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A138979
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Number of 4 X n matrices containing a 1 in the top left entry, all entries are integer values and adjacent entries differ by at most 1.
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5
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27, 771, 22979, 690437, 20780181, 625649047, 18838482047, 567241901289, 17080173559277, 514300085627023, 15486061794514775, 466299978310573033, 14040733816061115637, 422779788989982722559, 12730299739840800975879, 383321378409770250813777
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OFFSET
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1,1
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COMMENTS
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Horizontally or vertically adjacent entries can differ by at most 1. Diagonally adjacent entries thus differ by at most 2.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (45,-528,2592,-5997,5689,812,-4760,1942,278,-112).
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FORMULA
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a(n)=b(n)+c(n)+d(n)+e(n)+f(n)+g(n)+h(n)+j(n)+k(n)+l(n), where
b(1)=2,c(1)=4,d(1)=4,e(1)=2,f(1)=4,g(1)=2,h(1)=2,j(1)=4,k(1)=2,l(1)=1
b(n+1)=3*b(n)+2*c(n)+1*d(n)+2*e(n)+1*f(n)+0*g(n)+1*h(n)+0*j(n)+1*k(n)+0*l(n)
c(n+1)=4*b(n)+5*c(n)+3*d(n)+4*e(n)+3*f(n)+2*g(n)+2*h(n)+3*j(n)+4*k(n)+4*l(n)
d(n+1)=2*b(n)+3*c(n)+5*d(n)+2*e(n)+4*f(n)+4*g(n)+4*h(n)+4*j(n)+4*k(n)+4*l(n)
e(n+1)=2*b(n)+2*c(n)+1*d(n)+3*e(n)+2*f(n)+2*g(n)+2*h(n)+2*j(n)+2*k(n)+2*l(n)
f(n+1)=2*b(n)+3*c(n)+4*d(n)+4*e(n)+7*f(n)+6*g(n)+6*h(n)+6*j(n)+6*k(n)+8*l(n)
g(n+1)=0*b(n)+1*c(n)+2*d(n)+2*e(n)+3*f(n)+4*g(n)+2*h(n)+3*j(n)+2*k(n)+4*l(n)
h(n+1)=1*b(n)+1*c(n)+2*d(n)+2*e(n)+3*f(n)+2*g(n)+4*h(n)+3*j(n)+2*k(n)+4*l(n)
j(n+1)=0*b(n)+3*c(n)+4*d(n)+4*e(n)+6*f(n)+6*g(n)+6*h(n)+7*j(n)+6*k(n)+8*l(n)
k(n+1)=1*b(n)+2*c(n)+2*d(n)+2*e(n)+3*f(n)+2*g(n)+2*h(n)+3*j(n)+4*k(n)+4*l(n)
l(n+1)=0*b(n)+1*c(n)+1*d(n)+1*e(n)+2*f(n)+2*g(n)+2*h(n)+2*j(n)+2*k(n)+3*l(n).
G.f.: -x*(-27 +116*x^9 -206*x^8 +5284*x^6 -2464*x^7 -154*x^5 +6514*x^3 -6915*x^4 -2540*x^2 +444*x) / (1 -45*x -1942*x^8 +528*x^2 -278*x^9 -2592*x^3 +112*x^10 +5997*x^4 -5689*x^5 -812*x^6 +4760*x^7). - Alois P. Heinz, Sep 02 2014
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MAPLE
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a:= n-> (Matrix([2, 4, 4, 2, 4, 2, 2, 4, 2, 1]). Matrix([[3, 4, 2, 2, 2, 0, 1, 0, 1, 0], [2, 5, 3, 2, 3, 1, 1, 3, 2, 1], [1, 3, 5, 1, 4, 2, 2, 4, 2, 1], [2, 4, 2, 3, 4, 2, 2, 4, 2, 1], [1, 3, 4, 2, 7, 3, 3, 6, 3, 2], [0, 2, 4, 2, 6, 4, 2, 6, 2, 2], [1, 2, 4, 2, 6, 2, 4, 6, 2, 2], [0, 3, 4, 2, 6, 3, 3, 7, 3, 2], [1, 4, 4, 2, 6, 2, 2, 6, 4, 2], [0, 4, 4, 2, 8, 4, 4, 8, 4, 3]])^(n-1) .Matrix([[1], [1], [1], [1], [1], [1], [1], [1], [1], [1]]))[1, 1]: seq(a(n), n=1..20); # Alois P. Heinz, Aug 28 2008
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MATHEMATICA
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LinearRecurrence[{45, -528, 2592, -5997, 5689, 812, -4760, 1942, 278, -112}, {27, 771, 22979, 690437, 20780181, 625649047, 18838482047, 567241901289, 17080173559277, 514300085627023}, 20] (* Paolo Xausa, Mar 17 2024 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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