Stephen G Penrice - cp's OEIS frontend

A061021 a(n) = a(n-1)*a(n-2) - a(n-3) with a(0) = a(1) = a(2) = 3.

3, 3, 3, 6, 15, 87, 1299, 112998, 146784315, 16586334025071, 2434613678231239448367, 40381315689150066251526220641224742, 98312903521778500654864668915856114278134197773017871243

Offset: 0

Stephen G Penrice, May 23 2001

Any three consecutive terms are a solution to the Diophantine equation x^2 + y^2 + z^2 = xyz.

Harry J. Smith, Table of n, a(n) for n = 0..17
Loren C. Larson, Solution to Problem Proposal 701, College Mathematics Journal 33 (2002), pp. 241-242.
Edward T. H. Wang, Problem Proposal 701, College Mathematics Journal 32 (2001), p. 211.

Cf. A022405, A061292, A072878, A072879, A072880, A074394, A178768.

a061021 n = a061021_list !! n
a061021_list = 3 : 3 : 3 : zipWith (-)
(tail $ zipWith (*) (tail a061021_list) a061021_list) a061021_list
-- Reinhard Zumkeller, Mar 25 2015

RecurrenceTable[{a[n] == a[n - 1] a[n - 2] - a[n - 3], a[0] == a[1] == a[2] == 3}, a, {n, 0, 12}] (* Michael De Vlieger, Aug 21 2016 *)

for (n=0, 17, if (n>2, a=a1*a2 - a3; a3=a2; a2=a1; a1=a, if (n==0, a=a3=3, if (n==1, a=a2=3, a=a1=3))); write("b061021.txt", n, " ", a)) \\ Harry J. Smith, Jul 16 2009

From Jon E. Schoenfield, May 12 2019: (Start)
It appears that, for n >= 1,
  a(n) = ceiling(e^(c0*phi^n - c1/(-phi)^n))
where
  phi = (1 + sqrt(5))/2,
   c0 = 0.4004033011137849744572073756789830081726425559860...
   c1 = 0.2798639753144007577581523025628820390768226527315...
(End)

More terms from Erich Friedman, Jun 03 2001

Name clarified by Petros Hadjicostas, May 11 2019

A061292 a(n) = a(n-1)a(n-2)a(n-3) - a(n-4) for n>3 with a(0) = a(1) = a(2) = a(3) = 2.

Original entry on oeis.org

2, 2, 2, 2, 6, 22, 262, 34582, 199330642, 1806032092550706, 12449434806576800059248920402, 4481765860945171681908664776799089162954814190172722

Offset: 0

Stephen G Penrice, Jun 04 2001

Any four consecutive terms are a solution to the Diophantine equation w^2 + x^2 + y^2 + z^2 = wxyz.

a(n) = 2 * A072878(n+1).

Vincenzo Librandi, Table of n, a(n) for n = 0..18
Michael Magee and Brady Haran, A simple equation that behaves weirdly, Numberphile, Youtube video, 2025.

Cf. A022405, A061021, A072878, A072879, A072880.

a061292 n = a061292_list !! n
a061292_list = 2 : 2 : 2 : 2 : zipWith (-)
   (zipWith3 (((*) .) . (*)) (drop 2 xs) (tail xs) xs) a061292_list
   where xs = tail a061292_list
-- Reinhard Zumkeller, Mar 25 2015

I:=[2,2,2,2]; [n le 4 select I[n] else Self(n-1)*Self(n-2)*Self(n-3)-Self(n-4): n in [1..12]]; // Vincenzo Librandi, Sep 17 2011

a[1] := 2; a[2] := 2; a[3] := 2; a[4] := 2; a[n_] := a[n - 1]*a[n - 2]*a[n - 3] - a[n - 4]; Table[a[n], {n, 1, 15}] (* Stefan Steinerberger, Mar 31 2006 *)
RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==2,a[n]==a[n-1]a[n-2]a[n-3]- a[n-4]},a[n],{n,12}] (* Harvey P. Dale, Sep 15 2011 *)

More terms from Larry Reeves (larryr(AT)acm.org) and Jason Earls, Jun 05 2001

A055006 a(n) is the least multiple of n such that a(n) = 1 mod k for all integers k with 1 < k < n and k relatively prime to n.

Original entry on oeis.org

1, 2, 3, 4, 25, 6, 301, 736, 2241, 190, 25201, 4236, 83161, 19306, 64065, 135136, 7207201, 85086, 49008961, 49468420, 36951201, 27776386, 698377681, 855189336, 25700298625, 2445441076, 74364290001, 13624600276, 2248776129601, 6254036790, 39594522567601

Offset: 1

Stephen G Penrice, May 30 2000

nonn

Table[Block[{m = 1, t = Select[Range[2, n - 1], CoprimeQ[#, n] &]}, While[! AllTrue[t, Mod[m n, #] == 1 &], m++]; m n], {n, 20}] (* Michael De Vlieger, Mar 08 2022 *)

isok(m, n) = for (k=2, n-1, if ((gcd(k, n)==1) && ((m % k) !=1), return(0))); return(1);
a(n) = my(m=n); while (!isok(m,n), m+=n); m; \\ Michel Marcus, Mar 08 2022

More terms from Sean A. Irvine, Mar 07 2022

A050400 Number of independent sets of vertices in P_3 X C_n (n > 2).

Original entry on oeis.org

5, 1, 17, 43, 181, 621, 2309, 8303, 30277, 109753, 398857, 1447931, 5258725, 19095285, 69344061, 251811903, 914429445, 3320635025, 12058502657, 43789003563, 159014593621, 577442573597, 2096914206261, 7614694850543, 27651860345029, 100414447219721, 364643142303353

Offset: 0

Stephen G Penrice, Dec 21 1999

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,8,6,-1,-1).

Column 3 of A286513.

a:=[5,1,17,43,181];; for n in [6..30] do a[n]:=a[n-1]+8*a[n-2] +6*a[n-3] -a[n-4]-a[n-5]; od; a; # G. C. Greubel, Oct 30 2019

I:=[5,1,17,43,181]; [n le 5 select I[n] else Self(n-1) + 8*Self(n-2) + 6*Self(n-3) - Self(n-4) - Self(n-5): n in [1..30]]; // Vincenzo Librandi, May 11 2017

seq(coeff(series((5-4*x-24*x^2-12*x^3+x^4)/((1+x)*(1-2*x-6*x^2+x^4)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 30 2019

CoefficientList[Series[(5-4*x-24*x^2-12*x^3+x^4)/((1+x)*(1-2*x-6*x^2+ x^4)), {x, 0, 30}], x] (* Vincenzo Librandi, May 11 2017 *)

my(x='x+O('x^30)); Vec((5-4*x-24*x^2-12*x^3+x^4)/((1+x)*(1-2*x-6*x^2+x^4))) \\ G. C. Greubel, Oct 30 2019

def A077952_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((5-4*x-24*x^2-12*x^3+x^4)/((1+x)*(1-2*x-6*x^2+x^4))).list()
A077952_list(30) # G. C. Greubel, Oct 30 2019

a(n) = a(n-1) + 8*a(n-2) + 6*a(n-3) - a(n-4) - a(n-5).

G.f.: (5-4*x-24*x^2-12*x^3+x^4)/((1+x)*(1-2*x-6*x^2+x^4)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009

More terms from Michael Lugo (mlugo(AT)thelabelguy.com), Dec 22 1999

A050401 Number of independent sets of nodes in P_4 X C_n (n > 2).

Original entry on oeis.org

8, 1, 41, 142, 933, 4741, 26660, 143697, 788453, 4293286, 23454801, 127953981, 698467368, 3811712633, 20803963753, 113540081302, 619672701957, 3381980484909, 18457878595412, 100737602247769, 549796303339413

Offset: 0

Stephen G Penrice, Dec 21 1999

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,20,27,-14,-25,4,5,-1).

Column 4 of A286513.

a:=[8,1,41,142,933,4741,26660,143697];; for n in [9..30] do a[n]:= a[n-1]+20*a[n-2]+27*a[n-3]-14*a[n-4]-25*a[n-5]+4*a[n-6]+5*a[n-7]-a[n-8]; od; a; # G. C. Greubel, Oct 30 2019

I:=[8,1,41,142,933,4741,26660,143697]; [n le 8 select I[n] else Self(n-1)+20*Self(n-2)+27*Self(n-3)-14*Self(n-4)- 25*Self(n-5)+4*Self(n-6)+5*Self(n-7)-Self(n-8): n in [1..30]]; // Vincenzo Librandi, May 11 2017

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (8 -7*x -120*x^2 -135*x^3 +56*x^4 +75*x^5 -8*x^6 -5*x^7)/((1+x)*(1+2*x-x^2)*( 1-4*x-9*x^2+5*x^3+4*x^4-x^5)) )); // G. C. Greubel, Oct 30 2019

seq(coeff(series((8 -7*x -120*x^2 -135*x^3 +56*x^4 +75*x^5 -8*x^6 -5*x^7)/( (1+x)*(1+2*x-x^2)*(1-4*x-9*x^2+5*x^3+4*x^4-x^5)), x, n+1), x, n), n = 0 ..30); # G. C. Greubel, Oct 30 2019

CoefficientList[Series[(8-7*x-120*x^2-135*x^3+56*x^4+75*x^5-8*x^6-5*x^7) /( (1+x)*(1+2*x-x^2)*(1-4*x-9*x^2+5*x^3+4*x^4-x^5)), {x, 0, 50}], x] (* Vincenzo Librandi, May 11 2017 *)

my(x='x+O('x^30)); Vec((8 -7*x -120*x^2 -135*x^3 +56*x^4 +75*x^5 -8*x^6 -5*x^7)/((1+x)*(1+2*x-x^2)*(1-4*x-9*x^2+5*x^3+4*x^4-x^5))) \\ G. C. Greubel, Oct 30 2019

def A050401_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((8 -7*x -120*x^2 -135*x^3 +56*x^4 +75*x^5 -8*x^6 -5*x^7)/((1+x)*(1+2*x-x^2)*(1-4*x-9*x^2+5*x^3+4*x^4-x^5))).list()
A050401_list(30) # G. C. Greubel, Oct 30 2019

a(n) = a(n-1) + 20*a(n-2) + 27*a(n-3) - 14*a(n-4) - 25*a(n-5) + 4*a(n-6) + 5*a(n-7) - a(n-8).

G.f.: (8 -7*x -120*x^2 -135*x^3 +56*x^4 +75*x^5 -8*x^6 -5*x^7)/((1+x)*(1+2*x-x^2)*(1-4*x-9*x^2+5*x^3+4*x^4-x^5)). - Colin Barker, Aug 31 2012

More terms from Michael Lugo (mlugo(AT)thelabelguy.com), Dec 22 1999

A051928 Number of independent sets of vertices in graph K_3 X C_n (n > 2).

Original entry on oeis.org

4, 1, 13, 34, 121, 391, 1300, 4285, 14161, 46762, 154453, 510115, 1684804, 5564521, 18378373, 60699634, 200477281, 662131471, 2186871700, 7222746565, 23855111401, 78788080762, 260219353693, 859446141835, 2838557779204, 9375119479441, 30963916217533

Offset: 0

Stephen G Penrice, Dec 19 1999

Colin Barker, Table of n, a(n) for n = 0..1000
C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), Article 12.7.8.
Index entries for linear recurrences with constant coefficients, signature (2,4,1).

Row 3 of A287376.

LinearRecurrence[{2,4,1},{4,1,13},30] (* Harvey P. Dale, Nov 20 2021 *)

Vec((4-7*x-5*x^2)/((1+x)*(1-3*x-x^2)) + O(x^30)) \\ Colin Barker, May 11 2017

a(n) = 2*a(n-1) + 4*a(n-2) + a(n-3).
G.f.: (4-7*x-5*x^2)/((1+x)*(1-3*x-x^2)). - Colin Barker, May 22 2012
a(n) = 2*(-1)^n + ((3-sqrt(13))/2)^n + ((3+sqrt(13))/2)^n. - Colin Barker, May 11 2017
a(n) = A006497+2*(-1)^n. - R. J. Mathar, Oct 20 2017

A053763 a(n) = 2^(n^2 - n).

Original entry on oeis.org

1, 1, 4, 64, 4096, 1048576, 1073741824, 4398046511104, 72057594037927936, 4722366482869645213696, 1237940039285380274899124224, 1298074214633706907132624082305024, 5444517870735015415413993718908291383296, 91343852333181432387730302044767688728495783936

Offset: 0

Stephen G Penrice, Mar 29 2000

Nilpotent n X n matrices over GF(2). Also number of simple digraphs (without self-loops) on n labeled nodes (see also A002416).
For n >= 1 a(n) is the size of the Sylow 2-subgroup of the Chevalley group A_n(4) (sequence A053291). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001
(-1)^ceiling(n/2) * resultant of the Chebyshev polynomial of first kind of degree n and Chebyshev polynomial of first kind of degree (n+1) (cf. A039991). - Benoit Cloitre, Jan 26 2003
The number of reflexive binary relations on an n-element set. - Justin Witt (justinmwitt(AT)gmail.com), Jul 12 2005
From Rick L. Shepherd, Dec 24 2008: (Start)
Number of gift exchange scenarios where, for each person k of n people,
i) k gives gifts to g(k) of the others, where 0 <= g(k) <= n-1,
ii) k gives no more than one gift to any specific person,
iii) k gives no single gift to two or more people and
iv) there is no other person j such that j and k jointly give a single gift.
(In other words -- but less precisely -- each person k either gives no gifts or gives exactly one gift per person to 1 <= g(k) <= n-1 others.) (End)
In general, sequences of the form m^((n^2 - n)/2) enumerate the graphs with n labeled nodes with m types of edge. a(n) therefore is the number of labeled graphs with n nodes with 4 types of edge. To clarify the comment from Benoit Cloitre, dated Jan 26 2003, in this context: simple digraphs (without self-loops) have four types of edge. These types of edges are as follows: the absent edge, the directed edge from A -> B, the directed edge from B -> A and the bidirectional edge, A <-> B. - Mark Stander, Apr 11 2019

			a(2)=4 because there are four 2 x 2 nilpotent matrices over GF(2):{{0,0},{0,0}},{{0,1},{0,0}},{{0,0},{1,0}},{{1,1,},{1,1}} where 1+1=0. - _Geoffrey Critzer_, Oct 05 2012

J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 521.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 5, Eq. (1.1.5).

T. D. Noe, Table of n, a(n) for n = 0..35
Marcus Brinkmann, Extended Affine and CCZ Equivalence up to Dimension 4, Ruhr University Bochum (2019).
N. J. Fine and I. N. Herstein, The probability that a matrix be nilpotent, Illinois J. Math., 2 (1958), 499-504.
Murray Gerstenhaber, On the number of nilpotent matrices with coefficients in a finite field, Illinois J. Math., Vol. 5 (1961), 330-333.
Antal Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
Pakawut Jiradilok, Some Combinatorial Formulas Related to Diagonal Ramsey Numbers, arXiv:2404.02714 [math.CO], 2024. See p. 19.
Greg Kuperberg, Symmetry classes of alternating-sign matrices under one roof, Annals of mathematics, Second Series, Vol. 156, No. 3 (2002), pp. 835-866, arXiv preprint, arXiv:math/0008184 [math.CO], 2000-2001 (see Th. 3).
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Götz Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

Row sums of A123554, A189898, A346412, A346214.

Cf. A053773, A006125, A000273, A000984, A002416, A319016.

seq(4^(binomial(n, n-2)), n=0..12); # Zerinvary Lajos, Jun 16 2007
a:=n->mul(4^j, j=1..n-1): seq(a(n), n=0..12); # Zerinvary Lajos, Oct 03 2007

Table[2^(2*Binomial[n,2]), {n,0,20}] (* Geoffrey Critzer, Oct 04 2012 *)

a(n)=1<<(n^2-n) \\ Charles R Greathouse IV, Nov 20 2012

def A053763(n): return 1<Chai Wah Wu, Jul 05 2024

Sequence given by the Hankel transform (see A001906 for definition) of A059231 = {1, 1, 5, 29, 185, 1257, 8925, 65445, 491825, ...}; example: det([1, 1, 5, 29; 1, 5, 29, 185; 5, 29, 185, 1257; 29, 185, 1257, 8925]) = 4^6 = 4096. - Philippe Deléham, Aug 20 2005
a(n) = 4^binomial(n, n-2). - Zerinvary Lajos, Jun 16 2007
a(n) = Sum_{i=0..n^2-n} binomial(n^2-n, i). - Rick L. Shepherd, Dec 24 2008
G.f. A(x) satisfies: A(x) = 1 + x * A(4*x). - Ilya Gutkovskiy, Jun 04 2020
Sum_{n>=1} 1/a(n) = A319016. - Amiram Eldar, Oct 27 2020
Sum_{n>=0} a(n)*u^n/A002884(n) = Product_{r>=1} 1/(1-u/q^r). - Geoffrey Critzer, Oct 28 2021

A050402 Number of independent sets of nodes in C_4 X C_n (n > 2).

Original entry on oeis.org

7, 1, 35, 121, 743, 3561, 18995, 96433, 500871, 2573905, 13292995, 68492073, 353290343, 1821383097, 9392360019, 48428332641, 249716406791, 1287608913057, 6639354593123, 34234612471001, 176524935990503, 910219628918665, 4693389213891699, 24200638961917201

Offset: 0

Stephen G Penrice, Dec 21 1999

Colin Barker, Table of n, a(n) for n = 0..1000
C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), Article 12.7.8
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Torus Grid Graph
Index entries for linear recurrences with constant coefficients, signature (2,15,8,-7,-2,1).

a:=[7,1,35,121,743,3561];; for n in [7..30] do a[n]:=2*a[n-1] +15*a[n-2]+8*a[n-3]-7*a[n-4]-2*a[n-5]-a[n-6]; od; a; # G. C. Greubel, Oct 30 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (7 -13*x -72*x^2 -20*x^3 +17*x^4 +x^5)/((1+x)*(1+2*x-x^2)*(1-5*x-x^2+x^3)) )); // G. C. Greubel, Oct 30 2019

seq(coeff(series((7-13*x-72*x^2-20*x^3+17*x^4+x^5)/((1+x)*(1+2*x-x^2) *(1-5*x-x^2+x^3)), x, n+1), x, n), n = 0 ..30); # G. C. Greubel, Oct 30 2019

CoefficientList[Series[(7 -13*x -72*x^2 -20*x^3 +17*x^4 +x^5)/((1+x)*(1+2*x-x^2)*(1-5*x-x^2+x^3)), {x, 0, 30}], x]

Vec((7-13*x-72*x^2-20*x^3+17*x^4+x^5)/((1+x)*(1+2*x-x^2)*(1-5*x- x^2+x^3)) + O(x^30)) \\ Colin Barker, May 11 2017

def A050402_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((7 -13*x -72*x^2 -20*x^3 +17*x^4 +x^5)/((1+x)*(1+2*x-x^2)*(1-5*x-x^2+x^3))).list()
A050402_list(30) # G. C. Greubel, Oct 30 2019

a(n) = a(n-1) + 17*a(n-2) + 23*a(n-3) + a(n-4) - 9*a(n-5) - a(n-6) + a(n-7).

G.f.: (7 -13*x -72*x^2 -20*x^3 +17*x^4 +x^5)/((1+x)*(1+2*x-x^2)*(1-5*x-x^2+x^3)). - Colin Barker, Aug 31 2012

More terms from Michael Lugo (mlugo(AT)thelabelguy.com), Dec 22 1999

A051927 Number of independent vertex sets in the n-prism graph Y_n = K_2 X C_n (n > 2).

Original entry on oeis.org

3, 1, 7, 13, 35, 81, 199, 477, 1155, 2785, 6727, 16237, 39203, 94641, 228487, 551613, 1331715, 3215041, 7761799, 18738637, 45239075, 109216785, 263672647, 636562077, 1536796803, 3710155681, 8957108167, 21624372013, 52205852195

Offset: 0

Stephen G Penrice, Dec 19 1999

For n>1, a(n) is also the number of ways to place k non-attacking wazirs on a 2 X n horizontal cylinder, summed over all k>=0 (wazir is a leaper [0,1]). - Vaclav Kotesovec, May 08 2012

Also the number of vertex covers for Y_n. - Eric W. Weisstein, Jan 04 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, pp. 400-401.
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Prism Graph
Eric Weisstein's World of Mathematics, Vertex Cover
Index entries for linear recurrences with constant coefficients, signature (1,3,1).

Column 2 of A286513 and row 2 of A287376.

Cf. A000129, A002203.

I:=[3, 1, 7]; [n le 3 select I[n] else Self(n-1) + 3*Self(n-2) + Self(n-3): n in [1..30]]; // Vincenzo Librandi, May 04 2013

A051927 := x -> (1+sqrt(2))^x+(-1)^x+(1-sqrt(2))^x;
seq(simplify(A051927(i)),i=0..28); # Peter Luschny, Aug 13 2012

CoefficientList[Series[(3 - 2 x - 3 x^2) / ((1 - 2 x - x^2) (1 + x)), {x, 0, 40}], x] (* Vincenzo Librandi, May 04 2013 *)
Table[LucasL[n, 2] + (-1)^n, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 15 2016 *)
LinearRecurrence[{1, 3, 1}, {1, 7, 13}, {0, 20}] (* Eric W. Weisstein, Sep 27 2017 *)

a(n)=polcoeff((3-2*x-3*x^2)/(1-2*x-x^2)/(1+x)+x*O(x^n),n)

x='x+O('x^66); Vec( (3-2*x-3*x^2)/((1-2*x-x^2)*(1+x)) ) \\ Joerg Arndt, May 04 2013

def A051927(x) : return (1+sqrt(2))^x+(-1)^x+(1-sqrt(2))^x
[A051927(i).round() for i in (0..28)] # Peter Luschny, Aug 13 2012

a(n) = a(n-1) + 3*a(n-2) + a(n-3).
G.f.: (3-2x-3x^2)/((1-2x-x^2)(1+x)). - Michael Somos, Apr 07 2003
Let A=[0, 1, 1;1, 1, 1;1, 1, 0] be the adjacency matrix of a triangle with a loop at a vertex. Then a(n)=trace(A^n). a(n)=(-1)^n+(1-sqrt(2))^n+(1+sqrt(2))^n. - Paul Barry, Jul 22 2004
a(n) = A002203(n) + (-1)^n. - Vladimir Reshetnikov, Sep 15 2016
a(n)+a(n+1) = 4*A000129(n+1). - R. J. Mathar, Feb 13 2020
E.g.f.: cosh(x) + 2*exp(x)*cosh(sqrt(2)*x) - sinh(x). - Stefano Spezia, Mar 31 2024

A051926 Number of independent sets of nodes in graph C_4 X P_n (n>2).

Original entry on oeis.org

1, 7, 35, 181, 933, 4811, 24807, 127913, 659561, 3400911, 17536203, 90422365, 466247117, 2404121747, 12396433487, 63920042065, 329592522065, 1699486218903, 8763103574515, 45185411569413, 232990675202677, 1201375684008283, 6194683683674679, 31941803427179001

Offset: 0

Stephen G Penrice, Dec 19 1999

Number of ways zero or more black and white stones can be placed on the points of a 2 X n grid such that no white stones are adjacent to any black stones. A078057 is a related case, where the grid is 1 X n. - Wayne VanWeerthuizen, May 04 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), Article 12.7.8.
Sela Fried and Toufik Mansour, Staircase graph words, arXiv:2312.08273 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (5,1,-1).

Row 4 of A286513.

I:=[1, 7, 35]; [n le 3 select I[n] else 5*Self(n-1)+Self(n-2)-Self(n-3): n in [1..25]]; // Vincenzo Librandi, Apr 27 2012

CoefficientList[Series[(1+2*x-x^2)/(1-5*x-x^2+x^3),{x,0,30}],x] (* Vincenzo Librandi, Apr 27 2012 *)
LinearRecurrence[{5,1,-1},{1,7,35},40] (* Harvey P. Dale, Apr 29 2019 *)

a(n) = 5*a(n-1)+a(n-2)-a(n-3) for n>2. - Wayne VanWeerthuizen, May 04 2004

G.f.: (1+2*x-x^2)/(1-5*x-x^2+x^3). - Colin Barker, Apr 18 2012

More terms from James Sellers, Dec 20 1999

cp's OEIS Frontend

User: Stephen G Penrice

A061021 a(n) = a(n-1)*a(n-2) - a(n-3) with a(0) = a(1) = a(2) = 3.

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A061292 a(n) = a(n-1)*a(n-2)*a(n-3) - a(n-4) for n>3 with a(0) = a(1) = a(2) = a(3) = 2.

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A055006 a(n) is the least multiple of n such that a(n) = 1 mod k for all integers k with 1 < k < n and k relatively prime to n.

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A050400 Number of independent sets of vertices in P_3 X C_n (n > 2).

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A050401 Number of independent sets of nodes in P_4 X C_n (n > 2).

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A051928 Number of independent sets of vertices in graph K_3 X C_n (n > 2).

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A053763 a(n) = 2^(n^2 - n).

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A061292 a(n) = a(n-1)a(n-2)a(n-3) - a(n-4) for n>3 with a(0) = a(1) = a(2) = a(3) = 2.