Paul Raff - cp's OEIS frontend

A167061 Number of spanning trees in G X P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}}.

40, 47040, 48384000, 49461807360, 50545351901000, 51651393970176000, 52781550346052950760, 53936428658183506928640, 55116575633234676605184000, 56322544581812152703647896000, 57554900528304912551898910864840, 58814220831251084699615165546496000, 60101095479875496770600392870888679560

Offset: 1

Paul Raff

nonn

F. Faase, On the number of specific spanning subgraphs of the graphs A X P_n, Ars Combin. 49 (1998), 129-154.

P. Raff, Table of n, a(n) for n = 1..200
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
P. Raff, Spanning Trees in Grid Graphs, arXiv:0809.2551 [math.CO], 2008.
P. Raff, Analysis of the Number of Spanning Trees of G x P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}}. Contains sequence, recurrence, generating function, and more.
P. Raff, Analysis of the Number of Spanning Trees of Grid Graphs.
Index entries for sequences related to trees

a(n) = 1152 a(n-1)
- 138048 a(n-2)
+ 5263416 a(n-3)
- 72792384 a(n-4)
+ 279916416 a(n-5)
- 429599666 a(n-6)
+ 279916416 a(n-7)
- 72792384 a(n-8)
+ 5263416 a(n-9)
- 138048 a(n-10)
+ 1152 a(n-11)
- a(n-12)
G.f.: -40x(x^10 +24x^9 -7104x^8 +167016x^7 -378475x^6 +378475x^4 -167016x^3 +7104x^2 -24x -1)/ (x^12 -1152x^11 +138048x^10 -5263416x^9 +72792384x^8 -279916416x^7 +429599666x^6 -279916416x^5 +72792384x^4 -5263416x^3 +138048x^2 -1152x +1).

A167059 Number of spanning trees in G X P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}}.

Original entry on oeis.org

8, 4032, 1612800, 631427328, 246562692200, 96244833484800, 37566939748080392, 14663279200231130112, 5723424260979717196800, 2233987356983360324068800, 871977888467614764819315368, 340353508793721676084268236800, 132847991246505889127220947758952

Offset: 1

Paul Raff, Jun 01 2010

nonn

F. Faase, On the number of specific spanning subgraphs of the graphs A X P_n, Ars Combin. 49 (1998), 129-154.

P. Raff, Table of n, a(n) for n = 1..200
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
P. Raff, Spanning Trees in Grid Graphs, arXiv:0809.2551 [math.CO], 2008.
P. Raff, Analysis of the Number of Spanning Trees of G x P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}}. Contains sequence, recurrence, generating function, and more.
P. Raff, Analysis of the Number of Spanning Trees of Grid Graphs.
Index entries for sequences related to trees

a(n) = 504 a(n-1)
- 48706 a(n-2)
+ 1765008 a(n-3)
- 29021617 a(n-4)
+ 239655024 a(n-5)
- 1039298722 a(n-6)
+ 2447629128 a(n-7)
- 3242171236 a(n-8)
+ 2447629128 a(n-9)
- 1039298722 a(n-10)
+ 239655024 a(n-11)
- 29021617 a(n-12)
+ 1765008 a(n-13)
- 48706 a(n-14)
+ 504 a(n-15)
- a(n-16)
G.f.: -8x (x^14 -3710x^12 +104832x^11 -997954x^10 +3633840x^9 -4759203x^8 +4759203x^6 -3633840x^5 +997954x^4 -104832x^3 +3710x^2-1)/ (x^16 -504x^15 +48706x^14 -1765008x^13 +29021617x^12 -239655024x^11 +1039298722x^10 -2447629128x^9 +3242171236x^8 -2447629128x^7 +1039298722x^6 -239655024x^5 +29021617x^4 -1765008x^3 +48706x^2 -504x+1).

A167058 Number of spanning trees in (S_5 + e) X P_n.

Original entry on oeis.org

3, 945, 221184, 50055705, 11275732875, 2538325278720, 571357349020731, 128606300878893705, 28947814696524275712, 6515821689652895090625, 1466636804229895456081107, 330123137841949620861665280, 74306935243221668928140352051

Offset: 1

Paul Raff

nonn

F. Faase, On the number of specific spanning subgraphs of the graphs A X P_n, Ars Combin. 49 (1998), 129-154.

P. Raff, Table of n, a(n) for n = 1..200
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
P. Raff, Spanning Trees in Grid Graphs, arXiv:0809.2551 [math.CO], 2008.
P. Raff, Analysis of the Number of Spanning Trees of (S_5 + e) x P_n. Contains sequence, recurrence, generating function, and more.
P. Raff, Analysis of the Number of Spanning Trees of Grid Graphs.
Index entries for sequences related to trees
Index entries for linear recurrences with constant coefficients, signature (270,-10529,95310,-177156,95310,-10529,270,-1).

CoefficientList[Series[-3x (x^6+45x^5-793x^4+793x^2-45x-1)/(x^8-270x^7+ 10529x^6-95310x^5+177156x^4-95310x^3+10529x^2-270x+1),{x,0,30}],x] (* or *) LinearRecurrence[{270,-10529,95310,-177156,95310,-10529,270,-1},{0,3,945,221184,50055705,11275732875,2538325278720,571357349020731},30] (* Harvey P. Dale, Nov 22 2021 *)

a(n) = 270 a(n-1)
- 10529 a(n-2)
+ 95310 a(n-3)
- 177156 a(n-4)
+ 95310 a(n-5)
- 10529 a(n-6)
+ 270 a(n-7)
- a(n-8)
G.f.: -3x(x^6 +45x^5 -793x^4 +793x^2 -45x -1)/ (x^8 -270x^7 +10529x^6 -95310x^5 +177156x^4 -95310x^3 +10529x^2 -270x +1)

A167069 Number of spanning trees in G X P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {4, 5}}.

Original entry on oeis.org

3, 1005, 250848, 60075885, 14263332015, 3379514561280, 800337094071879, 189513130911442365, 44873808170614072416, 10625354802279238810125, 2515898969449422698378427, 595720806457312484163072000, 141056237447350542048435569739

Offset: 1

Paul Raff

nonn

F. Faase, On the number of specific spanning subgraphs of the graphs A X P_n, Ars Combin. 49 (1998), 129-154.

P. Raff, Table of n, a(n) for n = 1..200
P. Raff, Spanning Trees in Grid Graphs, arXiv:0809.2551 [math.CO], 2008.
P. Raff, Analysis of the Number of Spanning Trees of G x P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {4, 5}}. Contains sequence, recurrence, generating function, and more.
P. Raff, Analysis of the Number of Spanning Trees of Grid Graphs.
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
Index entries for sequences related to trees

a(n) = 335 a(n-1)
- 26224 a(n-2)
+ 744035 a(n-3)
- 10084457 a(n-4)
+ 72968360 a(n-5)
- 295849710 a(n-6)
+ 685799270 a(n-7)
- 909474816 a(n-8)
+ 685799270 a(n-9)
- 295849710 a(n-10)
+ 72968360 a(n-11)
- 10084457 a(n-12)
+ 744035 a(n-13)
- 26224 a(n-14)
+ 335 a(n-15)
- a(n-16)
G.f.: -3x(x^14 -2385x^12 +54940x^11 -451104x^10 +1542340x^9 -2024890x^8 +2024890x^6 -1542340x^5 +451104x^4 -54940x^3 +2385x^2 -1)/ (x^16 -335x^15 +26224x^14 -744035x^13 +10084457x^12 -72968360x^11 +295849710x^10 -685799270x^9 +909474816x^8 -685799270x^7 +295849710x^6 -72968360x^5 +10084457x^4 -744035x^3 +26224x^2 -335x +1).

A167068 Number of spanning trees in G X P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {4, 5}}.

Original entry on oeis.org

11, 6061, 2733511, 1215842661, 540144000000, 239933520731861, 106577890632874111, 47341582784338831461, 21028987835540967334811, 9341012640240002304000000, 4149249488236281570533713211, 1843084039808720108847180812661, 818692341198182161542031245824911

Offset: 1

Paul Raff

nonn

F. Faase, On the number of specific spanning subgraphs of the graphs A X P_n, Ars Combin. 49 (1998), 129-154.

P. Raff, Table of n, a(n) for n = 1..200
P. Raff, Spanning Trees in Grid Graphs, arXiv:0809.2551 [math.CO], 2008.
P. Raff, Analysis of the Number of Spanning Trees of G x P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {4, 5}}. Contains sequence, recurrence, generating function, and more.
P. Raff, Analysis of the Number of Spanning Trees of Grid Graphs.
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
Index entries for sequences related to trees

a(n) = 551 a(n-1)
- 51500 a(n-2)
+ 1873400 a(n-3)
- 31993500 a(n-4)
+ 271314053 a(n-5)
- 1157139603 a(n-6)
+ 2669595000 a(n-7)
- 3507446800 a(n-8)
+ 2669595000 a(n-9)
- 1157139603 a(n-10)
+ 271314053 a(n-11)
- 31993500 a(n-12)
+ 1873400 a(n-13)
- 51500 a(n-14)
+ 551 a(n-15)
- a(n-16)
G.f.: -11x (x^14 -3600x^12 +110200x^11 -1112601x^10 +3855898x^9 -4841800x^8 +4841800x^6 -3855898x^5 +1112601x^4 -110200x^3 +3600x^2-1)/(x^16 -551x^15 +51500x^14 -1873400x^13 +31993500x^12 -271314053x^11 +1157139603x^10 -2669595000x^9 +3507446800x^8 -2669595000x^7 +1157139603x^6 -271314053x^5 +31993500x^4 -1873400x^3 +51500x^2 -551x+1).

A167066 Number of spanning trees in G X P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 5}, {4, 5}}.

Original entry on oeis.org

24, 20160, 14515200, 10373448960, 7410329640120, 5293465841664000, 3781306797401609112, 2701118650243184317440, 1929502759140378901785600, 1378310758353447731649144000, 984575190426384431371033497336, 703316214957312006365562863616000, 502403171470887016026721609133115192

Offset: 1

Paul Raff

nonn

F. Faase, On the number of specific spanning subgraphs of the graphs A X P_n, Ars Combin. 49 (1998), 129-154.

P. Raff, Table of n, a(n) for n = 1..200
P. Raff, Spanning Trees in Grid Graphs, arXiv:0809.2551 [math.CO], 2008.
P. Raff, Analysis of the Number of Spanning Trees of G x P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 5}, {4, 5}}. Contains sequence, recurrence, generating function, and more.
P. Raff, Analysis of the Number of Spanning Trees of Grid Graphs.
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
Index entries for sequences related to trees

a(n) = 840 a(n-1)
- 95522 a(n-2)
+ 4231920 a(n-3)
- 87627601 a(n-4)
+ 863951760 a(n-5)
- 3862082882 a(n-6)
+ 9004563960 a(n-7)
- 11846119204 a(n-8)
+ 9004563960 a(n-9)
- 3862082882 a(n-10)
+ 863951760 a(n-11)
- 87627601 a(n-12)
+ 4231920 a(n-13)
- 95522 a(n-14)
+ 840 a(n-15)
- a(n-16)
G.f.: -24x(x^14 -5278x^12 +201600x^11 -2458194x^10 +8663760x^9 -10786195x^8 +10786195x^6 -8663760x^5 +2458194x^4 -201600x^3 +5278x^2 -1)/ (x^16 -840x^15 +95522x^14 -4231920x^13 +87627601x^12 -863951760x^11 +3862082882x^10 -9004563960x^9 +11846119204x^8 -9004563960x^7 +3862082882x^6 -863951760x^5 +87627601x^4 -4231920x^3 +95522x^2 -840x +1).

A167063 Number of spanning trees in G X P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 5}}.

Original entry on oeis.org

21, 16905, 11515392, 7766579625, 5234202655605, 3527304596766720, 2377020102892371573, 1601852459790100499625, 1079473906452564386072064, 727447713589013080159967625, 490220442215546503112745464469, 330355127203424593855513657344000, 222623335689469074506271256084716693

Offset: 1

Paul Raff

F. Faase, On the number of specific spanning subgraphs of the graphs a X P_n, Ars Combin. 49 (1998), 129-154.

P. Raff, Table of n, a(n) for n = 1..200
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
P. Raff, Spanning Trees in Grid Graphs, arXiv:0809.2551 [math.CO], 2008.
P. Raff, Analysis of the Number of Spanning Trees of G x P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 5}}. Contains sequence, recurrence, generating function, and more.
P. Raff, Analysis of the Number of Spanning Trees of Grid Graphs.
Index entries for sequences related to trees

a(n) = 805 a(n-1)
- 94300 a(n-2)
+ 4128845 a(n-3)
- 82955561 a(n-4)
+ 801676960 a(n-5)
- 3659544950 a(n-6)
+ 8726681390 a(n-7)
- 11584112776 a(n-8)
+ 8726681390 a(n-9)
- 3659544950 a(n-10)
+ 801676960 a(n-11)
- 82955561 a(n-12)
+ 4128845 a(n-13)
- 94300 a(n-14)
+ 805 a(n-15)
- a(n-16)
G.f.: -21x(x^14 -5373x^12 +196420x^11 -2311184x^10 +8452500x^9 -10863790x^8 +10863790x^6 -8452500x^5 +2311184x^4 -196420x^3 +5373x^2 -1)/ (x^16 -805x^15 +94300x^14 -4128845x^13 +82955561x^12 -801676960x^11 +3659544950x^10 -8726681390x^9 +11584112776x^8 -8726681390x^7 +3659544950x^6 -801676960x^5 +82955561x^4 -4128845x^3 +94300x^2 -805x +1).

A167072 Number of spanning trees in G X P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 5}, {4, 5}}.

Original entry on oeis.org

12, 6720, 3110400, 1423806720, 651286330860, 297900675072000, 136260356109480876, 62325740425973498880, 28507909150300692211200, 13039570449847302883368000, 5964323676112090939594326348, 2728092696767010687412666368000, 1247834652562251646622689145644236

Offset: 1

Paul Raff, Jun 01 2010

nonn

F. Faase, On the number of specific spanning subgraphs of the graphs A X P_n, Ars Combin. 49 (1998), 129-154.

P. Raff, Table of n, a(n) for n = 1..200
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
P. Raff, Spanning Trees in Grid Graphs, arXiv:0809.2551 [math.CO], 2008.
P. Raff, Analysis of the Number of Spanning Trees of G x P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 5}, {4, 5}}. Contains sequence, recurrence, generating function, and more.
P. Raff, Analysis of the Number of Spanning Trees of Grid Graphs.
Index entries for sequences related to trees

a(n) = 525 a(n-1)
- 32415 a(n-2)
+ 696920 a(n-3)
- 5936265 a(n-4)
+ 19827675 a(n-5)
- 29313582 a(n-6)
+ 19827675 a(n-7)
- 5936265 a(n-8)
+ 696920 a(n-9)
- 32415 a(n-10)
+ 525 a(n-11)
- a(n-12).
G.f.: -12x (x^10 +35x^9 -2385x^8 +26040x^7 -54030x^6 +54030x^4 -26040x^3 +2385x^2 -35x-1) / (x^12 -525x^11 +32415x^10 -696920x^9 +5936265x^8 -19827675x^7 +29313582x^6 -19827675x^5 +5936265x^4 -696920x^3 +32415x^2 -525x+1).

A167071 Number of spanning trees in G X P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 5}}.

Original entry on oeis.org

4, 1376, 361860, 92544256, 23575404820, 6002044445280, 1527898117755412, 388939442019315712, 99007542753465378420, 25203122804459545322080, 6415645979596681028789108, 1633151297922105531036929280, 415731036835959295502046104100, 105827485262836457484100780941664

Offset: 1

Paul Raff

nonn

F. Faase, On the number of specific spanning subgraphs of the graphs A X P_n, Ars Combin. 49 (1998), 129-154.

P. Raff, Table of n, a(n) for n = 1..200
P. Raff, Spanning Trees in Grid Graphs, arXiv:0809.2551 [math.CO], 2008.
P. Raff, Analysis of the Number of Spanning Trees of G x P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 5}}. Contains sequence, recurrence, generating function, and more.
P. Raff, Analysis of the Number of Spanning Trees of Grid Graphs.
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
Index entries for sequences related to trees

a(n) = 344 a(n-1)
- 25540 a(n-2)
+ 745448 a(n-3)
- 10445708 a(n-4)
+ 76194968 a(n-5)
- 303860988 a(n-6)
+ 687124520 a(n-7)
- 899525622 a(n-8)
+ 687124520 a(n-9)
- 303860988 a(n-10)
+ 76194968 a(n-11)
- 10445708 a(n-12)
+ 745448 a(n-13)
- 25540 a(n-14)
+ 344 a(n-15)
- a(n-16)
G.f.: -4x (x^14 -2331x^12 +56416x^11 -467115x^10 +1546624x^9 -1949983x^8 +1949983x^6 -1546624x^5 +467115x^4 -56416x^3 +2331x^2 -1)/ (x^16 -344x^15 +25540x^14 -745448x^13 +10445708x^12 -76194968x^11 +303860988x^10 -687124520x^9 +899525622x^8 -687124520x^7 +303860988x^6 -76194968x^5 +10445708x^4 -745448x^3 +25540x^2 -344x+1).

A167070 Number of spanning trees in G X P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {2, 5}}.

Original entry on oeis.org

1, 201, 27872, 3656793, 474581525, 61445719296, 7951276371389, 1028790034978377, 133107787044919648, 17221739109190982025, 2228177484370996025801, 288285215706960759705600, 37298804748402271018820409, 4825779209505263485071458889

Offset: 1

Paul Raff

nonn

F. Faase, On the number of specific spanning subgraphs of the graphs A X P_n, Ars Combin. 49 (1998), 129-154.

P. Raff, Table of n, a(n) for n = 1..200
P. Raff, Spanning Trees in Grid Graphs, arXiv:0809.2551 [math.CO], 2008.
P. Raff, Analysis of the Number of Spanning Trees of G x P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {2, 5}}. Contains sequence, recurrence, generating function, and more.
P. Raff, Analysis of the Number of Spanning Trees of Grid Graphs.
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
Index entries for sequences related to trees

a(n) = 201 a(n-1)
- 11104 a(n-2)
+ 259893 a(n-3)
- 3001225 a(n-4)
+ 18824856 a(n-5)
- 67848270 a(n-6)
+ 144802410 a(n-7)
- 186068896 a(n-8)
+ 144802410 a(n-9)
- 67848270 a(n-10)
+ 18824856 a(n-11)
- 3001225 a(n-12)
+ 259893 a(n-13)
- 11104 a(n-14)
+ 201 a(n-15)
- a(n-16)
G.f.: -x(x^14 -1425x^12 +26532x^11 -180448x^10 +545916x^9 -661242x^8 +661242x^6 -545916x^5 +180448x^4 -26532x^3 +1425x^2 -1)/ (x^16 -201x^15 +11104x^14 -259893x^13 +3001225x^12 -18824856x^11 +67848270x^10 -144802410x^9 +186068896x^8 -144802410x^7 +67848270x^6 -18824856x^5 +3001225x^4 -259893x^3 +11104x^2 -201x +1).

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User: Paul Raff

A167061 Number of spanning trees in G X P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}}.

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A167059 Number of spanning trees in G X P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}}.

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A167058 Number of spanning trees in (S_5 + e) X P_n.

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A167069 Number of spanning trees in G X P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {4, 5}}.

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A167068 Number of spanning trees in G X P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {4, 5}}.

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A167066 Number of spanning trees in G X P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 5}, {4, 5}}.

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A167063 Number of spanning trees in G X P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 5}}.

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A167072 Number of spanning trees in G X P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 5}, {4, 5}}.

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A167071 Number of spanning trees in G X P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 5}}.

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A167070 Number of spanning trees in G X P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {2, 5}}.

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