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A095797
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T(n,k) for k=0..3 is the k-th component of the vector obtained by multiplying the n-th power of the 4x4 matrix (1,1,1,1; 7,3,1,0; 12,2,0,0; 6,0,0,0) with the vector (1,1,1,1).
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| (n+1)-th 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.
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients, signature (0,0,0,4,0,0,0,24,0,0,0,-30,0,0,0,-12).
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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).
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EXAMPLE
| 3rd set of 4 terms = (35, 75, 70, 24) since M^2 * [1 1 1 1] = [35 75 70 24].
1,1,1,1;
4,11,14,6;
35,75,70,24;
204,540,570,210;
1524,3618,3528,1224;
9894,25050,25524,9144;
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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
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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
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CROSSREFS
| Cf. A028246, A091140, A091141, A091142, A095795, A053698.
Sequence in context: A098060 A154040 A066985 * A205846 A204542 A091436
Adjacent sequences: A095794 A095795 A095796 * A095798 A095799 A095800
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KEYWORD
| nonn,tabf,easy,uned
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 06 2004
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EXTENSIONS
| Name added by R. J. Mathar, several entries corrected by C. Greathouse, Jun 21 2011
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