x^3),{x,0,50}],x] (* Harvey P. Dale, Jul 07 2025 *)

a(n)=([0,1,0; 0,0,1; 1,0,1]^n*[1;2;3])[1,1] \\ Charles R Greathouse IV, Mar 25 2020

a(n) = a(n-1) + a(n-3) for n>3, see also A000930. - Reinhard Zumkeller, Oct 26 2005
For n>1, a(n) = 2*A000930(n) + A000930(n-2). - Gerald McGarvey, Sep 10 2008
a(n) = A058278(n+4) = A097333(n+1) for n >= 1. - Jianing Song, Aug 11 2023

More terms from Emeric Deutsch, Dec 11 2004

A207442 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 0 1 0 horizontally and 0 0 1 and 1 0 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 14, 81, 108, 81, 14, 21, 196, 323, 333, 196, 22, 31, 441, 1058, 1360, 1144, 484, 35, 46, 961, 3223, 6092, 6525, 4048, 1225, 56, 68, 2116, 9515, 25689, 41092, 32393, 14743, 3136, 90, 100, 4624, 28426, 105690, 243981, 287176

Offset: 1

R. H. Hardin Feb 17 2012

Table starts
..2....4.....6......9.......14........21.........31..........46............68
..4...16....36.....81......196.......441........961........2116..........4624
..6...36...108....323.....1058......3223.......9515.......28426.........84486
..9...81...333...1360.....6092.....25689.....105690......439332.......1821924
.14..196..1144...6525....41092....243981....1414091.....8282682......48412066
.22..484..4048..32393...287176...2405923...19685415...162644898....1341574108
.35.1225.14743.165626..2078416..24609889..284247216..3316383098...38630942068
.56.3136.54250.855471.15205846.254583643.4151434555.68394049216.1125157869978

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

R. H. Hardin, Table of n, a(n) for n = 1..390

Column 1 is A001611(n+2)
Column 2 is Column 1 squared
Row 1 is A038718(n+2)
Row 2 is A207069

A207068 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 0 1 0 horizontally and 0 0 1 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 8, 14, 81, 102, 64, 10, 21, 196, 288, 216, 100, 12, 31, 441, 896, 720, 390, 144, 14, 46, 961, 2499, 2688, 1485, 636, 196, 16, 68, 2116, 6634, 8799, 6398, 2709, 966, 256, 18, 100, 4624, 17848, 27063, 23856, 13132, 4536, 1392, 324, 20, 147

Offset: 1

R. H. Hardin Feb 14 2012

Table starts
..2...4....6....9....14.....21.....31......46.......68......100.......147
..4..16...36...81...196....441....961....2116.....4624....10000.....21609
..6..36..102..288...896...2499...6634...17848....47192...122200....315315
..8..64..216..720..2688...8799..27063...84502...257584...762900...2246895
.10.100..390.1485..6398..23856..82739..291364...997288..3297100..10818024
.12.144..636.2709.13132..54684.210118..818892..3093184.11234900..40417797
.14.196..966.4536.24304.111426.468348.1992996..8204200.32364700.126207438
.16.256.1392.7128.41664.208026.947298.4356936.19360144.82226100.344542569

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

R. H. Hardin, Table of n, a(n) for n = 1..1404

Column 2 is A016742
Column 3 is A086113
Row 1 is A038718(n+2)

Empirical for column k:
k=1: a(n) = 2*n
k=2: a(n) = 4*n^2
k=3: a(n) = 2*n^3 + 6*n^2 - 2*n
k=4: a(n) = (3/4)*n^4 + (15/2)*n^3 + (15/4)*n^2 - 3*n
k=5: a(n) = (7/30)*n^5 + 7*n^4 + (21/2)*n^3 - (56/15)*n
k=6: a(n) = (7/120)*n^6 + (147/40)*n^5 + (49/3)*n^4 + (91/8)*n^3 - (707/120)*n^2 - (91/20)*n
k=7: a(n) = (31/2520)*n^7 + (62/45)*n^6 + (4867/360)*n^5 + (2015/72)*n^4 + (1271/180)*n^3 - (4991/360)*n^2 - (713/140)*n

A207242 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 0 1 and 0 1 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 13, 81, 102, 81, 14, 18, 169, 283, 287, 196, 21, 25, 324, 699, 987, 882, 441, 31, 34, 625, 1526, 2884, 3866, 2491, 961, 46, 46, 1156, 3355, 7165, 13876, 13494, 6759, 2116, 68, 62, 2116, 6888, 17929, 40838, 58026, 44730, 18528

Offset: 1

R. H. Hardin Feb 16 2012

Table starts
..2....4.....6......9.....13......18.......25.......34........46........62
..4...16....36.....81....169.....324......625.....1156......2116......3844
..6...36...102....283....699....1526.....3355.....6888.....13954.....27816
..9...81...287....987...2884....7165....17929....40646.....90602....196548
.14..196...882...3866..13876...40838...122197...323039....839664...2118081
.21..441..2491..13494..58026..197256...683257..2037721...5952936..16732138
.31..961..6759..44730.228110..886959..3510185.11662500..37779767.116530737
.46.2116.18528.150608.919413.4132837.18982182.71594896.262849612.914523558

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

R. H. Hardin, Table of n, a(n) for n = 1..449

Column 1 is A038718(n+2)
Column 2 is A207069
Row 1 is A171861(n+1)
Row 2 is A207025

A207269 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 1 1 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 13, 81, 82, 81, 14, 18, 169, 217, 193, 196, 21, 25, 324, 499, 611, 488, 441, 31, 34, 625, 1014, 1602, 1910, 1087, 961, 46, 46, 1156, 2141, 3513, 5904, 5132, 2305, 2116, 68, 62, 2116, 4188, 8327, 14184, 18055, 13067, 4932, 4624, 100, 83

Offset: 1

R. H. Hardin Feb 16 2012

Table starts
..2....4....6.....9.....13.....18......25......34.......46.......62.......83
..4...16...36....81....169....324.....625....1156.....2116.....3844.....6889
..6...36...82...217....499...1014....2141....4188.....8150....15670....29517
..9...81..193...611...1602...3513....8327...17568....36988....76723...153865
.14..196..488..1910...5904..14184...38911...90670...211626...485438..1060958
.21..441.1087..5132..18055..45520..140295..349415...877101..2164569..5025447
.31..961.2305.13067..52240.135686..467686.1233127..3284096..8655283.21076355
.46.2116.4932.33937.156473.417772.1623957.4556171.12881467.36421182.93059536

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

R. H. Hardin, Table of n, a(n) for n = 1..881

Column 1 is A038718(n+2)
Column 2 is A207069
Row 1 is A171861(n+1)
Row 2 is A207025

A207426 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 1 and 0 1 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 15, 81, 114, 81, 14, 25, 225, 361, 351, 196, 21, 40, 625, 1425, 1521, 1162, 441, 31, 64, 1600, 5625, 8463, 6889, 3633, 961, 46, 104, 4096, 20550, 47089, 55361, 29929, 11067, 2116, 68, 169, 10816, 75076, 241087, 444889, 341329

Offset: 1

R. H. Hardin Feb 17 2012

Table starts
..2....4.....6......9.......15........25.........40...........64...........104
..4...16....36.....81......225.......625.......1600.........4096.........10816
..6...36...114....361.....1425......5625......20550........75076........282494
..9...81...351...1521.....8463.....47089.....241087......1234321.......6520459
.14..196..1162...6889....55361....444889....3210938.....23174596.....174570082
.21..441..3633..29929...341329...3892729...39698733....404854641....4315250265
.31..961.11067.127449..2048823..32936121..474552171...6837470721..102807481767
.46.2116.33994.546121.12436631.283215241.5755114104.116947584576.2485504480392

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

R. H. Hardin, Table of n, a(n) for n = 1..365

Column 1 is A038718(n+2)
Column 2 is A207069
Row 1 is A006498(n+2)
Row 2 is A189145(n+2)

A207729 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 15, 81, 78, 81, 14, 25, 225, 169, 189, 196, 21, 40, 625, 611, 441, 490, 441, 31, 64, 1600, 2209, 2163, 1225, 1113, 961, 46, 104, 4096, 6016, 10609, 8575, 2809, 2449, 2116, 68, 169, 10816, 16384, 33063, 60025, 27931, 6241, 5474, 4624

Offset: 1

R. H. Hardin Feb 19 2012

Table starts
..2....4....6.....9.....15......25.......40.......64........104.........169
..4...16...36....81....225.....625.....1600.....4096......10816.......28561
..6...36...78...169....611....2209.....6016....16384......51840......164025
..9...81..189...441...2163...10609....33063...103041.....418263.....1697809
.14..196..490..1225...8575...60025...211680...746496....4078944....22287841
.21..441.1113..2809..27931..277729..1029231..3814209...27996255...205492225
.31..961.2449..6241..88243.1247689..4799749.18464209..184732327..1848226081
.46.2116.5474.14161.288813.5890329.23473944.93547584.1307760792.18282014521

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

R. H. Hardin, Table of n, a(n) for n = 1..649

Column 1 is A038718(n+2)
Column 2 is A207069
Row 1 is A006498(n+2)
Row 2 is A189145(n+2)

A207123 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 0 1 0 horizontally and 0 0 1 and 0 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 14, 81, 102, 81, 12, 21, 196, 288, 289, 144, 16, 31, 441, 896, 1024, 612, 256, 20, 46, 961, 2499, 4096, 2560, 1296, 400, 25, 68, 2116, 6634, 14161, 12288, 6400, 2340, 625, 30, 100, 4624, 17848, 45796, 49861, 36864, 13200, 4225, 900

Offset: 1

R. H. Hardin Feb 15 2012

Table starts
..2...4....6.....9.....14......21......31.......46........68........100
..4..16...36....81....196.....441.....961.....2116......4624......10000
..6..36..102...288....896....2499....6634....17848.....47192.....122200
..9..81..289..1024...4096...14161...45796...150544....481636....1493284
.12.144..612..2560..12288...49861..186822...712756...2628872....9322638
.16.256.1296..6400..36864..175561..762129..3374569..14348944...58201641
.20.400.2340.13200..87744..475984.2330037.11635558..55554808..251535759
.25.625.4225.27225.208849.1290496.7123561.40119556.215091556.1087086841

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

R. H. Hardin, Table of n, a(n) for n = 1..545

Column 1 is A002620(n+2)
Column 2 is A030179(n+2)
Row 1 is A038718(n+2)
Row 2 is A207069
Row 3 is A207070

A207564 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 0 1 0 horizontally and 0 1 1 and 1 1 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 14, 81, 92, 81, 14, 21, 196, 241, 221, 196, 22, 31, 441, 720, 636, 618, 484, 35, 46, 961, 1889, 2234, 2135, 1690, 1225, 56, 68, 2116, 4719, 6315, 9568, 6709, 4861, 3136, 90, 100, 4624, 12102, 16812, 32823, 36426, 23276, 13900

Offset: 1

R. H. Hardin Feb 18 2012

Table starts
..2....4.....6.....9.....14......21.......31........46........68........100
..4...16....36....81....196.....441......961......2116......4624......10000
..6...36....92...241....720....1889.....4719.....12102.....30414......74588
..9...81...221...636...2234....6315....16812.....47596....129150.....337186
.14..196...618..2135...9568...32823...106833....378104...1270366....4115246
.22..484..1690..6709..36426..138017...493471...1980774...7259428...25276248
.35.1225..4861.23276.160652..738515..3275898..16648464..77299468..346860730
.56.3136.13900.78733.666212.3451393.17232331.100565794.513390118.2498101416

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

R. H. Hardin, Table of n, a(n) for n = 1..511

Column 1 is A001611(n+2)
Column 2 is A207436
Row 1 is A038718(n+2)
Row 2 is A207069
Row 3 is A207414

cp's OEIS Frontend

A171861 Expansion of x*(1+x+x^2) / ( (x-1)*(x^3+x^2-1) ).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A003410 Expansion of (1+x)(1+x^2)/(1-x-x^3).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A207442 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 0 1 0 horizontally and 0 0 1 and 1 0 0 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A207068 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 0 1 0 horizontally and 0 0 1 and 1 0 1 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A207242 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 0 1 and 0 1 0 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A207269 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 1 1 and 1 0 1 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A207426 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 1 and 0 1 0 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A207729 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A207123 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 0 1 0 horizontally and 0 0 1 and 0 1 1 vertically.

Views

Author

A171861 Expansion of x(1+x+x^2) / ( (x-1)(x^3+x^2-1) ).