A189237 -id:A189237 - cp's OEIS frontend

A189234 Expansion of (5-4x-12x^2+6x^3+3x^4)/(1-x-4x^2+3x^3+3*x^4-x^5).

5, 1, 9, 4, 25, 16, 78, 64, 257, 256, 874, 1013, 3034, 3953, 10684, 15229, 38017, 58056, 136338, 219508, 491870, 824737, 1782735, 3083887, 6484514, 11489516, 23652443, 42688039, 86459608, 158270401, 316576903, 585868009, 1160673633, 2166063365, 4259693562

Offset: 0

L. Edson Jeffery, Apr 18 2011

(Start) Let U be the unit-primitive matrix (see [Jeffery])
U=U_(11,1)=
(0 1 0 0 0)
(1 0 1 0 0)
(0 1 0 1 0)
(0 0 1 0 1)
(0 0 0 1 1).
Then a(n)=Trace(U^n). (End)
Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers of a unit-primitive matrix U_(N,r) (0

Michael De Vlieger, Table of n, a(n) for n = 0..3532
A. Akbary, Q. Wang, A generalized Lucas sequence and permutation binomials, Proc. Amer. Math. Soc. 134 (2006) 15-22, sequence a(n) with l=11.
L. E. Jeffery, Unit-primitive matrices
Genki Shibukawa, New identities for some symmetric polynomials and their applications, arXiv:1907.00334 [math.CA], 2019.
Q. Wang, On generalized Lucas sequences, Contemp. Math. 531 (2010) 127-141, Table 1 (k=5).
Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-3,1).

A189235, A189236, A189237.

Unsigned version of A094650.

CoefficientList[Series[(5-4x-12x^2+6x^3+3x^4)/(1-x-4x^2+3x^3+ 3x^4-x^5),{x,0,40}],x] (* or *) LinearRecurrence[{1,4,-3,-3,1},{5,1,9,4,25},40] (* Harvey P. Dale, Jan 18 2012 *)

Vec((5-4*x-12*x^2+6*x^3+3*x^4)/(1-x-4*x^2+3*x^3+3*x^4-x^5)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

G.f.: (5-4*x-12*x^2+6*x^3+3*x^4)/(1-x-4*x^2+3*x^3+3*x^4-x^5).
a(n)=a(n-1)+4*a(n-2)-3*a(n-3)-3*a(n-4)+a(n-5), {a(m)}={5,1,9,4,25}, m=0..4.
a(n)=Sum_{k=1..5} (x_k)^n; x_k=2*(-1)^(k-1)*cos(k*Pi/11).

A189235 Expansion of (5-16x+6x^2+10x^3-2x^4)/(1-4x+2x^2+5x^3-2x^4-x^5).

Original entry on oeis.org

5, 4, 12, 25, 64, 159, 411, 1068, 2808, 7423, 19717, 52529, 140251, 375015, 1003770, 2688570, 7204696, 19313075, 51782613, 138861732, 372414289, 998851473, 2679146955, 7186319506, 19276417059, 51707411684, 138702360471, 372064319188

Offset: 0

L. Edson Jeffery, Apr 18 2011

Same as A062883 preceded by 5.
Let U be the unit-primitive matrix (see [Jeffery])
U=U_(11,2)=
(0 0 1 0 0)
(0 1 0 1 0)
(1 0 1 0 1)
(0 1 0 1 1)
(0 0 1 1 1).
Then a(n)=Trace(U^n).
Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers of a unit-primitive matrix U_(N,r) (0Formulae given below are special cases of general one's defined and discussed in Witula-Slota's paper. For example a(n) = A(n;1), where A(n;d) := Sum_{k=1..5} (1 + 2d*cos(2Pi*k/11))^n, n=0,1,..., d in C. - Roman Witula, Jul 26 2012

R. Witula and D. Slota, Quasi-Fibonacci Numbers of Order 11, 10 (2007), J. Integer Seq., Article 07.8.5.

L. E. Jeffery, Unit-primitive matrices
Index entries for linear recurrences with constant coefficients, signature (4, -2, -5, 2, 1).

Cf. A189234, A189236, A189237.

u = {{0, 0, 1, 0, 0}, {0, 1, 0, 1, 0}, {1, 0, 1, 0, 1}, {0, 1, 0, 1, 1}, {0, 0, 1, 1, 1}}; a[n_] := Tr[ MatrixPower[u, n] ]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Oct 14 2013 *)

Vec((5-16*x+6*x^2+10*x^3-2*x^4)/(1-4*x+2*x^2+5*x^3-2*x^4-x^5)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

G.f.: (5-16*x+6*x^2+10*x^3-2*x^4)/(1-4*x+2*x^2+5*x^3-2*x^4-x^5).
a(n)=4*a(n-1)-2*a(n-2)-5*a(n-3)+2*a(n-4)+a(n-5), {a(m)}=5,4,12,25,64, m=0..4.
a(n)=Sum_{k=1..5} ((x_k)^2-1)^n; x_k=2*(-1)^(k-1)*cos(k*Pi/11).

A189236 Expansion of (5-8x-15x^2+4x^3+4x^4)/(1-2x-5x^2+2x^3+4x^4+x^5).

Original entry on oeis.org

5, 2, 14, 32, 114, 347, 1142, 3649, 11826, 38111, 123139, 397443, 1283406, 4143479, 13378435, 43194542, 139463234, 450284986, 1453839839, 4694021537, 15155624819, 48933074467, 157990585613, 510105367936, 1646980994190, 5317619734147

Offset: 0

L. Edson Jeffery, Apr 18 2011

(Start) Let U be the unit-primitive matrix (see [Jeffery])
U=U_(11,3)=
(0 0 0 1 0)
(0 0 1 0 1)
(0 1 0 1 1)
(1 0 1 1 1)
(0 1 1 1 1).
Then a(n)=Trace(U^n). (End)
Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers of a unit-primitive matrix U_(N,r) (0

L. E. Jeffery, Unit-primitive matrices
Index entries for linear recurrences with constant coefficients, signature (2, 5, -2, -4, -1).

Cf. A062883, A189234, A189235, A189237.

CoefficientList[Series[ (5-8x-15x^2+4x^3+4x^4)/ (1-2x-5x^2+2x^3+4x^4+x^5), {x,0,29}],x]  (* Harvey P. Dale, Apr 19 2011 *)
LinearRecurrence[{2, 5, -2, -4, -1}, {5, 2, 14, 32, 114}, 30] (* T. D. Noe, Apr 19 2011 *)

Vec((5-8*x-15*x^2+4*x^3+4*x^4)/(1-2*x-5*x^2+2*x^3+4*x^4+x^5)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

G.f.: (5-8*x-15*x^2+4*x^3+4*x^4)/(1-2*x-5*x^2+2*x^3+4*x^4+x^5).
a(n)=2*a(n-1)+5*a(n-2)-2*a(n-3)-4*a(n-4)-a(n-5), {a(m)}={5,2,14,32,114}, m=0..4.
a(n)=Sum_{k=1..5} ((x_k)^3-2*(x_k))^n; x_k=2*(-1)^(k-1)*cos(k*Pi/11).
Series expansion of g.f. at x=infinity gives -A062883 and all but the first term of -A189235.

A293312 Rectangular array read by antidiagonals: A(n,k) = tr((M_n)^k), k >= 0, where M_n is the n X n matrix M_1 = {{1}}, M_n = {{0,...,0,1},{0,...,0,1,1},...,{0,1,...,1},{1,...,1}}, n > 1, and tr(.) is the trace.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 3, 2, 4, 1, 4, 6, 2, 5, 1, 7, 11, 10, 3, 6, 1, 11, 26, 23, 15, 3, 7, 1, 18, 57, 70, 42, 21, 4, 8, 1, 29, 129, 197, 155, 69, 28, 4, 9, 1, 47, 289, 571, 533, 301, 106, 36, 5, 10, 1, 76, 650, 1640, 1884, 1223, 532, 154, 45, 5, 11

Offset: 1

L. Edson Jeffery, Oct 10 2017

Conjecture: For all n >= 1, for all k >= 2, A(n, k) = A293311(k, n); i.e., A(n, k) = number of magic labelings of the graph LOOP X C_k with magic sum n - 1.

			Array begins:
.   1 1  1   1    1     1      1       1       1        1         1
.   2 1  3   4    7    11     18      29      47       76       123
.   3 2  6  11   26    57    129     289     650     1460      3281
.   4 2 10  23   70   197    571    1640    4726    13604     39175
.   5 3 15  42  155   533   1884    6604   23219    81555    286555
.   6 3 21  69  301  1223   5103   21122   87677   363606   1508401
.   7 4 28 106  532  2494  11998   57271  274132  1310974   6271378
.   8 4 36 154  876  4654  25362  137155  743724  4029310  21836366
.   9 5 45 215 1365  8105  49347  298184 1806597 10936124  66220705
.  10 5 55 290 2035 13355  89848  599954 4016683 26868719 179784715
.  11 6 66 381 2926 21031 154935 1132942 8306078 60843972 445824731
.  ...

Cf. A293311.
Cf. A000012, A000032, A274975, A188128, A189237 (rows 1..5).
Cf. A000027, A000217, A019298, A006325, A244497, A244879, A244873, A244880, A293310, A293309 (columns k = 0,2..10 (conjectured)).

s[0, x_] := 1; s[1, x_] := x; s[k_, x_] := x*s[k - 1, x] - s[k - 2, x]; c[n_, j_] := 2 (-1)^(j - 1) Cos[j*Pi/(2 n + 1)]; a[n_, k_] := Round[Sum[s[n - 1, c[n, j]]^(k), {j, n}]];
(* Array: *)
Grid[Table[a[n, k], {n, 11}, {k, 0, 10}]]
(* Array antidiagonals flattened (gives this sequence): *)
Flatten[Table[a[n, k - n], {k, 11}, {n, k}]]

Let S(0, x) = 1, S(1, x) = x, S(k, x) = x*S(k - 1, x) - S(k - 2, x) (the S-polynomials of Wolfdieter Lang) and c(n, j) = 2*(-1)^(j - 1)*cos(j*Pi/(2*n + 1)). Then A(n, k) = Sum_{j=1..n} S(n - 1, c(n, j))^(k), n >= 1, k >= 0.

Showing 1-4 of 4 results.

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A189234 Expansion of (5-4*x-12*x^2+6*x^3+3*x^4)/(1-x-4*x^2+3*x^3+3*x^4-x^5).

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A189235 Expansion of (5-16*x+6*x^2+10*x^3-2*x^4)/(1-4*x+2*x^2+5*x^3-2*x^4-x^5).

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A189236 Expansion of (5-8*x-15*x^2+4*x^3+4*x^4)/(1-2*x-5*x^2+2*x^3+4*x^4+x^5).

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A293312 Rectangular array read by antidiagonals: A(n,k) = tr((M_n)^k), k >= 0, where M_n is the n X n matrix M_1 = {{1}}, M_n = {{0,...,0,1},{0,...,0,1,1},...,{0,1,...,1},{1,...,1}}, n > 1, and tr(.) is the trace.

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A189234 Expansion of (5-4x-12x^2+6x^3+3x^4)/(1-x-4x^2+3x^3+3*x^4-x^5).

A189235 Expansion of (5-16x+6x^2+10x^3-2x^4)/(1-4x+2x^2+5x^3-2x^4-x^5).

A189236 Expansion of (5-8x-15x^2+4x^3+4x^4)/(1-2x-5x^2+2x^3+4x^4+x^5).