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

0,5

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.

Links

Table of n, a(n) for n=0..41.

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).

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).

Example

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;

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A347561 A234903 A369615 * A205846 A204542 A247521

Adjacent sequences: A095794 A095795 A095796 * A095798 A095799 A095800

Keyword

nonn,tabf,easy

Author

Gary W. Adamson, Jun 06 2004

Extensions

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

Status

approved