A171861 -id:A171861 - cp's OEIS frontend

A207500 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 0 1 and 1 0 0 vertically.

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 13, 81, 102, 81, 14, 18, 169, 283, 297, 196, 22, 25, 324, 699, 1004, 932, 484, 35, 34, 625, 1526, 2942, 3939, 2974, 1225, 56, 46, 1156, 3355, 7305, 14253, 15495, 9723, 3136, 90, 62, 2116, 6888, 17911, 41938, 68745, 62530, 32164

Offset: 1

R. H. Hardin Feb 18 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...297...1004....2942....7305....17911.....40262.....87990.....187012
.14..196...932...3939...14253...41938...122061....319798....813724....2009628
.22..484..2974..15495...68745..237576...804887...2408502...6896518...18962878
.35.1225..9723..62530..343310.1413961..5715255..20090516..67370124..216796950
.56.3136.32164.253747.1714707.8384076.40046975.163471950.628620128.2300337450

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

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

Column 1 is A001611(n+2)
Column 2 is A207436
Row 1 is A171861(n+1)
Row 2 is A207025
Row 3 is A207243

A207682 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 0 1 horizontally and 0 0 1 and 1 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 13, 81, 72, 81, 13, 19, 169, 164, 166, 169, 18, 28, 361, 336, 436, 360, 324, 25, 41, 784, 702, 964, 1030, 660, 625, 34, 60, 1681, 1488, 2132, 2310, 2032, 1292, 1156, 46, 88, 3600, 3164, 4846, 5704, 4728, 4174, 2400, 2116, 62, 129

Offset: 1

R. H. Hardin Feb 19 2012

Table starts
..2....4....6....9....13....19.....28.....41.....60......88.....129......189
..4...16...36...81...169...361....784...1681...3600....7744...16641....35721
..6...36...72..164...336...702...1488...3164...6612...13916...29532....62032
..9...81..166..436...964..2132...4846..11301..25496...57407..131250...299913
.13..169..360.1030..2310..5704..14090..34245..81936..201951..490844..1183972
.18..324..660.2032..4728.11994..29982..77141.193184..485658.1213580..3090430
.25..625.1292.4174..9450.25076..65564.171382.433468.1150747.2968284..7722348
.34.1156.2400.8266.18940.52632.139730.379618.990368.2737864.7139792.19521802

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

R. H. Hardin, Table of n, a(n) for n = 1..839

Column 1 is A171861(n+1)
Column 2 is A207025
Column 3 is A207509
Row 1 is A000930(n+3)
Row 2 is A207170

A207693 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 1 and 1 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 15, 81, 60, 81, 13, 25, 225, 100, 144, 169, 18, 40, 625, 240, 256, 312, 324, 25, 64, 1600, 576, 768, 576, 612, 625, 34, 104, 4096, 1296, 2304, 1872, 1156, 1250, 1156, 46, 169, 10816, 2916, 5856, 6084, 4216, 2500, 2516, 2116, 62, 273

Offset: 1

R. H. Hardin Feb 19 2012

Table starts
..2....4....6....9....15.....25.....40......64.....104......169.......273
..4...16...36...81...225....625...1600....4096...10816....28561.....74529
..6...36...60..100...240....576...1296....2916....6804....15876.....36288
..9...81..144..256...768...2304...5856...14884...42700...122500....320600
.13..169..312..576..1872...6084..18564...56644..177548...556516...1724752
.18..324..612.1156..4216..15376..50096..163216..578528..2050624...6804864
.25..625.1250.2500.10000..40000.145200..527076.2056032..8020224..29854944
.34.1156.2516.5476.24420.108900.453420.1887876.8263236.36168196.155101060

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

R. H. Hardin, Table of n, a(n) for n = 1..648

Column 1 is A171861(n+1)
Column 2 is A207025
Column 3 is A207584
Row 1 is A006498(n+2)
Row 2 is A189145(n+2)

A207908 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 0 and 0 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 15, 81, 102, 81, 13, 25, 225, 289, 261, 169, 18, 40, 625, 1071, 841, 611, 324, 25, 64, 1600, 3969, 4089, 2209, 1278, 625, 34, 104, 4096, 13230, 19881, 13865, 5041, 2625, 1156, 46, 169, 10816, 44100, 80511, 87025, 39831, 11025

Offset: 1

R. H. Hardin Feb 21 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..102...289...1071....3969....13230.....44100.....153090......531441
..9...81..261...841...4089...19881....80511....326041....1428071.....6255001
.13..169..611..2209..13865...87025...417425...2002225...10896915....59305401
.18..324.1278..5041..39831..314721..1726758...9474084...62431074...411400089
.25..625.2625.11025.110775.1113025..6835345..41977441..336253621..2693506201
.34.1156.5134.22801.289467.3674889.24832818.167806116.1627138986.15777620881

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

R. H. Hardin, Table of n, a(n) for n = 1..513

Column 1 is A171861(n+1)
Column 2 is A207025
Row 1 is A006498(n+2)
Row 2 is A189145(n+2)
Row 3 is A207704

A070550 a(n) = a(n-1) + a(n-3) + a(n-4), starting with a(0..3) = 1, 2, 2, 3.

Original entry on oeis.org

1, 2, 2, 3, 6, 10, 15, 24, 40, 65, 104, 168, 273, 442, 714, 1155, 1870, 3026, 4895, 7920, 12816, 20737, 33552, 54288, 87841, 142130, 229970, 372099, 602070, 974170, 1576239, 2550408, 4126648, 6677057, 10803704, 17480760, 28284465, 45765226

Offset: 0

Sreyas Srinivasan (sreyas_srinivasan(AT)hotmail.com), May 02 2002

Shares some properties with Fibonacci sequence.
The sum of any two alternating terms (terms separated by one other term) produces a Fibonacci number (e.g., 2+6=8, 3+10=13, 24+65=89). The product of any two consecutive or alternating Fibonacci terms produces a term from this sequence (e.g., 5*8 = 40, 13*5 = 65, 21*8 = 168).
In Penney's game (see A171861), the number of ways that HTH beats HHH on flip 3,4,5,... - Ed Pegg Jr, Dec 02 2010
The Ca2 sums (see A180662 for the definition of these sums) of triangle A035607 equal the terms of this sequence. - Johannes W. Meijer, Aug 05 2011

			G.f.: 1 + 2*x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 15*x^6 + 24*x^7 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
David Applegate, Marc LeBrun and N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq., Vol. 14 (2011), Article # 11.9.8.
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,1).

Bisections: A001654, A059929.

Cf. A049853, A290565.

a070550 n = a070550_list !! n
a070550_list = 1 : 2 : 2 : 3 :
   zipWith (+) a070550_list
               (zipWith (+) (tail a070550_list) (drop 3 a070550_list))
-- Reinhard Zumkeller, Aug 06 2011

with(combinat): A070550 := proc(n): fibonacci(floor(n/2)+1) * fibonacci(ceil(n/2)+2) end: seq(A070550(n),n=0..37); # Johannes W. Meijer, Aug 05 2011

LinearRecurrence[{1, 0, 1, 1}, {1, 2, 2, 3}, 40] (* Jean-François Alcover, Jan 27 2018 *)
nxt[{a_,b_,c_,d_}]:={b,c,d,a+b+d}; NestList[nxt,{1,2,2,3},40][[;;,1]] (* Harvey P. Dale, Jul 16 2024 *)

A070550(n) = fibonacci(n\2+1)*fibonacci((n+5)\2) \\ M. F. Hasler, Aug 06 2011

x='x+O('x^100); Vec((1+x)/(1-x-x^3-x^4)) \\ Altug Alkan, Dec 24 2015

a(n) = F(floor(n/2)+1)*F(ceiling(n/2)+2), with F(n) = A000045(n). - Ralf Stephan, Apr 14 2004
G.f.: (1+x)/(1-x-x^3-x^4) = (1+x)/((1+x^2)*(1-x-x^2))
a(n) = A126116(n+4) - F(n+3). - Johannes W. Meijer, Aug 05 2011
a(n) = (1+3*i)/10*(-i)^n + (1-3*i)/10*(i)^n + (2+sqrt(5))/5*((1+sqrt(5))/2)^n + (2-sqrt(5))/5*((1-sqrt(5))/2)^n, where i = sqrt(-1). - Sergei N. Gladkovskii, Jul 16 2013
a(n+1)*a(n+3) = a(n)*a(n+2) + a(n+1)*a(n+2) for all n in Z. - Michael Somos, Jan 19 2014
Sum_{n>=1} 1/a(n) = A290565. - Amiram Eldar, Feb 17 2021

A207785 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 0 0 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 13, 81, 98, 81, 13, 18, 169, 257, 253, 169, 19, 25, 324, 611, 791, 621, 361, 28, 34, 625, 1326, 2173, 2200, 1575, 784, 41, 46, 1156, 2815, 5241, 6766, 6418, 4000, 1681, 60, 62, 2116, 5718, 12567, 17763, 22322, 19041, 10057, 3600, 88, 83

Offset: 1

R. H. Hardin Feb 20 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....98...257....611...1326....2815....5718....11362....22164.....42507
..9...81...253...791...2173...5241...12567...28101....61397...131083....272789
.13..169...621..2200...6766..17763...46199..110732...257118...582114...1274558
.19..361..1575..6418..22322..65094..185613..486006..1224264..2997835...7098270
.28..784..4000.19041..75557.246087..786907.2283773..6379507.17308075..45325727
.41.1681.10057.55101.246840.887406.3121285.9855231.29800935.87179997.244806537

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

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

Column 1 is A000930(n+3)
Column 2 is A207170
Column 3 is A203084(n-2)
Row 1 is A171861(n+1)
Row 2 is A207025
Row 3 is A202371(n-2)

A208420 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 0 and 0 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 9, 16, 100, 102, 81, 13, 26, 256, 378, 261, 169, 18, 42, 676, 1260, 1269, 611, 324, 25, 68, 1764, 4374, 5139, 3835, 1278, 625, 34, 110, 4624, 14946, 22509, 18395, 10098, 2625, 1156, 46, 178, 12100, 51384, 95265, 100113, 55404, 26375

Offset: 1

R. H. Hardin Feb 26 2012

Table starts
..2....4....6....10.....16......26.......42........68........110.........178
..4...16...36...100....256.....676.....1764......4624......12100.......31684
..6...36..102...378...1260....4374....14946.....51384.....176238......605022
..9...81..261..1269...5139...22509....95265....409239....1746639.....7475751
.13..169..611..3835..18395..100113...512525...2702193...14044303....73505289
.18..324.1278.10098..55404..365094..2187162..13759164...84389022...524422458
.25..625.2625.26375.161975.1297475..8948825..67061525..479579725..3521095775
.34.1156.5134.65178.440436.4270706.33334858.295643872.2428615750.20882937190

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

R. H. Hardin, Table of n, a(n) for n = 1..928

Column 1 is A171861(n+1)
Column 2 is A207025
Column 3 is A207903
Row 1 is A006355(n+2)
Row 2 is A206981
Row 3 is A060521

A207514 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 0 1 and 1 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 13, 81, 72, 81, 13, 18, 169, 166, 166, 169, 18, 25, 324, 360, 438, 360, 324, 25, 34, 625, 660, 1042, 1042, 660, 625, 34, 46, 1156, 1292, 1992, 2638, 1992, 1292, 1156, 46, 62, 2116, 2400, 4168, 5334, 5334, 4168, 2400, 2116, 62, 83, 3844

Offset: 1

R. H. Hardin Feb 18 2012

Table starts
..2....4....6....9....13....18.....25.....34.....46.....62......83.....111
..4...16...36...81...169...324....625...1156...2116...3844....6889...12321
..6...36...72..166...360...660...1292...2400...4396...8096...14580...26346
..9...81..166..438..1042..1992...4168...8174..15881..30801...58335..111083
.13..169..360.1042..2638..5334..11465..23663..47894..96640..191362..379376
.18..324..660.1992..5334.10340..22960..46826..94601.192192..375592..746568
.25..625.1292.4168.11465.22960..50322.105370.214519.434467..859485.1700263
.34.1156.2400.8174.23663.46826.105370.221672.455400.933158.1848347.3679944

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

R. H. Hardin, Table of n, a(n) for n = 1..1151

Column 1 is A171861(n+1)

Column 2 is A207025

A164315 Number of binary strings of length n with no substrings equal to 000 or 011.

Original entry on oeis.org

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: 0

R. H. Hardin, Aug 12 2009

			All solutions for N=6
001001 001010 010010 010100 010101 100100 100101 101001 101010 110010
110100 110101 111001 111010 111100 111101 111110 111111

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 500 terms from R. H. Hardin)
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1).

Cf. A171861 (essentially the same sequence).

CoefficientList[Series[(x^2 + x + 1)/((x - 1) (x^3 + x^2 - 1)), {x, 0, 44}], x] (* Michael De Vlieger, Oct 11 2017 *)

G.f.: (x^2+x+1)/((x-1)*(x^3+x^2-1)). - R. J. Mathar, Nov 28 2011

Edited by Alois P. Heinz, Oct 11 2017

A238644 Number of binary words on {H,T} that end in THTH but do not contain the contiguous subsequence HTHH.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 6, 11, 19, 34, 62, 112, 202, 365, 659, 1189, 2146, 3874, 6993, 12623, 22786, 41131, 74245, 134019, 241917, 436683, 788254, 1422873, 2568420, 4636240, 8368850, 15106563, 27268770, 49222700, 88851613, 160385536, 289511009, 522594658, 943332613, 1702804277

Offset: 0

Geoffrey Critzer, Mar 01 2014

nonn

In the Penney game THTH beats HTHH 9 times out of 14 yet the expected wait time for THTH is 20 while that for HTHH is only 18.

			a(7)=6 because we have: TTTTHTH, THTTHTH, THHTHTH, HTTTHTH, HHTTHTH, HHHTHTH.

Penney Ante, Counterintuitive Probabilities in Coin Tossing, Bay Area Circle for Teachers Summer Workshop. [broken link]
Raymond S. Nickerson, Penney Ante: Counterintuitive Probabilities in Coin Tossing, 2008.
Eric Weisstein's World of Mathematics, Coin Tossing
Wikipedia, Penney's game
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,0,0,1).

Cf. A171861.

nn=40;CoefficientList[Series[(x^4+x^7)/(1-2x+x^2-x^3-x^6),{x,0,nn}],x]
LinearRecurrence[{2,-1,1,0,0,1},{0,0,0,0,1,2,3,6},50]

G.f.: G(x) = (x^4 + x^7)/(1 - 2x + x^2 - x^3 - x^6). We note G(1/2) = 9/14.

cp's OEIS Frontend

A207500 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 0 1 and 1 0 0 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A207682 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 0 1 horizontally and 0 0 1 and 1 1 1 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A207693 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 1 and 1 1 1 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A207908 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 0 and 0 1 1 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A070550 a(n) = a(n-1) + a(n-3) + a(n-4), starting with a(0..3) = 1, 2, 2, 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Formula

A207785 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 0 0 and 1 0 1 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A208420 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 0 and 0 1 1 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A207514 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 0 1 and 1 1 1 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A164315 Number of binary strings of length n with no substrings equal to 000 or 011.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs