_N. J. A. Sloane (njas(AT)research.att.com), _, Jul 23 2008

editing

approved

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approved

editing

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N. J. A. Sloane (njas, (AT)research.att.com), Jul 23 2008

Triangular sequence produced from symmetrical power of two matrices of the general type: M={{1, 2, 4, 8}, {2, 1, 2, 4}, {4, 2, 1, 2}, {8, 4, 2, 1}};.

A007376(4n+3).

3, 7, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 1, 1, 0, 5, 1, 0, 9, 1, -1, -3, -1, 1, 7, 1, 2, 1, 1, 2, 5, 1, 2, 9, 21, 1, 3, 3, 1, 3, -7, 1, 4, 1, 1, 4, 5, 1, -27, -108, -102, -4, 9, 1, 5, 3, 1, 81, 459, 810, 438, 5, -7, 1, -243, -1782, -4617, -4932, -1797, -6, 1, 729, 6561, 22437, 36045, 26811, 7251, 7, -1, -2187, -23328, -99144, -213192, -242190, -136080, -29088, -8, 6, 5, 1, 6561, 80919, 411156, 1109052, 1702782, 1475010, 660420, 116460, 6, 9, 1, 7, 3, 1, 7

1,4

0,1

These matrices are related to symmetrical autocorrelation matrices. First column is alternating powers of 3. Absolute value row sum is ( new to OEIS too): Flatten[Join[{{1}}, Table[Apply[Plus,Abs[CoefficientList[CharacteristicPolynomial[ M[d, a], x], x]]], {d, 1, 10}]]]; {1, 2, 6, 34, 242, 1794, 13378, 99842, 745218, 5562370, 41518082}

MathRev083099.nb at Mathematica tutorial: http://www.arec.umd.edu/mathrev/LecturesMR99.ZIP/

I'm sorry the definition isn't easier, but this is what seems to work: Function of matrix definition: ( needs all four variables) t(n, m, d, a) := If[n == m, 1, If[n - m <= d - 1 || m - n <= d - 1, a^Abs[n - m], 0]]; Matrix definition for general constant "a": M(d, a) := Table[t[n, m, d, a], {n, 1, d}, {m, 1, d}]; Constant: a=2; a(n)=CoefficientList(CharacteristicPloynomial(M(d,2))

Triangle begins:

{1},

{1, -1},

{-3, -2, 1},

{9, 21, 3, -1},

{-27, -108, -102, -4, 1},

{81, 459, 810, 438, 5, -1},

{-243, -1782, -4617, -4932, -1797, -6, 1}

t[n_, m_, d_, a_] := If[n == m, 1, If[n - m <= d - 1 || m - n <= d - 1, a^Abs[n - m], 0]]; M[d_, a_] := Table[t[n, m, d, a], {n, 1, d}, {m, 1, d}]; a = 2; a0 = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[M[d, a], x], x], {d, 1, 10}]]; Flatten[a0]

nonn,tabl,uned,new

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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jun 10 2007

njas, Jul 23 2008

Least prime number of the form k 2^n + 2^n - 1 (k=1,1,2,3,..; n=1,2,3,..).

Triangular sequence produced from symmetrical power of two matrices of the general type: M={{1, 2, 4, 8}, {2, 1, 2, 4}, {4, 2, 1, 2}, {8, 4, 2, 1}};.

1, 1, -1, -3, -2, 1, 9, 21, 3, -1, -27, -108, -102, -4, 1, 81, 459, 810, 438, 5, -1, -243, -1782, -4617, -4932, -1797, -6, 1, 729, 6561, 22437, 36045, 26811, 7251, 7, 23, 31, 127, 127, 383, 1279, 3583, 5119, 6143, 8191, 73727, 81919, 131071, 131071, 524287, 524287, 14680063, 14680063, 14680063, 109051903, 109051903, 654311423, 738197503, 738197503, 2147483647, 2147483647, 2147483647-1, -2187, -23328, -99144, -213192, -242190, -136080, -29088, -8, 1, 6561, 80919, 411156, 1109052, 1702782, 1475010, 660420, 116460, 9

0,1

1,4

These matrices are related to symmetrical autocorrelation matrices. First column is alternating powers of 3. Absolute value row sum is ( new to OEIS too): Flatten[Join[{{1}}, Table[Apply[Plus,Abs[CoefficientList[CharacteristicPolynomial[ M[d, a], x], x]]], {d, 1, 10}]]]; {1, 2, 6, 34, 242, 1794, 13378, 99842, 745218, 5562370, 41518082}

MathRev083099.nb at Mathematica tutorial: http://www.arec.umd.edu/mathrev/LecturesMR99.ZIP/

I'm sorry the definition isn't easier, but this is what seems to work: Function of matrix definition: ( needs all four variables) t(n, m, d, a) := If[n == m, 1, If[n - m <= d - 1 || m - n <= d - 1, a^Abs[n - m], 0]]; Matrix definition for general constant "a": M(d, a) := Table[t[n, m, d, a], {n, 1, d}, {m, 1, d}]; Constant: a=2; a(n)=CoefficientList(CharacteristicPloynomial(M(d,2))

a(3)=7 because A007522(1)=7

Triangle begins:

{1},

{1, -1},

{-3, -2, 1},

{9, 21, 3, -1},

{-27, -108, -102, -4, 1},

{81, 459, 810, 438, 5, -1},

{-243, -1782, -4617, -4932, -1797, -6, 1}

a = {}; Do[k = 1; While[ !PrimeQ[k 2^n + 2^n - 1], k++ ]; AppendTo[a, k 2^n + 2^n - 1], {n, 1, 50}]; a

t[n_, m_, d_, a_] := If[n == m, 1, If[n - m <= d - 1 || m - n <= d - 1, a^Abs[n - m], 0]]; M[d_, a_] := Table[t[n, m, d, a], {n, 1, d}, {m, 1, d}]; a = 2; a0 = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[M[d, a], x], x], {d, 1, 10}]]; Flatten[a0]

Cf. A007522, A127575, A127576, A127577, A127578, A127580, A127581, A127582, A127583, A127586, A127587.

nonn,tabl,uned,new

Artur Jasinski Roger L. Bagula (grafixrlbagulatftn(AT)cslyahoo.plcom), Jan 19 Jun 10 2007

Something seems to have gone wrong with A127581-A127584 - they need editing. - njas, Jan 21 2007

Least prime number of the form k 2^n + 2^n - 1 (k=1,1,2,3,..; n=1,2,3,..).

3, 7, 23, 31, 127, 127, 383, 1279, 3583, 5119, 6143, 8191, 73727, 81919, 131071, 131071, 524287, 524287, 14680063, 14680063, 14680063, 109051903, 109051903, 654311423, 738197503, 738197503, 2147483647, 2147483647, 2147483647

0,1

a(3)=7 because A007522(1)=7

a = {}; Do[k = 1; While[ !PrimeQ[k 2^n + 2^n - 1], k++ ]; AppendTo[a, k 2^n + 2^n - 1], {n, 1, 50}]; a

Cf. A007522, A127575, A127576, A127577, A127578, A127580, A127581, A127582, A127583, A127586, A127587.

nonn,uned

Artur Jasinski (grafix(AT)csl.pl), Jan 19 2007

Something seems to have gone wrong with A127581-A127584 - they need editing. - njas, Jan 21 2007

approved