A006053 -id:A006053 - cp's OEIS frontend

A188107 Triangle T(n,k) with the coefficient [x^k] of 1/(1 - x - 2*x^2 + x^3)^(n-k+1) in row n, column k.

1, 1, 1, 1, 2, 3, 1, 3, 7, 4, 1, 4, 12, 14, 9, 1, 5, 18, 31, 35, 14, 1, 6, 25, 56, 87, 70, 28, 1, 7, 33, 90, 175, 207, 154, 47, 1, 8, 42, 134, 310, 476, 504, 306, 89, 1, 9, 52, 189, 504, 941, 1274, 1137, 633, 155, 1, 10, 63, 256, 770, 1680, 2745, 3188, 2571

Offset: 0

L. Edson Jeffery, Mar 20 2011

Modified versions of the generating function for the diagonal, A006053, are related to rhombus substitution tilings (see A187065, A187066 and A187067).

			The triangle starts in row n=0 as
  1;
  1,   1;
  1,   2,   3;
  1,   3,   7,   4;
  1,   4,  12,  14,   9;
  1,   5,  18,  31,  35,  14;
  1,   6,  25,  56,  87,  70,  28;
  1,   7,  33,  90, 175, 207, 154,  47;
  1,   8,  42, 134, 310, 476, 504, 306,  89;

Nathaniel Johnston, Rows n = 0..100, flattened

Cf. A001654, A187065, A187066, A187067.

Cf. Columns: A000012, A000027, A055998.

A188107 := proc(n,k) 1/(1-x-2*x^2+x^3)^(n-k+1) ; coeftayl(%,x=0,k) ; end proc:
seq(seq(A188107(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Mar 22 2011

Sum_{k=0..n} T(n,k) = A001654(n+1).

T(n,k) = T(n-1,k) + T(n-1,k-1) + 2*T(n-2,k-2) - T(n-3,k-3). - Philippe Deléham, Feb 24 2012

A099918 A Chebyshev transform related to the 7th cyclotomic polynomial.

Original entry on oeis.org

1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1

Offset: 0

Paul Barry, Oct 30 2004

The g.f. is a Chebyshev transform of 1/(1+x-2x^2-x^3) under the Chebyshev mapping g(x)->(1/(1+x^2))g(x/(1+x^2)). The denominator is the 7th cyclotomic polynomial. The inverse of the 7 cyclotomic polynomial A014016 is given by sum{k=0..n, A099918(n-k)(k/2+1)(-1)^(k/2)(1+(-1)^k)/2}.

Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1,-1,-1).

Cf. A099860.

LinearRecurrence[{-1,-1,-1,-1,-1,-1},{1,-1,2,-2,1,-1},90] (* Harvey P. Dale, May 25 2019 *)

G.f.: (1+x^2)^2/(1+x+x^2+x^3+x^4+x^5+x^6).

a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)^(-1)^k*b(n-2k), where b(n)=A094790(n/2+1)(1+(-1)^n)/2+A094789((n+1)/2)(1-(-1)^n)/2=(-1)^n*A006053(n+2).

A122161 Expansion of x(1 - 3x + x^2) / (1 - x - 2*x^2 + x^3).

Original entry on oeis.org

1, -2, 1, -4, 0, -9, -5, -23, -24, -65, -90, -196, -311, -613, -1039, -1954, -3419, -6288, -11172, -20329, -36385, -65871, -118312, -213669, -384422, -693448, -1248623, -2251097, -4054895, -7308466, -13167159, -23729196, -42755048, -77046281, -138827181, -250164695, -450772776

Offset: 1

Gary W. Adamson and Roger L. Bagula, Oct 17 2006

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Index entries for linear recurrences with constant coefficients, signature (1,2,-1).

M = {{0, -1, -1}, {-1, 0, 0}, {-1, 0, 1}}; v[1] = {1, 1, 1}; v[n_] := v[n] = M.v[n - 1]; a1 = Table[v[n][[1]], {n, 1, 50}]

Vec( x*(1 - 3*x + x^2) / (1 - x - 2*x^2 + x^3) + O(x^50)) \\ Michel Marcus, Sep 19 2017

a(n) = A006053(n+1) - 3*A006053(n) + A006053(n-1). - R. J. Mathar, Nov 07 2011

A215139 a(n) = (a(n-1) - a(n-3))*7^((1+(-1)^n)/2) with a(6)=5, a(7)=4, a(8)=22.

Original entry on oeis.org

5, 4, 22, 17, 91, 69, 364, 273, 1428, 1064, 5537, 4109, 21315, 15778, 81683, 60368, 312130, 230447, 1190553, 878423, 4535832, 3345279, 17267992, 12732160, 65708167, 48440175, 249956105, 184247938, 950654341, 700698236, 3615152086, 2664497745, 13746596563, 10131444477

Offset: 6

Roman Witula, Aug 04 2012

The Ramanujan-type sequence the number 9 for the argument 2*Pi/7. The sequence is connecting with the following decomposition: (s(4)/s(1))^(1/3)*s(1)^n + (s(1)/s(2))^(1/3)*s(2)^n + (s(2)/s(4))^(1/3)*s(4)^n = x(n)*(4-3*7^(1/3))^(1/3) + y(n)*(11-3*49^(1/3))^(1/3), where s(j) := sin(2*Pi*j/7), x(0)=1, x(1)=-7^(1/6)/2, x(2)=y(0)=y(1)=0, y(2)=7^(1/3)/4 and X(n)=sqrt(7)*(X(n-1)-X(n-3)) for every n=3,4,..., and X=x or X=y. It could be deduced the formula 4*y(n) = a(n)*7^(1/3 + (3+(-1)^n)/4), which implies a(0)=0, a(1)= 0, a(2)= 1/7, a(3)=1/7, a(4)=1, a(5)=6/7, i.e., A163260(n)=7*a(n) for every n=0,1,...,5. The sequence a(n) is discussed in third Witula paper.

			From values of x(2),y(2) and the identity 2*sin(t)^2=1-cos(2*t) we obtain (s(4)/s(1))^(1/3)*c(1) + (s(1)/s(2))^(1/3)*c(4) + (s(2)/s(4))^(1/3)*c(1) = (4-3*7^(1/3))^(1/3) - (1/2)*(7*(11-3*49^(1/3)))^(1/3), where c(j):=cos(2*Pi*j/7). Further, from values of x(1),x(3),y(1),y(3) and the identity 4*sin(t)^3=3*sin(t)-sin(3*t) we obtain (s(4)/s(1))^(1/3)*s(4) + (s(1)/s(2))^(1/3)*s(1) + (s(2)/s(4))^(1/3)*s(2) = (-3*7^(1/6)/2 +4*7^(1/2))*(4-3*7^(1/3))^(1/3) - 7^(5/6)*(11-3*49^(1/3))^(1/3).

R. Witula, E. Hetmaniok and D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, Proceedings of the Fifteenth International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.

G. C. Greubel, Table of n, a(n) for n = 6..1005
Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6.
Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5.
Roman Witula, Full Description of Ramanujan Cubic Polynomials, Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.7.
Roman Witula, Ramanujan Cubic Polynomials of the Second Kind, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.5.
Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796.
Index entries for linear recurrences with constant coefficients, signature (0,7,0,-14,0,7).

Cf. A214683, A215112, A006053, A006054, A215076, A215100, A120757, A215560, A215569, A215572, A214699.

I:=[5,4,22,17,91,69]; [n le 6 select I[n] else 7*Self(n-2) - 14*Self(n-4) + 7*Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 19 2018

LinearRecurrence[{0,7,0,-14,0,7}, {5,4,22,17,91,69}, {1,50}] (* G. C. Greubel, Apr 19 2018 *)

Vec(-x*(1+x)*(6*x^4+x^3-12*x^2-x+5)/(-1+7*x^2-14*x^4+7*x^6) + O(x^50)) \\ Michel Marcus, Apr 20 2016

G.f.: -x*(1+x)*(6*x^4+x^3-12*x^2-x+5) / ( -1+7*x^2-14*x^4+7*x^6 ). - R. J. Mathar, Sep 14 2012

More terms from Michel Marcus, Apr 20 2016

A265755 a(n) = a(n-1) + a(n-2) if n is even and a(n) = a(n-3) + a(n-4) if n is odd, with a(0) = a(1) = a(2) = 0 and a(3) = 1.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 1, 2, 3, 1, 4, 5, 9, 5, 14, 14, 28, 19, 47, 42, 89, 66, 155, 131, 286, 221, 507, 417, 924, 728, 1652, 1341, 2993, 2380, 5373, 4334, 9707, 7753, 17460, 14041, 31501, 25213, 56714, 45542, 102256, 81927, 184183, 147798, 331981, 266110, 598091, 479779, 1077870, 864201, 1942071, 1557649

Offset: 0

Nicholas Drozd, Dec 15 2015

			a(8) = a(7) + a(6)
     = a(4) + a(3) + a(5) + a(4)
     = (a(3) + a(2)) + a(3) + (a(2) + a(1)) + (a(3) + a(2))
     = 1 + 1 + 0 + 1
     = 3

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,2,0,-1).

Interleaves A006053 and A052547, with an extra leading 0.

A187066 with even values and odd values swapped and an extra leading 0.

a[0] = a[1] = a[2] = 0; a[3] = 1; a[n_] := a[n] = If[EvenQ@ n, a[n - 1] + a[n - 2], a[n - 3] + a[n - 4]]; Table[a@ n, {n, 0, 55}] (* Michael De Vlieger, Dec 15 2015 *)
nxt[{n_,a_,b_,c_,d_}]:={n+1,b,c,d,If[OddQ[n],c+d,a+b]}; NestList[nxt,{1,0,0,0,1},60][[All,2]] (* or *) LinearRecurrence[{0,1,0,2,0,-1},{0,0,0,1,1,0},60] (* Harvey P. Dale, Nov 10 2017 *)

concat(vector(3), Vec(x^3*(1+x-x^2)/(1-x^2-2*x^4+x^6) + O(x^70))) \\ Colin Barker, Dec 16 2015

From Colin Barker, Dec 16 2015: (Start)
a(n) = a(n-2) + 2*a(n-4) - a(n-6) for n>5.
G.f.: x^3*(1+x-x^2) / (1-x^2-2*x^4+x^6).
(End)

More terms from Michael De Vlieger, Dec 15 2015

A122500 Semiprimes in A006054.

Original entry on oeis.org

25, 183922, 413269, 2086561, 268550439, 77664004259, 1286498670037058, 835992504648428541, 69918194872608910320341, 157104757807495406675035, 353011186440414668176111, 13139449230370726052061293111

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 15 2006

nonn

Cf. A006054, A006053.

semiPrimeQ[n_Integer] := Plus @@ Last /@ FactorInteger[n] == 2; s = CoefficientList[ Series[x/(x^3 - x^2 - 2*x + 1), {x, 0, 100}], x]; Select[s, SemiprimeQ@# &]
Select[LinearRecurrence[{2,1,-1},{0,0,1},100],PrimeOmega[#]==2&] (* Harvey P. Dale, Aug 21 2013 *)

Edited by N. J. A. Sloane, Sep 17 2006

More terms from Robert G. Wilson v, Sep 18 2006

A122504 a(n) = -a(n-6) + 3a(n-5) + a(n-4) - 7a(n-3) + a(n-2) + 3*a(n-1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, -3, -13, -39, -107, -273, -675, -1624, -3847, -8995, -20851, -47995, -109915, -250695, -570024, -1292915, -2926953, -6616051, -14936895, -33690357, -75931283, -171029936, -385046687, -866536007, -1949510615, -4384874471, -9860587191, -22170707871, -49842661456

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 15 2006

sign

Original name started "Bi_Steinbach heptagon recursion".

Index entries for linear recurrences with constant coefficients, signature (3, 1, -7, 1, 3, -1).

Cf. A006054, A006053.

a[0] = 1; a[1] = 1; a[2] = 1; a[3] = 1; a[4] = 1; a[5] = 1; a[n_] := a[n] = -a[n - 6] + 3 a[n - 5] + a[n - 4] - 7 a[n - 3] + a[n - 2] + 3 a[n - 1] Table[a[n], {n, 0, 30}]
M = {{0, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1}, {-1, 3, 1, -7, 1, 3}} v[1] = {1, 1, 1, 1, 1, 1} v[n_] := v[n] = M.v[n - 1] a = Table[Floor[v[n][[1]]], {n, 1, 50}]
LinearRecurrence[{3,1,-7,1,3,-1},{1,1,1,1,1,1},40] (* or *) Rest[ CoefficientList[ Series[x(1-x-3x^2)(1-x-x^2)/((1-2x-x^2+x^3)(1-x-2x^2+x^3)),{x,0,40}],x]] (* Harvey P. Dale, Jun 24 2011 *)

O.g.f.: x*(1-x-3*x^2)*(1-x-x^2)/((1-2*x-x^2+x^3)*(1-x-2*x^2+x^3)). - R. J. Mathar, Aug 22 2008

A122517 G.f.: 1/(1 - x^3 - 2 x^4 + x^5).

Original entry on oeis.org

1, 0, 0, 1, 2, -1, 1, 4, 2, -3, 7, 9, -3, -1, 26, 8, -16, 27, 61, -26, -13, 131, 69, -126, 131, 344, -119, -190, 732, 438, -772, 471, 2092, -628, -1511, 3806, 3085, -4859, 1412, 12208, -2495, -11391, 19891, 20509, -28589, -396, 71682, -7462, -78083, 99479

Offset: 0

Roger L. Bagula, Sep 16 2006

sign

Based on characteristic polynomial x^5 - x^2 - 2x + 1.

Cf. A006054, A006053.

p[x_] := x^5 - x^2 - 2x + 1 q[x_] := ExpandAll[x^5*p[1/x]] Table[ SeriesCoefficient[ Series[x/q[x], {x, 0, 50}], n], {n, 0, 50}]

Edited by N. J. A. Sloane, Oct 01 2006

A122518 G.f.: 1/(1 -2 x^3 - x^4 + x^5).

Original entry on oeis.org

1, 0, 0, 2, 1, -1, 4, 4, -3, 6, 13, -6, 5, 35, -5, -9, 81, 20, -58, 158, 130, -177, 238, 476, -382, 169, 1367, -526, -520, 3285, 146, -2933, 6576, 4097, -9005, 10073, 17703, -20489, 7044, 54484, -33348, -24104, 136501, -19256, -136040, 282246, 122093, -427837, 447708, 662472

Offset: 0

Roger L. Bagula, Sep 16 2006

Based on characteristic polynomial x^5 - 2x^2 - x + 1.

Index entries for linear recurrences with constant coefficients, signature (0,0,2,1,-1).

Cf. A006054, A006053.

p[x_] := x^5 - 2x^2 - x + 1 q[x_] := ExpandAll[x^5*p[1/x]] Table[ SeriesCoefficient[ Series[x/q[x], {x, 0, 50}], n], {n, 0, 50}]

Edited by N. J. A. Sloane, Oct 01 2006

A218664 Coefficients of cubic polynomials p(x+n), where p(x) = x^3 + x^2 - 2*x - 1.

Original entry on oeis.org

1, 1, -2, -1, 1, 4, 3, -1, 1, 7, 14, 7, 1, 10, 31, 29, 1, 13, 54, 71, 1, 16, 83, 139, 1, 19, 118, 239, 1, 22, 159, 377, 1, 25, 206, 559, 1, 28, 259, 791, 1, 31, 318, 1079, 1, 34, 383, 1429, 1, 37, 454, 1847, 1, 40, 531, 2339, 1, 43, 614, 2911, 1, 46, 703, 3569, 1, 49, 798, 4319

Offset: 0

Roman Witula, Nov 04 2012

sign

We have p(x) = (x - c(1))*(x - c(2))*(x - c(4)), where c(j) := 2*cos(2*Pi*j/7). We note that c(4) = c(3) = -c(1/2), c(1) = s(3) and c(2) = -s(1), where s(j) := 2*sin(Pi*j/14). Moreover we obtain -p(-x) = x^3 - x^2 - 2*x + 1 = (x + c(1))*(x + c(2))*(x + c(4)), q(x) := -x^3*p(1/x) = x^3 + 2*x^2 + x - 1 = (x - c(1)^(-1))*(x - c(2)^(-1))*(x - c(4)^(-1)), and -q(-x) = x^3 - 2*x^2 + x + 1 = (x + c(1)^(-1))*(x + c(2)^(-1))*(x + c(4)^(-1)).
We also have p(x+2) =  x^3 + 7*x^2 + 14*x + 7 = (x + s(2)^2)*(x + s(4)^2)*(x + s(6)^2). The polynomial -p(-x-2) = x^3 - 7*x^2 + 14*x - 7 = (x - s(2)^2)*(x - s(4)^2)*(x - s(6)^2) is known as Johannes Kepler's cubic polynomial (see Witula's book).
Let us set r(x) := p(x+1). It can be verified that -x^3*r(1/x) = x^3 - 3*x^2 - 4*x - 1 = (x - c(1)/c(4))*(x - c(4)/c(2))*(x - c(2)/c(1)); for example, we have c(1)^3 + c(1)^2 - 2*c(1) - 1 = 0 which implies that c(1)^2 + 2*c(1) = 1/(c(1) - 1), and then c(1)^2 + 2*c(1) = c(4)/c(2) since c(4)/c(2) = (c(1)^4 - 4*c(1)^2 + 2)/(c(1)^2 - 2).
The polynomials p(x+n) and the ones obtained as above (i.e., after simple algebraic transformations) are the characteristic polynomials of many sequences in the OEIS; see crossrefs.

R. Witula, Complex Numbers, Polynomials and Partial Fraction Decomposition, Part 3, Wydawnictwo Politechniki Slaskiej, Gliwice 2010 (Silesian Technical University publishers).

Cf. A094648, A096975, A033304, A214683, A006053, A215076, A215100, A215007, A215008, A215143, A215493, A215494, A215510.

We have a(4*k) = 1, a(4*k + 1) = 3*k + 1, a(4*k + 2) = 3*k^2 + 2*k - 2, a(4*k + 3) = k^3 + k^2 - 2*k - 1. Further, the following relations hold true: b(k+1) = b(k) + 3, c(k+1) = 2*b(k) -2*c(k) + 3, d(k+1) = b(k) - 2*c(k) - d(k) + 1, where p(x + k) = x^3 + b(k)*x^2 + c(k)*x + d(k).

Empirical g.f.: -(x^15 - x^14 - 2*x^13 + x^12 - 5*x^11 + 10*x^10 + 3*x^9 - 3*x^8 - 3*x^7 - 11*x^6 + 3*x^4 + x^3 + 2*x^2 - x - 1) / ((x-1)^4*(x+1)^4*(x^2+1)^4). - Colin Barker, May 17 2013

cp's OEIS Frontend

A188107 Triangle T(n,k) with the coefficient [x^k] of 1/(1 - x - 2*x^2 + x^3)^(n-k+1) in row n, column k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A099918 A Chebyshev transform related to the 7th cyclotomic polynomial.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A122161 Expansion of x*(1 - 3*x + x^2) / (1 - x - 2*x^2 + x^3).

Views

Author

Keywords

Links

Programs

Mathematica

PARI

Formula

A215139 a(n) = (a(n-1) - a(n-3))*7^((1+(-1)^n)/2) with a(6)=5, a(7)=4, a(8)=22.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A265755 a(n) = a(n-1) + a(n-2) if n is even and a(n) = a(n-3) + a(n-4) if n is odd, with a(0) = a(1) = a(2) = 0 and a(3) = 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A122500 Semiprimes in A006054.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Extensions

A122504 a(n) = -a(n-6) + 3*a(n-5) + a(n-4) - 7*a(n-3) + a(n-2) + 3*a(n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A122517 G.f.: 1/(1 - x^3 - 2 x^4 + x^5).

Views

Author

Keywords

A122161 Expansion of x(1 - 3x + x^2) / (1 - x - 2*x^2 + x^3).

A122504 a(n) = -a(n-6) + 3a(n-5) + a(n-4) - 7a(n-3) + a(n-2) + 3*a(n-1).