A095797 Four-column array read by rows: T(n,k) for k=0..3 is the k-th component of the vector obtained by multiplying the n-th power of the 4 X 4 matrix (1,1,1,1; 7,3,1,0; 12,2,0,0; 6,0,0,0) and the vector (1,1,1,1).

1, 1, 1, 1, 4, 11, 14, 6, 35, 75, 70, 24, 204, 540, 570, 210, 1524, 3618, 3528, 1224, 9894, 25050, 25524, 9144, 69612, 169932, 168828, 59364, 467736, 1165908, 1175208, 417672, 3226524, 7947084, 7944648, 2806416, 21924672, 54371568, 54612456, 19359144, 150267840, 371199864

Offset: 0

Author

Gary W. Adamson, Jun 06 2004

Keywords

• nonn
• tabf
• easy

Comments

(n+1)-st set of 4 terms = leftmost finite differences of sequences generated from 3rd degree polynomials having n-th row coefficients, (given n = 1,2,3...) For example, first row is (1 1 1 1) with a corresponding polynomial x^3 + x^2 + x + 1. (f(x),x = 1,2,3...) = 4, 15, 40, 85, 156...Leftmost term of the sequence = 4, with finite difference rows: 11, 25, 45, 71...; 14, 20, 26, 32...; and 6, 6, 6, 6. Thus leftmost terms of the sequence 4, 15, 40...and the finite difference rows are (4 11 14 6) which is the second row.

The matrix generator is discussed in A028246, while 2nd degree polynomial examples are A091140, A091141 and A091140. The first degree case is A095795.

Examples

			3rd set of 4 terms = (35, 75, 70, 24) since M^2 * [1 1 1 1] = [35 75 70 24].
Array begins:
     1,    1,    1,   1;
     4,   11,   14,   6;
    35,   75,   70,  24;
   204,  540,  570, 210;
  1524, 3618, 3528,1224;
  9894,25050,25524,9144;
		

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,0,4,0,0,0,24,0,0,0,-30,0,0,0,-12).

Crossrefs

Cf. A028246, A091140, A091141, A091142, A095795, A053698.

Programs

Maple

M := Matrix(4,4,[1,1,1,1,7,3,1,0,12,2,0,0,6,0,0,0]) ;
v := Vector(4,[1,1,1,1]) ;
for i from 0 to 20 do
        Mpr := (M ^ i).v ;
        for j from 1 to 4 do
                printf("%d,", Mpr[j]) ;
        end do;
end do; # R. J. Mathar, Jun 20 2011

Mathematica

LinearRecurrence[{0,0,0,4,0,0,0,24,0,0,0,-30,0,0,0,-12},{1,1,1,1,4,11,14,6,35,75,70,24,204,540,570,210},50] (* Harvey P. Dale, Feb 08 2013 *)

PARI

Vec((1+x+x^2+x^3+7*x^5+10*x^6+2*x^7-5*x^8+7*x^9-10*x^10-2*x^12 +6*x^13-16*x^14-24*x^11) / (1-4*x^4-24*x^8+30*x^12+12*x^16)+O(x^99)) \\ Charles R Greathouse IV, Jun 21 2011

Formula

G.f.: ( 1 +x +x^2 +x^3 +7*x^5 +10*x^6 +2*x^7 -5*x^8 +7*x^9 -10*x^10 -2*x^12 +6*x^13 -16*x^14 -24*x^11 ) / ( 1-4*x^4-24*x^8+30*x^12+12*x^16 ). - R. J. Mathar, Jun 20 2011

a(n) = +4*a(n-4) +24*a(n-8) -30*a(n-12) -12*a(n-16).

Extensions

Name added by R. J. Mathar, several entries corrected by Charles R Greathouse IV, Jun 21 2011