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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).

%I A095797 #27 Jan 06 2025 01:21:49
%S A095797 1,1,1,1,4,11,14,6,35,75,70,24,204,540,570,210,1524,3618,3528,1224,
%T A095797 9894,25050,25524,9144,69612,169932,168828,59364,467736,1165908,
%U A095797 1175208,417672,3226524,7947084,7944648,2806416,21924672,54371568,54612456,19359144,150267840,371199864
%N 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).
%C A095797 (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.
%C A095797 The matrix generator is discussed in A028246, while 2nd degree polynomial examples are A091140, A091141 and A091140. The first degree case is A095795.
%H A095797 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,4,0,0,0,24,0,0,0,-30,0,0,0,-12).
%F A095797 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
%F A095797 a(n) = +4*a(n-4) +24*a(n-8) -30*a(n-12) -12*a(n-16).
%e A095797 3rd set of 4 terms = (35, 75, 70, 24) since M^2 * [1 1 1 1] = [35 75 70 24].
%e A095797 Array begins:
%e A095797      1,    1,    1,   1;
%e A095797      4,   11,   14,   6;
%e A095797     35,   75,   70,  24;
%e A095797    204,  540,  570, 210;
%e A095797   1524, 3618, 3528,1224;
%e A095797   9894,25050,25524,9144;
%p A095797 M := Matrix(4,4,[1,1,1,1,7,3,1,0,12,2,0,0,6,0,0,0]) ;
%p A095797 v := Vector(4,[1,1,1,1]) ;
%p A095797 for i from 0 to 20 do
%p A095797         Mpr := (M ^ i).v ;
%p A095797         for j from 1 to 4 do
%p A095797                 printf("%d,", Mpr[j]) ;
%p A095797         end do;
%p A095797 end do; # _R. J. Mathar_, Jun 20 2011
%t A095797 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 *)
%o A095797 (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
%Y A095797 Cf. A028246, A091140, A091141, A091142, A095795, A053698.
%K A095797 nonn,tabf,easy
%O A095797 0,5
%A A095797 _Gary W. Adamson_, Jun 06 2004
%E A095797 Name added by _R. J. Mathar_, several entries corrected by _Charles R Greathouse IV_, Jun 21 2011