A066170 -id:A066170 - cp's OEIS frontend

A124039 Triangle read by rows: T(n, k) = (-1)^floor((n+k+2)/2)(2 - (-1)^(n+k))A046854(n-1, k-1) with T(1, 1) = 3.

3, 3, -1, -1, -3, 1, -3, 2, 3, -1, 1, 6, -3, -3, 1, 3, -3, -9, 4, 3, -1, -1, -9, 6, 12, -5, -3, 1, -3, 4, 18, -10, -15, 6, 3, -1, 1, 12, -10, -30, 15, 18, -7, -3, 1, 3, -5, -30, 20, 45, -21, -21, 8, 3, -1, -1, -15, 15, 60, -35, -63, 28, 24, -9, -3, 1

Offset: 1

Roger L. Bagula and Gary W. Adamson, Nov 03 2006

			Triangle begins as:
   3;
   3,  -1;
  -1,  -3,   1;
  -3,   2,   3,  -1;
   1,   6,  -3,  -3,   1;
   3,  -3,  -9,   4,   3,  -1;
  -1,  -9,   6,  12,  -5,  -3,   1;
  -3,   4,  18, -10, -15,   6,   3, -1;
   1,  12, -10, -30,  15,  18,  -7, -3,  1;
   3,  -5, -30,  20,  45, -21, -21,  8,  3, -1;
  -1, -15,  15,  60, -35, -63,  28, 24, -9, -3,  1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A046854, A066170.

Columns include: (-1)^n*A112030(n-1) (k=1), (-1)^floor((n+1)/2)*A064455(n) (k=2).

A124039:= func< n,k | (-1)^Floor((n+k+2)/2)*(2-(-1)^(n+k))*Binomial(Floor((n+k-2)/2), k-1) + 2*0^(n-1) >;
[A124039(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jan 30 2025

(* First program *)
f[n_, m_, d_]:= If[n==m && n>1 && m>1, 0, If[n==m-1 || n==m+1, -1, If[n==m== 1, 3, 0]]];
M[d_]:= Table[T[n,m,d], {n,d}, {m,d}];
A124039[n_]:= Join[{M[1]}, CoefficientList[Det[M[n] - x*IdentityMatrix[n]], x]];
Table[A124039[n], {n,12}]//Flatten
(* Second program *)
A124039[n_, k_]:= (-1)^Floor[(n+k+2)/2]*(2-(-1)^(n-k))*Binomial[Floor[(n+k- 2)/2], k-1] +2*Boole[n==1];
Table[T[n,k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jan 30 2025 *)

@CachedFunction
def t(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = 3*t(n-1,k) if n==1 else 0
    return t(n-1,k-1) - t(n-2,k) - h
def A124039(n,k): return t(n,k) + 2*0^n
print([[A124039(n,k) for k in range(n+1)] for n in range(13)]) # Peter Luschny, Nov 20 2012

def A124039(n,k): return (-1)^((n+k+2)//2)*(2-(-1)^(n+k))*binomial((n+k-2)//2, k-1) + 2*0^(n-1)
print(flatten([[A124039(n,k) for k in range(1,n+1)] for n in range(1,13)])) # G. C. Greubel, Jan 30 2025

T(n, k) = (-1)^floor((n+k+2)/2)*(2 - (-1)^(n+k))*A046854(n-1,k-1) + 2*[n=1]. - G. C. Greubel, Jan 30 2025

Edited by G. C. Greubel, Jan 30 2025

A136571 Irregular triangle of coefficients of the minimal polynomial of 2cos(2Pi/n) in decreasing powers.

Original entry on oeis.org

1, -2, 1, 2, 1, 1, 1, 0, 1, 1, -1, 1, -1, 1, 1, -2, -1, 1, 0, -2, 1, 0, -3, 1, 1, -1, -1, 1, 1, -4, -3, 3, 1, 1, 0, -3, 1, 1, -5, -4, 6, 3, -1, 1, -1, -2, 1, 1, -1, -4, 4, 1, 1, 0, -4, 0, 2, 1, 1, -7, -6, 15, 10, -10, -4, 1, 1, 0, -3, -1, 1, 1, -8, -7, 21

Offset: 1

T. D. Noe, Jan 07 2008

The degree of the n-th polynomial is A023022(n), the half-totient function for n>2. These polynomials are integral, monic and irreducible over the integers. Hence 2*cos(2*Pi/n) is an algebraic integer. When n is prime, the n-th row is the same as the n-th row of A066170. Carlitz and Thomas give an algorithm for computing these polynomials.

			x-2, x+2, x+1, x, x^2+x-1, x-1, x^3+x^2-2x-1, x^2-2, x^3-3x+1, x^2-x-1

T. D. Noe, Rows n=1..100 of triangle, flattened
Scott Beslin and Valerio de Angelis, The minimal polynomials of sin(2 pi/p) and cos(2 pi/n), Math. Mag., 77 (2004), 146-149.
L. Carlitz and J. M. Thomas, Rational tabulated values of trigonometric functions, Amer. Math. Monthly, 69 (1962), 789-793.
G. P. Dresden, On the middle coefficient of a cyclotomic polynomial, Amer. Math. Monthly, 111 (No. 6, 2004), 531-533.
D. H. Lehmer, A note on trigonometric algebraic numbers, Amer. Math. Monthly, 40 (1933), 165-166.
William Watkins and Joel Zeitlin, The minimal polynomial of cos(2 pi/n), Amer. Math. Monthly, 100 (1993), 471-474.
Eric Weisstein's World of Mathematics, Trigonometry Angles

Flatten[Table[Reverse[CoefficientList[MinimalPolynomial[2Cos[2Pi/n],x],x]], {n,25}]]

A091650 Let M = the 4 X 4 matrix [0 1 0 0 / 0 0 1 0 / 0 0 0 1 / -1 -1 3 2]. Set seed vector = [1 1 1 1] = first row, then take M[1 1 1 1] = [1 1 1 3] then M [1 1 1 3], etc. Sequence gives terms in rightmost column.

Original entry on oeis.org

1, 3, 7, 21, 59, 171, 491, 1415, 4073, 11729, 33771, 97241, 279993, 806209, 2321385, 6684163, 19246279, 55417453, 159568195, 459458307, 1322957467, 3809304207, 10968454313, 31582405473, 90937912211, 261845282321, 753953441489, 2170922412257, 6250921954449

Offset: 1

Gary W. Adamson, Jan 25 2004

a(n)/a(n-1) tends to a 9-Gon diagonal.
The other 3 columns are offsets of 1, 3, 7, 21, 59, ... starting with 1's.
The characteristic equation of the 4 X 4 matrix is x^4 - 2x^3 - 3x^4 + x + 1 (coefficients may be found in A066170) with roots 2.879385241..., -1, -.5320888862... and .65270364466... An alternative matrix giving the same eigenvalues (refer to A046854) relates to the 9-Gon: [1 1 1 1 / 1 1 1 0 / 1 1 0 0 / 1 0 0 0] since the eigenvalue 2.8793852...is the longest diagonal of the 9-Gon given edge = 1. Or, 2.879385... = 1/(2*cos(k*Pi/9)), k = 4.

			a(5) = 59 since M*[1 1 1 1] then 4 iterates = [3 7 21 59]. a(5) = rightmost term.
a(10)/a(9) = 11729/4073 = 2.8796955...

Index entries for linear recurrences with constant coefficients, signature (2,3,-1,-1).

Cf. A066170, A046854.

Rest[CoefficientList[Series[x (1+x-2x^2-x^3)/(1-2x-3x^2+x^3+x^4),{x,0,40}],x]] (* or *) LinearRecurrence[{2,3,-1,-1},{1,3,7,21},40] (* Harvey P. Dale, Feb 17 2012 *)

Vec((1+x-2*x^2-x^3)/(1-2*x-3*x^2+x^3+x^4)+O(x^99)) \\ Charles R Greathouse IV, Jan 31 2012

G.f.: x*(1+x-2*x^2-x^3)/(1-2*x-3*x^2+x^3+x^4). - Colin Barker, Jan 31 2012

a(1)=1, a(2)=3, a(3)=7, a(4)=21, a(n)=2*a(n-1)+3*a(n-2)-a(n-3)-a(n-4). - Harvey P. Dale, Feb 17 2012

More terms from Harvey P. Dale, Feb 17 2012

A122602 a(1) = 1; a(2) = 0; a(3) = 0; a(4) = 0; a(5) = 0; a(6) = 0; a(7) = 0; a(8) = 0; a(9) = 0; a(10) = 0; a(n) = a(n - 1) + 9a(n - 2) - 8a(n - 3) - 28a(n - 4) + 21a(n - 5) + 35a(n - 6) - 20a(n - 7) - 15a(n - 8) + 5a(n - 9) + a(n - 10) for n >= 11.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 10, 11, 65, 77, 350, 440, 1700, 2244, 7752, 10659, 33915, 48279, 144210, 211508, 600875, 904475, 2466750, 3798795, 10015005, 15737864, 40320149, 64512209, 161280568, 262255753, 641885440, 1059105390, 2544612396

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 20 2006

nonn

Index entries for linear recurrences with constant coefficients, signature (1, 9, -8, -28, 21, 35, -20, -15, 5, 1).

Cf. A066170.

a[1]:=1: a[2]:=0: a[3]:=0: a[4]:=0: a[5]:=0: a[6]:=0: a[7]:=0: a[8]:=0: a[9]:=0: a[10]:=0: for n from 11 to 39 do a[n]:=a[n-1]+9*a[n-2]-8*a[n-3]-28*a[n-4]+21*a[n-5]+35*a[n-6]-20*a[n-7]-15*a[n-8]+5*a[n-9]+a[n-10] od: seq(a[n],n=1..39);

M = {{0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 5, -15, -20, 35, 21, -28, -8, 9, 1}}; v[1] = {1, 0, 0, 0, 0, 0, 0, 0, 0, 0}; v[n_] := v[n] = M.v[n - 1] a = Table[Floor[v[n][[1]]], {n, 1, 50}]
LinearRecurrence[{1,9,-8,-28,21,35,-20,-15,5,1},{1,0,0,0,0,0,0,0,0,0},50] (* Harvey P. Dale, Dec 03 2014 *)

G.f.:((5*x^4-5*x^2+1)*(x^5-3*x^4-3*x^3+4*x^2+x-1))/((x-1)*(x^3-2*x^2-x+1)*(x^6+8*x^5+8*x^4-6*x^3-6*x^2+x+1)). [Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009]

Edited by N. J. A. Sloane, Oct 08 2006

A122605 Expansion of -x(2x - 1)(2x^2 - 1)(x^3 + 2x^2 - x - 1)/((x - 1)(x^2 + x - 1)(x^4 - 4x^3 - 4x^2 + x + 1)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, -1, -1, -7, -8, -35, -44, -154, -208, -637, -910, -2548, -3808, -9996, -15504, -38760, -62015, -149225, -245135, -572010, -961125, -2186886, -3746886, -8348172, -14547183, -31842580, -56309764, -121415344, -217478888, -462925232, -838520240, -1765205473, -3228800413

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 20 2006

Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. Vol. 70, No. 1, Feb. 1997, 22-31.
Index entries for linear recurrences with constant coefficients, signature (1,6,-5,-10,6,4,-1).

Cf. A066170.

LinearRecurrence[{1,6,-5,-10,6,4,-1},{1,0,0,0,0,0,0},60] (* Harvey P. Dale, May 02 2011 *)

a(0)=1, a(1)=a(2)=a(3)=a(4)=a(5)=a(6)=0; for n>6, a(n) = a(n-1) + 6*a(n-2) - 5*a(n-3) - 10*a(n-4) + 6*a(n-5) + 4*a(n-6) - a(n-7). - Harvey P. Dale, May 02 2011

G.f.: -x*(2*x-1)*(2*x^2-1)*(x^3+2*x^2-x-1)/((x-1)*(x^2+x-1)*(x^4-4*x^3-4*x^2+x+1)). - Colin Barker, Nov 08 2012

Edited by N. J. A. Sloane, Feb 11 2007

Definition changed using Barker's g.f. by Bruno Berselli, Sep 19 2017

A122607 Expansion of x(8x^5 + 5x^4 - x^3 - 5x^2 - 1)/(x^6 + 3x^5 + 6x^4 + 4x^3 - 5x^2 + x - 1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 10, 19, -17, -62, 163, 550, -548, -3050, 2665, 19450, -7550, -113534, 8308, 667423, 187462, -3800825, -2366747, 21303154, 21068938, -116488961, -162036530, 621601885, 1153785034, -3216794309, -7799929064, 16026195376, 50784142789, -75764359214, -320876463932

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 20 2006

Obtained as the top element of the vector resulting from multiplying the n-th power of the 6 X 6 matrix [[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, 6, 4, -5, 1]] with the column vector which contains only 1's.

Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. Vol. 70, No. 1, Feb. 1997, 22-31.
Index entries for linear recurrences with constant coefficients, signature (1,-5,4,6,3,1).

Cf. A066170.

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, 6, 4, -5, 1}}; 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[{1,-5,4,6,3,1},{1,1,1,1,1,1},40] (* Harvey P. Dale, Feb 17 2024 *)

G.f.: x*(8*x^5+5*x^4-x^3-5*x^2-1)/(x^6+3*x^5+6*x^4+4*x^3-5*x^2+x-1). - Colin Barker, Nov 08 2012

Edited by N. J. A. Sloane, Sep 24 2006

Definition changed using Barker's g.f. by Bruno Berselli, Sep 19 2017

A122608 a(1) = 1; a(2) = 1; a(3) = 1; a(4) = 1; a(5) = 1; a(n) = a(n-1)+4a(n-2)-3a(n-3)-3a(n-4)+a(n-5) for n >= 6.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, -1, -6, -12, -32, -59, -134, -244, -519, -948, -1949, -3586, -7225, -13397, -26640, -49744, -98024, -184114, -360455, -680247, -1325397, -2510702, -4874298, -9260629, -17929771, -34142684, -65967689, -125841523, -242755543, -463720287, -893457507, -1708515146

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 20 2006

sign

Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. Vol. 70, No. 1, Feb. 1997, 22-31.

Cf. A066170.

a[1]:=1: a[2]:=1: a[3]:=1: a[4]:=1: a[5]:=1: for n from 6 to 37 do a[n]:=a[n-1]+4*a[n-2]-3*a[n-3]-3*a[n-4]+a[n-5] od: seq(a[n],n=1..37);

M = {{0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}, {1, -3, -3, 4, 1}}; v[1] = {1, 1, 1, 1, 1}; v[n_] := v[n] = M.v[n - 1] a = Table[Floor[v[n][[1]]], {n, 1, 50}]
nxt[{a_,b_,c_,d_,e_}]:={b,c,d,e,e+4d-3c-3b+a}; NestList[nxt,{1,1,1,1,1},50][[;;,1]] (* Harvey P. Dale, Aug 02 2024 *)

G.f.: -(2*x-1)*(x+1)*(x^2-x-1)/(-1+x^5-3*x^4-3*x^3+4*x^2+x). [Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009]

Edited by N. J. A. Sloane, Oct 08 2006

A122611 G.f.: 1/(1 - 7 x + 15 x^2 - 6 x^3 - 11 x^4 + 6 x^5 + x^6).

Original entry on oeis.org

1, 7, 34, 139, 516, 1802, 6039, 19657, 62634, 196404, 608361, 1866462, 5683236, 17200871, 51807242, 155421557, 464755958, 1386055506, 4124569714, 12251283960, 36334754000, 107624090145, 318444202635, 941387240040

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 20 2006

Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. Vol. 70, No. 1, Feb. 1997, 22-31.
Index entries for linear recurrences with constant coefficients, signature (7,-15,6,11,-6,-1).

Cf. A066170.

b = {1, -7, 15, -6, -11, 6, 1}; p[x_] := Sum[x^(n - 1)*b[[8 - n]], {n, 1, 7}] q[x_] := ExpandAll[x^6*p[1/x]] Table[ SeriesCoefficient[ Series[x/q[x], {x, 0, 30}], n], {n, 0, 30}]

Edited by N. J. A. Sloane, Sep 21 2006

cp's OEIS Frontend

A124039 Triangle read by rows: T(n, k) = (-1)^floor((n+k+2)/2)*(2 - (-1)^(n+k))*A046854(n-1, k-1) with T(1, 1) = 3.

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A136571 Irregular triangle of coefficients of the minimal polynomial of 2*cos(2*Pi/n) in decreasing powers.

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A091650 Let M = the 4 X 4 matrix [0 1 0 0 / 0 0 1 0 / 0 0 0 1 / -1 -1 3 2]. Set seed vector = [1 1 1 1] = first row, then take M*[1 1 1 1] = [1 1 1 3] then M * [1 1 1 3], etc. Sequence gives terms in rightmost column.

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A122602 a(1) = 1; a(2) = 0; a(3) = 0; a(4) = 0; a(5) = 0; a(6) = 0; a(7) = 0; a(8) = 0; a(9) = 0; a(10) = 0; a(n) = a(n - 1) + 9a(n - 2) - 8a(n - 3) - 28a(n - 4) + 21a(n - 5) + 35a(n - 6) - 20a(n - 7) - 15a(n - 8) + 5a(n - 9) + a(n - 10) for n >= 11.

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A122605 Expansion of -x*(2*x - 1)*(2*x^2 - 1)*(x^3 + 2*x^2 - x - 1)/((x - 1)*(x^2 + x - 1)*(x^4 - 4*x^3 - 4*x^2 + x + 1)).

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A122607 Expansion of x*(8*x^5 + 5*x^4 - x^3 - 5*x^2 - 1)/(x^6 + 3*x^5 + 6*x^4 + 4*x^3 - 5*x^2 + x - 1).

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A122608 a(1) = 1; a(2) = 1; a(3) = 1; a(4) = 1; a(5) = 1; a(n) = a(n-1)+4a(n-2)-3a(n-3)-3a(n-4)+a(n-5) for n >= 6.

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A124039 Triangle read by rows: T(n, k) = (-1)^floor((n+k+2)/2)(2 - (-1)^(n+k))A046854(n-1, k-1) with T(1, 1) = 3.

A136571 Irregular triangle of coefficients of the minimal polynomial of 2cos(2Pi/n) in decreasing powers.

A091650 Let M = the 4 X 4 matrix [0 1 0 0 / 0 0 1 0 / 0 0 0 1 / -1 -1 3 2]. Set seed vector = [1 1 1 1] = first row, then take M[1 1 1 1] = [1 1 1 3] then M [1 1 1 3], etc. Sequence gives terms in rightmost column.

A122605 Expansion of -x(2x - 1)(2x^2 - 1)(x^3 + 2x^2 - x - 1)/((x - 1)(x^2 + x - 1)(x^4 - 4x^3 - 4x^2 + x + 1)).

A122607 Expansion of x(8x^5 + 5x^4 - x^3 - 5x^2 - 1)/(x^6 + 3x^5 + 6x^4 + 4x^3 - 5x^2 + x - 1).