A020874 -id:A020874 - cp's OEIS frontend

A288516 Number of (undirected) paths in the ladder graph P_2 X P_n.

1, 12, 49, 146, 373, 872, 1929, 4118, 8589, 17644, 35889, 72538, 146021, 293200, 587801, 1177278, 2356541, 4715412, 9433537, 18870210, 37744021, 75492152, 150988969, 301983206, 603972333, 1207951292, 2415909969, 4831828138, 9663665349, 19327340704

Offset: 1

Andrew Howroyd, Jun 10 2017

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Graph Path
Eric Weisstein's World of Mathematics, Ladder Graph
Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-9,2).

Row 2 of A288518.

Cf. A288032, A137882, A287992, A020874.

Table[18 (2^n - 1) - n (n^2 + 9 n + 41)/3, {n, 20}] (* Eric W. Weisstein, Jun 30 2017 *)
LinearRecurrence[{6, -14, 16, -9, 2}, {1, 12, 49, 146, 373}, 20] (* Eric W. Weisstein, Jun 30 2017 *)
CoefficientList[Series[(-1 - 6 x + 9 x^2 - 4 x^3)/((-1 + x)^4 (-1 + 2 x)), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 30 2017 *)

Vec((1+6*x-9*x^2+4*x^3)/((1-x)^4*(1-2*x))+O(x^25))

a(n) = 18*(2^n - 1) - n*(n^2 + 9*n + 41)/3 \\ Charles R Greathouse IV, Jun 30 2017

a(n) = 18*(2^n - 1) - n*(n^2 + 9*n + 41)/3. - Eric W. Weisstein, Jun 30 2017
a(n) = 6*a(n-1)-14*a(n-2)+16*a(n-3)-9*a(n-4)+2*a(n-5) for n > 5.
G.f.: x*(1+6*x-9*x^2+4*x^3)/((1-x)^4*(1-2*x)).
a(n) =  18*(2^n-1) - (41*n)/3 - 3*n^2 - n^3/3. - Colin Barker, Jun 11 2017

A288148 Number of (undirected) paths in the n-triangular grid graph.

Original entry on oeis.org

0, 6, 108, 2598, 123750, 12994248, 3114709914, 1730766715308, 2248937669398650, 6877862090075063484, 49790967547432817528562, 857275977287938332061154856, 35233501393224883314185777947590

Offset: 0

Eric W. Weisstein, Jun 05 2017

Paths of length zero are not counted here.

			a(1) = 6:
:    o    :    o    :    o    :    o    :    o    :    o
:         :   /     :     \   :   / \   :     \   :   /
:  o---o  :  o   o  :  o   o  :  o   o  :  o---o  :  o---o  .

Eric Weisstein's World of Mathematics, Graph Path
Eric Weisstein's World of Mathematics, Triangular Grid Graph

Cf. A020874, A266513, A112675, A112676.

a(7)-a(12) from Andrew Howroyd, Jun 07 2017

A287992 Number of (undirected) paths in the prism graph Y_n.

Original entry on oeis.org

1, 26, 129, 444, 1285, 3366, 8281, 19544, 44829, 100770, 223201, 488916, 1061749, 2289854, 4910505, 10480176, 22275661, 47178234, 99605809, 209704940, 440390181, 922733526, 1929364729, 4026514824, 8388588925, 17448283346, 36238762881, 75161901444, 155692535509, 322122515310

Offset: 1

Eric W. Weisstein, Jun 04 2017

Extended to a(1)-a(2) using the formula.

Colin Barker, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Graph Path.
Eric Weisstein's World of Mathematics, Prism Graph.
Index entries for linear recurrences with constant coefficients, signature (8,-26,44,-41,20,-4).

Cf. A077265, A124350, A124349, A287988, A020874, A288516.

Table[(5 2^(n + 1) - 5 n - n^2 - 13) n, {n, 20}]
LinearRecurrence[{8, -26, 44, -41, 20, -4}, {1, 26, 129, 444, 1285, 3366}, 20]
CoefficientList[Series[(1 + 18 x - 53 x^2 + 44 x^3 - 16 x^4)/((1 - x)^4 (1 - 2 x)^2), {x, 0, 20}], x]

Vec(x*(1 + 18*x - 53*x^2 + 44*x^3 - 16*x^4) / ((1 - x)^4*(1 - 2*x)^2) + O(x^30)) \\ Colin Barker, Jun 04 2017

a(n) = (5*2^(n + 1) - 5*n - n^2 - 13)*n.
From Colin Barker, Jun 04 2017: (Start)
G.f.: x*(1 + 18*x - 53*x^2 + 44*x^3 - 16*x^4) / ((1 - x)^4*(1 - 2*x)^2).
a(n) = 8*a(n-1) - 26*a(n-2) + 44*a(n-3) - 41*a(n-4) + 20*a(n-5) - 4*a(n-6) for n>6. (End)

cp's OEIS Frontend

A288516 Number of (undirected) paths in the ladder graph P_2 X P_n.

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A288148 Number of (undirected) paths in the n-triangular grid graph.

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A287992 Number of (undirected) paths in the prism graph Y_n.

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