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%I #15 Mar 17 2024 15:41:57
%S 27,771,22979,690437,20780181,625649047,18838482047,567241901289,
%T 17080173559277,514300085627023,15486061794514775,466299978310573033,
%U 14040733816061115637,422779788989982722559,12730299739840800975879,383321378409770250813777
%N 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.
%C Horizontally or vertically adjacent entries can differ by at most 1. Diagonally adjacent entries thus differ by at most 2.
%H Alois P. Heinz, <a href="/A138979/b138979.txt">Table of n, a(n) for n = 1..600</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (45,-528,2592,-5997,5689,812,-4760,1942,278,-112).
%F a(n)=b(n)+c(n)+d(n)+e(n)+f(n)+g(n)+h(n)+j(n)+k(n)+l(n), where
%F 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
%F 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)
%F 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)
%F 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)
%F 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 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)
%F 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)
%F 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)
%F 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)
%F 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)
%F 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).
%F 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
%p 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
%t 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 *)
%Y Cf. A138977, A138978.
%K nonn,easy
%O 1,1
%A _Wayne VanWeerthuizen_, Apr 05 2008
%E More terms from _Alois P. Heinz_, Aug 28 2008
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