A138979 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.

27, 771, 22979, 690437, 20780181, 625649047, 18838482047, 567241901289, 17080173559277, 514300085627023, 15486061794514775, 466299978310573033, 14040733816061115637, 422779788989982722559, 12730299739840800975879, 383321378409770250813777

Offset: 1

Author

Wayne VanWeerthuizen, Apr 05 2008

Keywords

• nonn
• easy

Comments

Horizontally or vertically adjacent entries can differ by at most 1. Diagonally adjacent entries thus differ by at most 2.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..600
• Index entries for linear recurrences with constant coefficients, signature (45,-528,2592,-5997,5689,812,-4760,1942,278,-112).

Crossrefs

Cf. A138977, A138978.

Programs

Maple

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

Mathematica

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 *)

Formula

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

Extensions

More terms from Alois P. Heinz, Aug 28 2008