Cesar Bautista - cp's OEIS frontend

A182141 Number of independent sets of nodes in the armchair (3,3) carbon nanotorus graph of breadth n (n>=1).

27, 18, 322, 2787, 37730, 486773, 6616216, 89809934, 1226678898, 16759965210, 229174768672, 3134027776854, 42863602781324, 586250943722267, 8018366958787066, 109670557564651352, 1500014136347328018, 20516391520781511387, 280612359537735848734

Offset: 0

Cesar Bautista, Apr 14 2012

Cesar Bautista, Table of n, a(n) for n = 0..500
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 (8,-1,-1018,1836,20616,-43461,-185682,384405,762090,-1499721,-1538730, 2873116,1499424,-2714609,-574862,1107300,18144,-108864).

a[0]:27; a[1]:18; a[2]:322; a[3]:2787; a[4]:37730; a[5]:486773; a[6]:6616216; a[7]:89809934; a[8]:1226678898; a[9]:16759965210; a[10]:229174768672; a[11]:3134027776854; a[12]:42863602781324; a[13]:586250943722267; a[14]:8018366958787066; a[15]:109670557564651352; a[16]:1500014136347328018; a[17]:20516391520781511387; a[18]:280612359537735848734;
a[n]:=18*a[n-1]-a[n-2]-1018*a[n-3]+1836*a[n-4]+20616*a[n-5]-43461*a[n-6]-185682*a[n-7]+384405*a[n-8]+762090*a[n-9]-1499721*a[n-10]-1538730*a[n-11]+2873116*a[n-12]+1499424*a[n-13]-2714609*a[n-14]-574862*a[n-15]+1107300*a[n-16]+18144*a[n-17]-108864*a[n-18];
makelist(a[k],k,0,25);

a(n) = 18*a(n-1) -a(n-2) -1018*a(n-3) +1836*a(n-4) +20616*a(n-5) -43461*a(n-6) -185682*a(n-7) +384405*a(n-8) +762090*a(n-9) -1499721*a(n-10) -1538730*a(n-11) +2873116*a(n-12) +1499424*a(n-13) -2714609*a(n-14) -574862*a(n-15) +1107300*a(n-16) +18144*a(n-17) -108864*a(n-18).

G.f.: (979776*x^18 -75600*x^17 -12197940*x^16 +5916552*x^15 +35833019*x^14 -19220271*x^13 -44070216*x^12 +23310438*x^11 +26177559*x^10 -13274349*x^9 -7520073*x^8 +3654387*x^7 +940365*x^6 -451464*x^5 -43362*x^4 +24495*x^3 +25*x^2 -468*x+27)/( (x-1) *(x+1) *(3*x^3-5*x^2-5*x+1) *(36*x^4-x^3-20*x^2-x+1) *(36*x^4+x^3-20*x^2+x+1) *(28*x^5+42*x^4-109*x^3+17*x^2+13*x-1)).

A182130 Number of independent sets of nodes in the armchair (3,3) carbon nanotube graph of breadth n (n>=1).

Original entry on oeis.org

27, 322, 4556, 61814, 847098, 11580788, 158413552, 2166639646, 29634348798, 405322443028, 5543789598764, 75825036741014, 1037095063081722, 14184841682767868, 194012817135153904, 2653605447140034790, 36294622054374551742, 496418784252960527212

Offset: 0

Cesar Bautista, Apr 13 2012

Cesar Bautista, Table of n, a(n) for n = 0..500
C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), #12.7.8.
Index entries for linear recurrences with constant coefficients, signature (13,17,-109,42,28).

LinearRecurrence[{13, 17, -109, 42, 28}, {27, 322, 4556, 61814,
847098}, 20] (* Wesley Ivan Hurt, Oct 10 2021 *)

a[0]:27; a[1]:322; a[2]:4556; a[3]:61814; a[4]:847098;
a[n]:=13*a[n-1]+17*a[n-2]-109*a[n-3]+42*a[n-4]+28*a[n-5];
makelist(a[k],k,0,25);

a(n) = 13*a(n-1)+17*a(n-2)-109*a(n-3)+42*a(n-4)+28*a(n-5) with a(0)=27, a(1)=322, a(2)=4556, a(3)=61814, a(4)=847098.

G.f.: (27-29*x-89*x^2+55*x^3+28*x^4)/(1-13*x-17*x^2+109*x^3-42*x^4-28*x^5).

A182143 Number of independent vertex sets in the Moebius ladder graph with 2n nodes (n >= 0).

Original entry on oeis.org

1, 3, 5, 15, 33, 83, 197, 479, 1153, 2787, 6725, 16239, 39201, 94643, 228485, 551615, 1331713, 3215043, 7761797, 18738639, 45239073, 109216787, 263672645, 636562079, 1536796801, 3710155683, 8957108165, 21624372015, 52205852193, 126036076403, 304278004997

Offset: 0

Cesar Bautista, Apr 14 2012

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

Cesar Bautista, 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, Moebius Ladder
Eric Weisstein's World of Mathematics, Vertex Cover
Index entries for linear recurrences with constant coefficients, signature (1,3,1).

Cf. A000129, A001333, A002203, A098600, A100227.

I:=[1,3,5]; [n le 3 select I[n] else Self(n-1)+3*Self(n-2)+Self(n-3): n in [1..31]]; // Bruno Berselli, Apr 14 2012

Table[(1 + Sqrt[2])^n + (1 - Sqrt[2])^n - (-1)^n, {n, 0, 30}] (* Bruno Berselli, Apr 14 2012 *)
Table[LucasL[n, 2] - (-1)^n, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 15 2016 *)
LinearRecurrence[{1, 3, 1}, {1, 3, 5}, 20] (* Eric W. Weisstein, Mar 31 2017 *)
CoefficientList[Series[(-1 - 2 x + x^2)/(-1 + x + 3 x^2 + x^3), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 21 2017 *)

Vec((x^2-2*x-1)/((x+1)*(x^2+2*x-1))+O(x^31)) \\ Bruno Berselli, Apr 14 2012

G.f.: (x^2-2*x-1)/((x+1)*(x^2+2*x-1)).
a(n) = (1+sqrt(2))^n + (1-sqrt(2))^n - (-1)^n = A002203(n) - (-1)^n.
a(n) = a(n-1) + 3*a(n-2) + a(n-3) with a(0)=1, a(1)=3, a(2)=5.
From Peter Bala, Jun 29 2015: (Start)
a(n) = Pell(n-1) + Pell(n+1) - (-1)^n.
a(n) = [x^n] ( (1 + 2*x + sqrt(1 + 8*x + 8*x^2))/2 )^n.
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 3*x^2 + 7*x^3 + 17*x^4 + 41*x^5 + ... = Sum_{n >= 0} A001333*x^n. Cf. A098600. (End)

A182077 Number of independent sets of nodes in the generalized Petersen graph G(2n+1,2) (n>=1).

Original entry on oeis.org

13, 76, 435, 2461, 13971, 79197, 449188, 2547179, 14445169, 81917079, 464547653, 2634418076, 14939621779, 84721638085, 480451043995, 2724607324221, 15451075136020, 87622065595371, 496899168779481, 2817883624638175, 15980039054921477, 90621786488479756

Offset: 0

Cesar Bautista, Apr 10 2012

Cesar Bautista, Table of n, a(n) for n = 0..499
C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), Article 12.7.8.
Stephan G. Wagner, The Fibonacci Number of Generalized Petersen Graphs, Fibonacci Quarterly, 44 (2006), 362-367.
Index entries for linear recurrences with constant coefficients, signature (3, 15, 3, -13, 4).

LinearRecurrence[{3,15,3,-13,4},{13,76,435,2461,13971},30] (* Harvey P. Dale, Jul 22 2013 *)

a(n) = 3*a(n-1)+15*a(n-2)+3*a(n-3)-13*a(n-4)+4*a(n-5) with a(0)=13,a(1)=76,a(2)=435,a(3)=2461,a(4)=13971.

G.f.: (-4*x^4+23*x^3-12*x^2-37*x-13)/(4*x^5-13*x^4+3*x^3+15*x^2+3*x-1).

A182054 Number of independent sets of nodes in the generalized Petersen graph G(2n,2) (n>=0).

Original entry on oeis.org

8, 3, 39, 171, 1055, 5828, 33327, 188499, 1069855, 6065487, 34399844, 195074223, 1106262671, 6273528979, 35576813647, 201753798116, 1144133068159, 6488305791115, 36794770328583, 208660804936031, 1183302172416580, 6710431459264095, 38054430587741959

Offset: 0

Cesar Bautista, Apr 08 2012

Cesar Bautista, Table of n, a(n) for n = 0..499
C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), Article 12.7.8.
Stephan G. Wagner, The Fibonacci Number of Generalized Petersen Graphs, Fibonacci Quarterly, 44 (2006), 362-367.
Index entries for linear recurrences with constant coefficients, signature (3,15,3,-13,4).

Cf. A182077.

a(n) = 3*a(n-1)+15*a(n-2)+3*a(n-3)-13*a(n-4)+4*a(n-5) with a(0)=8, a(1)=3, a(2)=39, a(3)=171, a(4)=1055, a(5)=5828.

G.f.: ((6*x^2-11*x-8)*(2*x^3-5*x^2-4*x+1)) / (4*x^5-13*x^4+3*x^3+15*x^2+3*x-1).

A182052 Number of independent sets of nodes in C_6 X C_n (n >= 1).

Original entry on oeis.org

18, 1, 199, 1300, 18995, 199821, 2406862, 27285777, 317960739, 3658040968, 42338077399, 488631332773, 5646974285234, 65218753680549, 753462136109959, 8703368091760320, 100541026090416195, 1161408360176875825, 13416320242101088558, 154981059170079355117

Offset: 0

Cesar Bautista, Apr 08 2012

M. Golin, Y. C. Leung, Y. J. Wang and X. R. Yong, Counting structures in grid-graphs, cylinders and tori using transfer matrices: Survey and new results. In: C. Demetrescu, R. Sedgewick and R.Tamassia, (eds.) The Proceedings of the Second Workshop on Analytic Algorithmics and Combinatorics (ANALCO05), SIAM, Philadelphia, (2005), 250-258.

Cesar Bautista, Table of n, a(n) for n = 0..399
C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), #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 (4,84,89,-575,-360,1301,115,-1032,295,119,-36,-4,1).

LinearRecurrence[{4,84,89,-575,-360,1301,115,-1032,295,119,-36,-4,1},{18,1,199,1300,18995,199821,2406862,27285777,317960739,3658040968,42338077399,488631332773,5646974285234},20](* Harvey P. Dale, Nov 24 2012 *)

a(n) = 4*a(n-1) +84*a(n-2) +89*a(n-3) -575*a(n-4) -360*a(n-5) +1301*a(n-6) +115*a(n-7) -1032*a(n-8) +295*a(n-9) +119*a(n-10) -36*a(n-11) -4*a(n-12) +a(n-13) with a(0)=18, a(1)=1, a(2)=199, a(3)=1300, a(4)=18995, a(5)=199821, a(6)=2406862, a(7)=27285777, a(8)=317960739, a(9)=3658040968, a(10)=42338077399, a(11)=488631332773, a(12)=5646974285234.

G.f: (-9*x^12 -67*x^11 +556*x^10 +1162*x^9 -6841*x^8 +1421*x^7 +12335*x^6 -3985*x^5 -7340*x^4 +1182*x^3 +1317*x^2 +71*x-18) / ((x-1) *(x^2-3*x-1) *(x^2-x-1) *(x^3+3*x^2-5*x-1) *(x^5-2*x^4-25*x^3-3*x^2+12*x-1)).

A182041 Number of independent sets of nodes in C_5 X C_n (n >= 1).

Original entry on oeis.org

11, 1, 81, 391, 3561, 25531, 199821, 1511931, 11589281, 88389661, 675443291, 5157630831, 39394699881, 300868345701, 2297915763861, 17550293888221, 134040955378561, 1023739686467981, 7818833928607761, 59716490127924211, 456085875187977011, 3483364700645591901

Offset: 0

Cesar Bautista, Apr 07 2012

M. Golin, Y. C. Leung, Y. J. Wang and X. R. Yong, Counting structures in grid-graphs, cylinders and tori using transfer matrices: Survey and new results. In: Demetrescu, C., Sedgewick, R., Tamassia, R., (eds.) The Proceedings of the Second Workshop on Analytic Algorithmics and Combinatorics (ANALCO05), SIAM, Philadelphia, (2005), 250-258.

Cesar Bautista, Table of n, a(n) for n = 0..399
C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), #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 (4,27,10,-30,-7,8,-1).

LinearRecurrence[{4,27,10,-30,-7,8,-1},{11,1,81,391,3561,25531,199821},30] (* Harvey P. Dale, Mar 06 2013 *)

a(n)=4*a(n-1)+27*a(n-2)+10*a(n-3)-30*a(n-4)-7*a(n-5)+8*a(n-6)-a(n-7) with a(0)=11, a(1)=1, a[2]=81, a(3)=391, a(4)=3561, a(5)=25531, a(6)=199821.

G.f.: (-11*x^6+27*x^5+130*x^4-70*x^3-220*x^2-43*x+11)/((x^3-5*x^2-7*x+1)*(x^4-3*x^3-x^2+3*x+1)).

A182019 Number of independent sets of nodes in graph C_8 x P_n (n>=0).

Original entry on oeis.org

1, 47, 1155, 30277, 788453, 20546803, 535404487, 13951571713, 363549830913, 9473376491295, 246857112567171, 6432599206076589, 167620580643483109, 4367854759124964451, 113817498564834289095, 2965854794621630365713, 77284202988962060229833

Offset: 0

Cesar Bautista, Apr 06 2012

Cesar Bautista, Table of n, a(n) for n = 0..499
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 (29,-65,-317,334,187,-109,5,1).

Row 8 of A286513.

a(n) = 29*a(n-1)-65*a(n-2)-317*a(n-3)+334*a(n-4)+187*a(n-5)-109*a(n-6)+5*a(n-7)+a(n-8) with a(0)=1, a(1)=47,a(2)=1155,a(3)=30277,a(4)=788453, a(5)=20546803, a(6)=535404487, a(7)=13951571713.

G.f.: -(x^7 +4*x^6 -79*x^5 +60*x^4 +154*x^3 -143*x^2 +18*x +1)/(x^8 +5*x^7 -109*x^6 +187*x^5 +334*x^4 -317*x^3 -65*x^2 +29*x -1). [Colin Barker, Aug 31 2012]

A182014 Number of independent sets of nodes in graph C_7 x P_n (n>=0).

Original entry on oeis.org

1, 29, 477, 8303, 143697, 2488431, 43089985, 746156517, 12920616493, 223736359029, 3874270087045, 67087749098875, 1161706844818941, 20116382073294655, 348339884131004417, 6031933298656980345, 104450339960964929961, 1808686034441106749965

Offset: 0

Cesar Bautista, Apr 06 2012

Cesar Bautista, Table of n, a(n) for n = 0..399
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 (17,8,-44,5,1).

Row 7 of A286513.

LinearRecurrence[{17,8,-44,5,1},{1,29,477,8303,143697},30] (* Harvey P. Dale, Aug 27 2012 *)

Vec((x^4+6*x^3-24*x^2+12*x+1)/(-x^5-5*x^4+44*x^3-8*x^2-17*x+1)+O(x^99)) \\ Charles R Greathouse IV, Apr 06 2012

a(n) = 17*a(n-1) + 8*a(n-2) - 44*a(n-3) + 5*a(n-4) + a(n-5) with a(0)=1, a(1)=29, a(2)=477, a(3)=8303, a(4)=143697.

G.f.: (x^4+6*x^3-24*x^2+12*x+1)/(-x^5-5*x^4+44*x^3-8*x^2-17*x+1).

A181989 Number of independent sets of nodes in graph C_5 x P_n (n >= 0).

Original entry on oeis.org

1, 11, 81, 621, 4741, 36211, 276561, 2112241, 16132281, 123210611, 941023441, 7187084861, 54891500621, 419234905211, 3201914754721, 24454686308481, 186773143027761, 1426483517982011, 10894795654704401, 83209214029813101, 635511992964231701, 4853735225243983011

Offset: 0

Cesar Bautista, Apr 04 2012

Cesar Bautista, Table of n, a(n) for n = 0..299
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 (7,5,-1).

Row 5 of A286513.

a(n) = 7*a(n-1) + 5*a(n-2) - a(n-3) with a(0)=1, a(1)=11, a(2)=81.

G.f.: (1+4*x-x^2)/(x^3-5*x^2-7*x+1).

cp's OEIS Frontend

User: Cesar Bautista

A182141 Number of independent sets of nodes in the armchair (3,3) carbon nanotorus graph of breadth n (n>=1).

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A182130 Number of independent sets of nodes in the armchair (3,3) carbon nanotube graph of breadth n (n>=1).

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A182143 Number of independent vertex sets in the Moebius ladder graph with 2n nodes (n >= 0).

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A182077 Number of independent sets of nodes in the generalized Petersen graph G(2n+1,2) (n>=1).

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A182054 Number of independent sets of nodes in the generalized Petersen graph G(2n,2) (n>=0).

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A182052 Number of independent sets of nodes in C_6 X C_n (n >= 1).

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A182041 Number of independent sets of nodes in C_5 X C_n (n >= 1).

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A182019 Number of independent sets of nodes in graph C_8 x P_n (n>=0).

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A182014 Number of independent sets of nodes in graph C_7 x P_n (n>=0).

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