A006355 -id:A006355 - cp's OEIS frontend

A197054 T(n,k)=Number of nXk 0..4 arrays with each element equal to the number of its horizontal and vertical zero neighbors.

1, 2, 2, 2, 2, 2, 3, 4, 4, 3, 4, 6, 10, 6, 4, 5, 10, 18, 18, 10, 5, 7, 16, 38, 42, 38, 16, 7, 9, 26, 78, 108, 108, 78, 26, 9, 12, 42, 156, 274, 358, 274, 156, 42, 12, 16, 68, 320, 692, 1132, 1132, 692, 320, 68, 16, 21, 110, 654, 1754, 3580, 4468, 3580, 1754, 654, 110, 21, 28

Offset: 1

R. H. Hardin, Oct 09 2011

Every 0 is next to 0 0's, every 1 is next to 1 0's, every 2 is next to 2 0's, every 3 is next to 3 0's, every 4 is next to 4 0's

Also, the number of maximal independent vertex sets in the grid graph P_n X P_k. - Andrew Howroyd, May 16 2017

			Table starts
..1...2....2.....3......4.......5........7.........9.........12..........16
..2...2....4.....6.....10......16.......26........42.........68.........110
..2...4...10....18.....38......78......156.......320........654........1326
..3...6...18....42....108.....274......692......1754.......4442.......11248
..4..10...38...108....358....1132.....3580.....11382......36270......114992
..5..16...78...274...1132....4468....17742.....70616.....281202.....1117442
..7..26..156...692...3580...17742....88056....439338....2192602....10912392
..9..42..320..1754..11382...70616...439338...2745186...17155374...106972582
.12..68..654..4442..36270..281202..2192602..17155374..134355866..1049189170
.16.110.1326.11248.114992.1117442.10912392.106972582.1049189170.10264692132
...
Some solutions for n=6 k=4
..0..2..1..0....0..2..0..1....2..0..2..0....0..3..0..2....0..2..1..0
..2..0..1..2....1..1..1..1....0..2..1..1....2..0..4..0....3..0..1..2
..1..1..2..0....1..0..2..0....2..1..0..2....1..2..0..2....0..3..1..0
..0..3..0..3....1..1..1..1....0..2..2..0....0..1..1..1....2..0..1..2
..3..0..4..0....0..3..0..2....3..0..1..2....1..1..1..0....1..1..2..0
..0..3..0..2....2..0..3..0....0..2..1..0....1..0..1..1....0..2..0..2

R. H. Hardin, Table of n, a(n) for n = 1..364
Oh, Seungsang. "Maximal independent sets on a grid graph." Discrete Mathematics 340.12 (2017): 2762-2768. Also arXiv:1709.03678.
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set

Column 1 is A000931(n+6).
Column 2 is A006355(n+1).
Columns 3-7 are A197049, A197050, A197051, A197052, A197053.
Main diagonal is A197048.
Cf. A089934 (independent sets), A218354 (dominating sets).

A006367 Number of binary vectors of length n+1 beginning with 0 and containing just 1 singleton.

Original entry on oeis.org

1, 0, 2, 2, 5, 8, 15, 26, 46, 80, 139, 240, 413, 708, 1210, 2062, 3505, 5944, 10059, 16990, 28646, 48220, 81047, 136032, 228025, 381768, 638450, 1066586, 1780061, 2968040, 4944519, 8230370, 13689118, 22751528, 37786915, 62716752, 104028245

Offset: 0

David M. Bloom

Number of compositions of n+1 containing exactly one 1. - Emeric Deutsch, Mar 08 2002
Number of permutations with one fixed point avoiding 231 and 321.
A singleton is a run of length 1. - Michael Somos, Nov 29 2014
Second column of A105422. - Michael Somos, Nov 29 2014
Number of weak compositions of n with one 0 and no 1's. Example: Combine one 0 with the compositions of 5 without 1 to get a(5) = 8 weak compositions: 0,5; 5,0; 0,2,3; 0,3,2; 2,0,3; 3,0,2; 2,3,0; 3,2,0. - Gregory L. Simay, Mar 21 2018

			a(4) = 5 because among the 2^4 compositions of 5 only 4+1,1+4,2+2+1,2+1+2,1+2+2 contain exactly one 1.
a(4) = 5 because the binary vectors of length 4+1 beginning with 0 and with exactly one singleton are: 00001, 00100, 00110, 01100, 01111. - _Michael Somos_, Nov 29 2014
G.f. = 1 + 2*x^2 + 2*x^3 + 5*x^4 + 8*x^5 + 15*x^6 + 26*x^7 + 46*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Ricardo Gómez Aíza, Symbolic dynamical scales: modes, orbitals, and transversals, arXiv:2009.02669 [math.DS], 2020.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 11.
Mengmeng Liu and Andrew Yezhou Wang, The Number of Designated Parts in Compositions with Restricted Parts, J. Int. Seq., Vol. 23 (2020), Article 20.1.8.
J. J. Madden, A generating function for the distribution of runs in binary words, arXiv:1707.04351 [math.CO], 2017, Theorem 1.1, r=k=1.
T. Mansour and A. Robertson, Refined restricted permutations avoiding subsets of patterns of length three, arXiv:math/0204005 [math.CO], 2002.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000032, A000045, A004798, A006355, A010049, A105422, A139821, A212804.

I:=[1,0]; [n le 2 select I[n] else Self(n-1)+Self(n-2)+Fibonacci(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 20 2014

nn=36; CoefficientList[Series[1/(1 -x/(1-x) +x)^2, {x, 0, nn}], x] (* Geoffrey Critzer, Feb 18 2014 *)
a[n_]:= If[ n<0, SeriesCoefficient[((1-x)/(1+x-x^2))^2, {x, 0, -2-n}], SeriesCoefficient[((1-x)/(1-x-x^2))^2, {x, 0, n}]]; (* Michael Somos, Nov 29 2014 *)

Vec( (1-x)^2/(1-x-x^2)^2 + O(x^66) ) \\ Joerg Arndt, Feb 20 2014

{a(n) = if( n<0, n = -2-n; polcoeff( (1 - x)^2 / (1 + x - x^2)^2 + x * O(x^n), n), polcoeff( (1 - x)^2 / (1 - x - x^2)^2 + x * O(x^n), n))}; /* Michael Somos, Nov 29 2014 */

from sympy import fibonacci
from sympy.core.cache import cacheit
@cacheit
def a(n): return 1 if n==0 else 0 if n==1 else a(n - 1) + a(n - 2) + fibonacci(n - 3)
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 20 2017

def A006367(n): return (1/5)*(n*lucas_number2(n-2, 1, -1) + fibonacci(n+1) + 4*fibonacci(n-1))
[A006367(n) for n in (0..40)] # G. C. Greubel, Apr 06 2022

a(n) = a(n-1) + a(n-2) + Fibonacci(n-3).
G.f.: (1-x)^2/(1-x-x^2)^2. - Emeric Deutsch, Mar 08 2002
a(n) = A010049(n+1) - A010049(n). - R. J. Mathar, May 30 2014
Convolution square of A212804. - Michael Somos, Nov 29 2014
a(n) = -(-1)^n * A004798(-1-n) for all n in Z. - Michael Somos, Nov 29 2014
0 = a(n)*(-2*a(n) - 7*a(n+1) + 2*a(n+2) + a(n+3)) + a(n+1)*(-4*a(n+1) + 10*a(n+2) - 2*a(n+3)) + a(n+2)*(+4*a(n+2) - 7*a(n+3)) + a(n+3)*(+2*a(n+3)) for all n in Z. - Michael Somos, Nov 29 2014
a(n) = (n*Lucas(n-2) + Fibonacci(n))/5 + Fibonacci(n-1). - Ehren Metcalfe, Jul 29 2017

A118658 a(n) = 2*F(n-1) = L(n) - F(n), where F(n) and L(n) are Fibonacci and Lucas numbers respectively.

Original entry on oeis.org

2, 0, 2, 2, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338

Offset: 0

Bill Jones (b92057(AT)yahoo.com), May 18 2006

Essentially the same as A006355, A047992, A054886, A055389, A068922, A078642, A090991. - Philippe Deléham, Sep 20 2006 and Georg Fischer, Oct 07 2018
Also the number of matchings in the (n-2)-pan graph. - Eric W. Weisstein, Jun 30 2016
Also the number of maximal independent vertex sets (and minimal vertex covers) in the (n-1)-ladder graph. - Eric W. Weisstein, Jun 30 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Ladder Graph
Eric Weisstein's World of Mathematics, Matching
Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set
Eric Weisstein's World of Mathematics, Minimal Vertex Cover
Eric Weisstein's World of Mathematics, Pan Graph
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A000045, A003714, A212804.

List([0..40],n->2*Fibonacci(n-1)); # Muniru A Asiru, Oct 07 2018

[Lucas(n) - Fibonacci(n): n in [0..40]]; // Vincenzo Librandi, Sep 14 2014

with(combinat): seq(2*fibonacci(n-1),n=0..40); # Muniru A Asiru, Oct 07 2018
a := n -> -2*I^n*ChebyshevU(n-2, -I/2):
seq(simplify(a(n)), n = 0..39);  # Peter Luschny, Dec 03 2023

LinearRecurrence[{1, 1}, {2, 0}, 100] (* Vladimir Joseph Stephan Orlovsky, Jun 05 2011 *)
Table[LucasL[n] - Fibonacci[n], {n, 0, 40}] (* Vincenzo Librandi, Sep 14 2014 *)
Table[2 Fibonacci[n - 1], {n, 0, 20}] (* Eric W. Weisstein, Jun 30 2017 *)
2 Fibonacci[Range[0, 20] - 1] (* Eric W. Weisstein, Jun 30 2017 *)
Subtract @@@ (Through[{LucasL, Fibonacci}[#]] & /@ Range[0, 20]) (* Eric W. Weisstein, Jun 30 2017 *)
CoefficientList[Series[(2 (-1 + x))/(-1 + x + x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 30 2017 *)

a(n)=fibonacci(n-1)<<1 \\ Charles R Greathouse IV, Jun 05 2011

From Philippe Deléham, Sep 20 2006: (Start)
a(0)=2, a(1)=0; for n > 1, a(n) = a(n-1) + a(n-2).
G.f. (2 - 2*x)/(1 - x - x^2).
a(0)=2 and a(n) = 2*A000045(n-1) for n > 0. (End)
a(n) = A006355(n) + 0^n. - M. F. Hasler, Nov 05 2014
a(n) = Lucas(n-2) + Fibonacci(n-2). - Bruno Berselli, May 27 2015
a(n) = 3*Fibonacci(n-2) + Fibonacci(n-5). - Bruno Berselli, Feb 20 2017
a(n) = 2*A212804(n). - Bruno Berselli, Feb 21 2017
E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) - sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Apr 18 2022

More terms from Philippe Deléham, Sep 20 2006

Corrected by T. D. Noe, Nov 01 2006

A220062 Number A(n,k) of n length words over k-ary alphabet, where neighboring letters are neighbors in the alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 4, 2, 0, 0, 1, 5, 6, 6, 2, 0, 0, 1, 6, 8, 10, 8, 2, 0, 0, 1, 7, 10, 14, 16, 12, 2, 0, 0, 1, 8, 12, 18, 24, 26, 16, 2, 0, 0, 1, 9, 14, 22, 32, 42, 42, 24, 2, 0, 0, 1, 10, 16, 26, 40, 58, 72, 68, 32, 2, 0, 0

Offset: 0

Alois P. Heinz, Dec 03 2012

Equivalently, the number of walks of length n-1 on the path graph P_k. - Andrew Howroyd, Apr 17 2017

			A(5,3) = 12: there are 12 words of length 5 over 3-ary alphabet {a,b,c}, where neighboring letters are neighbors in the alphabet: ababa, ababc, abcba, abcbc, babab, babcb, bcbab, bcbcb, cbaba, cbabc, cbcba, cbcbc.
Square array A(n,k) begins:
  1,  1,  1,  1,  1,   1,   1,   1, ...
  0,  1,  2,  3,  4,   5,   6,   7, ...
  0,  0,  2,  4,  6,   8,  10,  12, ...
  0,  0,  2,  6, 10,  14,  18,  22, ...
  0,  0,  2,  8, 16,  24,  32,  40, ...
  0,  0,  2, 12, 26,  42,  58,  74, ...
  0,  0,  2, 16, 42,  72, 104, 136, ...
  0,  0,  2, 24, 68, 126, 188, 252, ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0, 2-10 give: A000007, A040000, A029744(n+2) for n>0, A006355(n+3) for n>0, A090993(n+1) for n>0, A090995(n-1) for n>2, A129639, A153340, A153362, A153360.
Rows 0-6 give: A000012, A001477, A005843(k-1) for k>0, A016825(k-2) for k>1, A008590(k-2) for k>2, A113770(k-2) for k>3, A063164(k-2) for k>4.
Main diagonal gives: A102699.
Cf. A198632, A188866, A276562, A208727, A208671.

b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i=0, add(b(n-1, j, k), j=1..k),
      `if`(i>1, b(n-1, i-1, k), 0)+
      `if`(i b(n, 0, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i == 0, Sum[b[n-1, j, k], {j, 1, k}], If[i>1, b[n-1, i-1, k], 0] + If[iJean-François Alcover, Jan 19 2015, after Alois P. Heinz *)

TransferGf(m,u,t,v,z)=vector(m,i,u(i))*matsolve(matid(m)-z*matrix(m,m,i,j,t(i,j)),vectorv(m,i,v(i)));
ColGf(m,z)=1+z*TransferGf(m, i->1, (i,j)->abs(i-j)==1, j->1, z);
a(n,k)=Vec(ColGf(k,x) + O(x^(n+1)))[n+1];
for(n=0, 7, for(k=0, 7, print1( a(n,k), ", ") ); print(); );
\\ Andrew Howroyd, Apr 17 2017

A002062 a(n) = Fibonacci(n) + n.

Original entry on oeis.org

0, 2, 3, 5, 7, 10, 14, 20, 29, 43, 65, 100, 156, 246, 391, 625, 1003, 1614, 2602, 4200, 6785, 10967, 17733, 28680, 46392, 75050, 121419, 196445, 317839, 514258, 832070, 1346300, 2178341, 3524611, 5702921, 9227500, 14930388, 24157854, 39088207, 63246025

Offset: 0

N. J. A. Sloane

Let A006355(n+4)_0%20-%20A066982(n+1)_1%20(conjecture);%20(a(n))%20=%20em%5BK*%20%5Dseq(%20.25'i%20-%20.25'j%20-%20.25'k%20-%20.25i'%20+%20.25j'%20-%20.75k'%20-%20.25'ii'%20-%20.25'jj'%20-%20.25'kk'%20-%20.25'ij'%20-%20.25'ik'%20-%20.75'ji'%20+%20.25'jk'%20-%20.25'ki'%20-%20.75'kj'%20+%20.75e),%20apart%20from%20initial%20term.%20-%20_Creighton%20Dement">x indicate the sequence offset. Then a(n+2)_0 = A006355(n+4)_0 - A066982(n+1)_1 (conjecture); (a(n)) = em[K* ]seq( .25'i - .25'j - .25'k - .25i' + .25j' - .75k' - .25'ii' - .25'jj' - .25'kk' - .25'ij' - .25'ik' - .75'ji' + .25'jk' - .25'ki' - .75'kj' + .75e), apart from initial term. - _Creighton Dement, Nov 19 2004

R. Honsberger, Ingenuity in Math., Random House, 1970, p. 96.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..500
Hung Viet Chu, A Note on the Fibonacci Sequence and Schreier-type Sets, arXiv:2205.14260 [math.CO], 2022.
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 (3,-2,-1,1).

Cf. A000045, A001611, A160536, A212272.

List([0..50], n-> Fibonacci(n)+n); # G. C. Greubel, Jul 09 2019

a002062 n = a000045 n + toInteger n
a002062_list = 0 : 2 : 3 : (map (subtract 1) $
   zipWith (-) (map (* 2) $ drop 2 a002062_list) a002062_list)
-- Reinhard Zumkeller, Oct 03 2012

[Fibonacci(n)+n: n in [0..50]]; // G. C. Greubel, Jul 09 2019

a:= n-> combinat[fibonacci](n)+n: seq(a(n), n=0..50); # Zerinvary Lajos, Mar 20 2008

Table[Fibonacci[n]+n,{n,0,50}] (* Harvey P. Dale, Jul 27 2011 *)

numlib::fibonacci(n)+n $ n = 0..50; // Zerinvary Lajos, May 08 2008

a(n)=fibonacci(n) + n \\ Charles R Greathouse IV, Oct 03 2016

[fibonacci(n)+n for n in (0..50)] # G. C. Greubel, Jul 09 2019

G.f.: x*(-2+3*x) / ( (x^2+x-1)*(x-1)^2 ). - Simon Plouffe in his 1992 dissertation
From Wolfdieter Lang: (Start)
Convolution of natural numbers n >= 1 with Fibonacci numbers F(k), k >= -3, (F(-k)=(-1)^(k+1)*F(k));
G.f.: x*(2-3*x)/((1-x-x^2)*(1-x)^2). (End)
a(n) = 2*a(n-1) - a(n-3) - 1. - Kieren MacMillan, Nov 08 2008
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4). - Emmanuel Vantieghem, May 19 2016
E.g.f.: 2*exp(x/2)*sinh(sqrt(5)*x/2)/sqrt(5) + x*exp(x). - Ilya Gutkovskiy, Apr 11 2017

A200871 T(n,k)=Number of 0..k arrays x(0..n+1) of n+2 elements without any interior element greater than both neighbors or less than both neighbors.

Original entry on oeis.org

6, 17, 10, 36, 37, 16, 65, 94, 77, 26, 106, 195, 236, 163, 42, 161, 356, 567, 602, 343, 68, 232, 595, 1168, 1673, 1528, 723, 110, 321, 932, 2163, 3886, 4917, 3882, 1523, 178, 430, 1389, 3704, 7973, 12890, 14455, 9858, 3209, 288, 561, 1990, 5973, 14932, 29325

Offset: 1

R. H. Hardin Nov 23 2011

Table starts
...6....17.....36......65.....106......161......232.......321.......430
..10....37.....94.....195.....356......595......932......1389......1990
..16....77....236.....567....1168.....2163.....3704......5973......9184
..26...163....602....1673....3886.....7973....14932.....26073.....43066
..42...343...1528....4917...12890....29325....60112....113745....201994
..68...723...3882...14455...42744...107777...241718....495495....945790
.110..1523...9858...42479..141688...395929...971416...2156867...4424298
.178..3209..25038..124851..469726..1454643..3904290...9389377..20696974
.288..6761..63592..366959.1557320..5344795.15693816..40880321..96838448
.466.14245.161514.1078565.5163158.19638715.63085186.177996275.453123270

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

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

Column 1 is A006355(n+4)

Row 1 is A084990(n+1)

t[0,k_,x_,y_] := 1; t[n_,k_,x_,y_] := t[n,k,x,y] = Sum[If[z <= x <= y || y <= x <= z, t[n-1, k, z, x], 0], {z, k+1}]; t[n_, k_] := Sum[t[n, k, x, y], {x, k+1}, {y, k+1}]; TableForm@ Table[t[n, k], {n, 8}, {k, 8}] (* Giovanni Resta, Mar 05 2014 *)

Empirical for columns:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = 2*a(n-1) +a(n-4)
k=3: a(n) = 2*a(n-1) +a(n-2) +2*a(n-4) +a(n-5)
k=4: a(n) = 3*a(n-1) -a(n-2) +a(n-3) +4*a(n-4) +a(n-6) +a(n-7)
k=5: a(n) = 3*a(n-1) +a(n-3) +7*a(n-4) +3*a(n-5) +2*a(n-6) +3*a(n-7) +a(n-8)
k=6: a(n) = 4*a(n-1) -3*a(n-2) +4*a(n-3) +9*a(n-4) +7*a(n-6) +6*a(n-7) +a(n-8) +2*a(n-9) +a(n-10)
k=7: a(n) = 4*a(n-1) -2*a(n-2) +4*a(n-3) +15*a(n-4) +6*a(n-5) +12*a(n-6) +16*a(n-7) +7*a(n-8) +5*a(n-9) +4*a(n-10) +a(n-11)
Empirical for rows:
n=1: a(k) = (1/3)*k^3 + 2*k^2 + (8/3)*k + 1
n=2: a(k) = (1/12)*k^4 + (3/2)*k^3 + (47/12)*k^2 + (7/2)*k + 1
n=3: a(k) = (1/60)*k^5 + (3/4)*k^4 + (15/4)*k^3 + (25/4)*k^2 + (127/30)*k + 1
n=4: a(k) = (1/360)*k^6 + (7/24)*k^5 + (197/72)*k^4 + (185/24)*k^3 + (1667/180)*k^2 + 5*k + 1
n=5: a(k) = (1/2520)*k^7 + (17/180)*k^6 + (281/180)*k^5 + (64/9)*k^4 + (4927/360)*k^3 + (2303/180)*k^2 + (604/105)*k + 1
n=6: a(k) = (1/20160)*k^8 + (19/720)*k^7 + (211/288)*k^6 + (1889/360)*k^5 + (44167/2880)*k^4 + (15991/720)*k^3 + (5689/336)*k^2 + (391/60)*k + 1
n=7: a(k) = (1/181440)*k^9 + (131/20160)*k^8 + (8893/30240)*k^7 + (4621/1440)*k^6 + (118933/8640)*k^5 + (83957/2880)*k^4 + (763489/22680)*k^3 + (36343/1680)*k^2 + (9169/1260)*k + 1

A218064 T(n,k) = Number of n X k arrays of the minimum value of corresponding elements and their horizontal or antidiagonal neighbors in a random 0..1 n X k array.

Original entry on oeis.org

2, 2, 4, 4, 6, 8, 6, 18, 16, 16, 10, 42, 74, 42, 32, 16, 108, 260, 308, 110, 64, 26, 268, 1046, 1664, 1282, 288, 128, 42, 676, 3974, 10246, 10566, 5338, 754, 256, 68, 1694, 15578, 60804, 99934, 66978, 22228, 1974, 512, 110, 4258, 60242, 368220, 925904, 975296

Offset: 1

R. H. Hardin Oct 19 2012

Table starts
....2.....2........4..........6...........10..............16................26
....4.....6.......18.........42..........108.............268...............676
....8....16.......74........260.........1046............3974.............15578
...16....42......308.......1664........10246...........60804............368220
...32...110.....1282......10566........99934..........925904...........8679594
...64...288.....5338......66978.......975296........14096488.........204211050
..128...754....22228.....424332......9519284.......214514380........4803404048
..256..1974....92562....2687866.....92918894......3263893360......112984025006
..512..5168...385450...17025060....907013406.....49658510202.....2657601417086
.1024.13530..1605108..107835994...8853740672....755518606946....62512388662498
.2048.35422..6684066..683025864..86425318934..11494629017172..1470427119098160
.4096.92736.27834106.4326234664.843636853718.174881679122376.34587669928567490

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

R. H. Hardin, Table of n, a(n) for n = 1..262

Column 2 is A025169(n-1).

Row 1 is A006355(n+1).

A218897 T(n,k)=Number of nXk arrays of the minimum value of corresponding elements and their horizontal, diagonal or antidiagonal neighbors in a random 0..1 nXk array.

Original entry on oeis.org

2, 2, 4, 4, 6, 8, 6, 18, 16, 16, 10, 34, 72, 38, 32, 16, 86, 212, 286, 98, 64, 26, 180, 786, 1096, 1162, 244, 128, 42, 426, 2594, 6280, 6422, 4700, 614, 256, 68, 930, 9110, 28598, 54726, 35812, 19046, 1542, 512, 110, 2140, 31062, 149238, 373612, 463746, 202662

Offset: 1

R. H. Hardin Nov 08 2012

Table starts
....2.....2........4..........6...........10.............16...............26
....4.....6.......18.........34...........86............180..............426
....8....16.......72........212..........786...........2594.............9110
...16....38......286.......1096.........6280..........28598...........149238
...32....98.....1162.......6422........54726.........373612..........2880294
...64...244.....4700......35812.......463746........4575512.........52413310
..128...614....19046.....202662......3960146.......57116962........969423130
..256..1542....77198....1143320.....33785542......710022390......17870306268
..512..3872...312944....6451714....288335090.....8831593874.....329607859688
.1024..9726..1268694...36412818...2461034914...109858747456....6079480914622
.2048.24426..5143506..205486910..21006978672..1366468080924..112133703714680
.4096.61348.20852924.1159668470.179319456042.16997248406138.2068302082008760

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

R. H. Hardin, Table of n, a(n) for n = 1..199

Column 2 is A217631

Row 1 is A006355(n+1)

A245869 T(n,k)=Number of length n+2 0..k arrays with some pair in every consecutive three terms totalling exactly k.

Original entry on oeis.org

6, 19, 10, 36, 45, 16, 61, 100, 103, 26, 90, 193, 256, 239, 42, 127, 318, 549, 676, 553, 68, 168, 493, 960, 1629, 1764, 1281, 110, 217, 712, 1579, 3102, 4753, 4624, 2967, 178, 270, 993, 2368, 5515, 9726, 13961, 12100, 6873, 288, 331, 1330, 3433, 8840, 18505, 30900

Offset: 1

R. H. Hardin, Aug 04 2014

Table starts
.....6.......19........36.........61..........90..........127..........168
....10.......45.......100........193.........318..........493..........712
....16......103.......256........549.........960.........1579.........2368
....26......239.......676.......1629........3102.........5515.........8840
....42......553......1764.......4753........9726........18505........31176
....68.....1281......4624......13961.......30900........63241.......113024
...110.....2967.....12100......40901.......97602.......214315.......404264
...178.....6873.....31684.....119953......309078.......729097......1455496
...288....15921.....82944.....351649......977664......2475985......5223552
...466....36881....217156....1031057.....3094038......8415217.....18775816
...754....85435....568516....3022933.....9789654.....28590415.....67437448
..1220...197911...1488400....8863117....30977796.....97151683....242306240
..1974...458463...3896676...25986061....98020170....330100459....870461352
..3194..1062035..10201636...76189749...310161870...1121650903...3127322696
..5168..2460217..26708224..223384017...981426624...3811203385..11235107264
..8362..5699123..69923044..654949861..3105480558..12950003383..40363689352
.13530.13202089.183060900.1920277409..9826505742..44002376953.145010699592
.21892.30582803.479259664.5630150189.31093507092.149514426895.520968428032

			Some solutions for n=6 k=4
..1....4....0....4....0....1....2....3....1....2....0....3....3....0....2....4
..4....2....1....1....4....2....4....1....0....3....1....0....3....4....1....3
..0....2....4....0....3....2....0....3....3....1....3....1....1....2....3....1
..4....0....0....4....0....1....4....0....1....1....1....3....0....0....4....3
..4....2....4....4....4....2....1....4....4....3....2....1....4....4....1....2
..0....2....0....0....3....2....0....0....0....3....2....2....4....3....0....2
..2....0....4....4....0....1....4....4....3....1....4....3....0....1....4....2
..4....4....0....1....1....2....3....0....1....4....0....1....0....2....1....2

R. H. Hardin, Table of n, a(n) for n = 1..9999
Robert Israel, Proof of recurrences for columns
Robert Israel et al, A Pattern for OEIS Sequence A245869, Mathematics StackExchange, Jul 29-30, 2024.

Column 1 is A006355(n+4)
Column 3 is A206981(n+2)
Row 1 is A090381.

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = 2*a(n-1) +a(n-2) -a(n-4) -a(n-5)
k=3: a(n) = 2*a(n-1) +2*a(n-2) -a(n-3)
k=4: a(n) = 3*a(n-1) +a(n-2) -a(n-3) -5*a(n-4) -8*a(n-5) +3*a(n-6)
k=5: a(n) = 2*a(n-1) +4*a(n-2) -a(n-3)
k=6: a(n) = 3*a(n-1) +3*a(n-2) -a(n-3) -9*a(n-4) -24*a(n-5) +5*a(n-6)
k=7: a(n) = 2*a(n-1) +6*a(n-2) -a(n-3)
k=8: a(n) = 3*a(n-1) +5*a(n-2) -a(n-3) -13*a(n-4) -48*a(n-5) +7*a(n-6)
k=9: a(n) = 2*a(n-1) +8*a(n-2) -a(n-3)
Empirical for row n:
n=1: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4)
n=2: a(n) = 3*a(n-1) -2*a(n-2) -2*a(n-3) +3*a(n-4) -a(n-5)
n=3: a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +a(n-4) +2*a(n-5) -a(n-6)
n=4: a(n) = 3*a(n-1) -a(n-2) -5*a(n-3) +5*a(n-4) +a(n-5) -3*a(n-6) +a(n-7)
n=5: a(n) = 2*a(n-1) +2*a(n-2) -6*a(n-3) +6*a(n-5) -2*a(n-6) -2*a(n-7) +a(n-8)
n=6: a(n) = 3*a(n-1) -8*a(n-3) +6*a(n-4) +6*a(n-5) -8*a(n-6) +3*a(n-8) -a(n-9)
n=7: a(n) = 2*a(n-1) +3*a(n-2) -8*a(n-3) -2*a(n-4) +12*a(n-5) -2*a(n-6) -8*a(n-7) +3*a(n-8) +2*a(n-9) -a(n-10)
From Robert Israel, Aug 06 2024: (Start) For odd k, T(n,k) = 2 T(n-1,k) + (k-1) T(n-2,k) - T(n-3,k).
For even k, T(n,k) = 3 T(n-1,k) + (k-3) T(n-2,k) - T(n-3,k) + (2 k - 3) T(n-4,k) - k (k-2) T(n-5,k) + (k-1) T(n-6,k).
See links. (End)

A248461 T(n,k)=Number of length n+2 0..k arrays with no three consecutive terms having the sum of any two elements equal to twice the third.

Original entry on oeis.org

6, 18, 10, 48, 36, 16, 96, 148, 72, 26, 174, 380, 460, 144, 42, 282, 862, 1512, 1436, 288, 68, 432, 1652, 4272, 6040, 4488, 576, 110, 624, 2956, 9684, 21182, 24160, 14040, 1152, 178, 870, 4860, 20236, 56782, 105026, 96736, 43940, 2304, 288, 1170, 7642, 37868

Offset: 1

R. H. Hardin, Oct 06 2014

Table starts
...6...18......48.......96.......174........282.........432.........624
..10...36.....148......380.......862.......1652........2956........4860
..16...72.....460.....1512......4272.......9684.......20236.......37868
..26..144....1436.....6040.....21182......56782......138534......295078
..42..288....4488....24160....105026.....332940......948412.....2299356
..68..576...14040....96736....520788....1952254.....6493036....17917712
.110.1152...43940...387488...2582406...11447368....44452660...139623544
.178.2304..137532..1552448..12805334...67123652...304332258..1088015294
.288.4608..430508..6220480..63497776..393591402..2083523194..8478351478
.466.9216.1347652.24926080.314866606.2307892826.14264241960.66067495706

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

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

Column 1 is A006355(n+4)

Column 2 is A005010

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = 2*a(n-1)
k=3: a(n) = 2*a(n-1) +3*a(n-2) +4*a(n-3) -3*a(n-4) -12*a(n-5) -4*a(n-6)
k=4: a(n) = 3*a(n-1) +5*a(n-2) +2*a(n-3) -16*a(n-4) -28*a(n-5) -8*a(n-6)
k=5: [order 12]
k=6: [order 16]
k=7: [order 22]
Empirical for row n:
n=1: a(n) = 3*a(n-1) -2*a(n-2) -2*a(n-3) +3*a(n-4) -a(n-5); also a cubic polynomial plus a constant quasipolynomial with period 2
n=2: a(n) = 2*a(n-1) -a(n-3) -2*a(n-5) +2*a(n-6) +a(n-8) -2*a(n-10) +a(n-11); also a quartic polynomial plus a linear quasipolynomial with period 12
n=3: [order 27; also a degree 5 polynomial plus a quadratic quasipolynomial with period 840]
n=4: [order 61]

cp's OEIS Frontend

A197054 T(n,k)=Number of nXk 0..4 arrays with each element equal to the number of its horizontal and vertical zero neighbors.

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A006367 Number of binary vectors of length n+1 beginning with 0 and containing just 1 singleton.

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A118658 a(n) = 2*F(n-1) = L(n) - F(n), where F(n) and L(n) are Fibonacci and Lucas numbers respectively.

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A220062 Number A(n,k) of n length words over k-ary alphabet, where neighboring letters are neighbors in the alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A002062 a(n) = Fibonacci(n) + n.

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A200871 T(n,k)=Number of 0..k arrays x(0..n+1) of n+2 elements without any interior element greater than both neighbors or less than both neighbors.

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