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

1,1

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).

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

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 *)

Crossrefs

Cf. A138977, A138978.

Sequence in context: A158645 A348634 A232951 * A183505 A218717 A159234

Adjacent sequences: A138976 A138977 A138978 * A138980 A138981 A138982

Keyword

nonn,easy

Author

Wayne VanWeerthuizen, Apr 05 2008

Extensions

More terms from Alois P. Heinz, Aug 28 2008

Status

approved