A049310 -id:A049310 - cp's OEIS frontend

A131327 Triangle |4*|A049310(n,k)| - 3| read by rows, 0<=k<=n.

1, 3, 1, 1, 3, 1, 3, 5, 3, 1, 1, 3, 9, 3, 1, 3, 9, 3, 13, 3, 1, 1, 3, 21, 3, 17, 3, 1, 3, 13, 3, 37, 3, 21, 3, 1, 1, 3, 37, 3, 57, 3, 25, 3, 1, 3, 17, 3, 77, 3, 81, 3, 29, 3, 1, 1, 3, 57, 3, 137, 3, 109, 3, 33, 3, 1, 3, 21, 3, 137, 3, 221, 3, 141, 3, 37, 3, 1, 1, 3, 81, 3, 277, 3, 333, 3, 177

Offset: 0

Gary W. Adamson, Jun 28 2007

			First few rows of the triangle are:
1;
3, 1;
1, 3, 1;
3, 5, 3, 1;
1, 3, 9, 3, 1;
3, 9, 3, 13, 3, 1;
1, 3, 21, 3, 17, 3, 1;
...

Cf. A049310, A131328 (row sums), A131324, A131325, A131326.

A131327 := proc(n,k)
    abs(4*abs(A049310(n,k))-3) ;
end proc:
seq(seq(A131327(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Aug 13 2012

Definition corrected. - R. J. Mathar, Aug 13 2012

A131777 5A065941 - 4A049310.

Original entry on oeis.org

1, 5, 1, 1, 5, 1, 5, -3, 10, 1, 1, 5, 3, 10, 1, 5, -7, 20, -1, 15, 1, 1, 5, 1, 20, 10, 15, 1, 5, -11, 30, -15, 50, 6, 20, 1, 1, 5, -5, 30, 15, 50, 22, 20, 1, 5, -15, 40, -45, 105, -9, 100, 18, 25, 1

Offset: 1

Gary W. Adamson, Jul 14 2007

Row sums = A022096(n-1).

			First few rows of the triangle are:
1;
5, 1
1, 5, 1;
5, -3, 10, 1;
1, 5, 3, 10, 1;
5, -7, 20, -1, 15, 1;
1, 5, 1, 20, 10, 15, 1;
...

Cf. A131774, A131775, A131776, A065941, A049310, A022096.

5*A065941 - 4*A049310 as infinite lower triangular matrices.

Corrected A-number in row sums reference R. J. Mathar, Jun 16 2009

A131778 6A065941 - 5A049310.

Original entry on oeis.org

1, 6, 1, 1, 6, 1, 6, -4, 12, 1, 1, 6, 3, 12, 1, 6, -9, 24, -2, 18, 1, 1, 6, 0, 24, 11, 18, 1, 6, -14, 36, -20, 60, 6, 24, 1, 1, 6, -8, 36, 15, 60, 25, 24, 1, 6, -19, 48, -58, 126, -15, 120, 20, 30, 1

Offset: 1

Gary W. Adamson, Jul 14 2007

Row sums = A022097, a Fibonacci-like sequence starting (1, 7, 8, 15, 23, 38, ...).

			First few rows of the triangle:
  1;
  6,  1;
  1,  6,  1;
  6, -4, 12,  1;
  1,  6,  3, 12,  1;
  6, -9, 24, -2, 18,  1;
  1,  6,  0, 24, 11, 18,  1;
  ...

Cf. A065941, A049310, A099097, A131774, A131775, A131776, A131777.

6*A065941 - 5*A049310 as infinite lower triangular matrices.

A285872 a(n) is the number of zeros of the Chebyshev S(n, x) polynomial (A049310) in the open interval (-sqrt(3), +sqrt(3)).

Original entry on oeis.org

0, 1, 2, 3, 4, 3, 4, 5, 6, 7, 8, 7, 8, 9, 10, 11, 12, 11, 12, 13, 14, 15, 16, 15, 16, 17, 18, 19, 20, 19, 20, 21, 22, 23, 24, 23, 24, 25, 26, 27, 28, 27, 28, 29, 30, 31, 32, 31, 32, 33, 34, 35, 36, 35, 36, 37, 38, 39, 40, 39, 40, 41, 42, 43, 44, 43, 44, 45

Offset: 0

Wolfdieter Lang, May 12 2017

nonn

See a May 06 2017 comment on A049310 where these problems are considered which originated in a conjecture by Michel Lagneau (see A008611) on Fibonacci polynomials.

			n = 3: S(3, x) = x*(-2 + x^2), with all three zeros (-sqrt(2), 0, +sqrt(2)) in the interval (-sqrt(3), +sqrt(3)).
n = 4: S(4, x) = 1 - 3*x^2 + x^4, all four zeros  (-phi, -1/phi, 1/phi, phi) with phi = (1 + sqrt(5))/2, approximately 1.618, lie in the interval.
n = 6, two zeros of  S(6, x) = -1 + 6*x^2 - 5*x^4 + x^6 are out of the interval (-sqrt(3), +sqrt(3)), namely - 1.8019... and +1.8019... .

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Cf. A008611(n-1) (1), A049310, A285869 (sqrt(2)), A285870.

m:=80; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+x+x^2+x^3-x^4+x^5)/((1-x)^2*(1+x)*(1-x+x^2)*(1+x+x^2)))); // G. C. Greubel, Mar 08 2018

CoefficientList[Series[x*(1+x+x^2+x^3-x^4+x^5)/((1-x)^2*(1+x)*(1-x+x^2)*(1+x+x^2)), {x, 0, 50}], x] (* G. C. Greubel, Mar 08 2018 *)

concat(0, Vec(x*(1 + x + x^2 + x^3 - x^4 + x^5) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)) + O(x^100))) \\ Colin Barker, May 18 2017

a(n) = 2*b(n) if n is even and 1 + 2*b(n) if n is odd with b(n) = floor(n/2) - floor((n+1)/6) = A285870(n). See the g.f. for {b(n)}_{n>=0} there.
From Colin Barker, May 18 2017: (Start)
G.f.: x*(1 + x + x^2 + x^3 - x^4 + x^5) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-6) - a(n-7) for n>6.
(End)

A131324 2*A049310 - A000012(signed).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 7, 1, 1, 1, 1, 11, 1, 9, 1, 1, 1, 7, 1, 19, 1, 11, 1, 1, 1, 1, 19, 1, 29, 1, 13, 1, 1, 1, 9, 1, 39, 1, 41, 1, 15, 1, 1

Offset: 0

Gary W. Adamson, Jun 28 2007

Row sums = A062114: (1, 2, 3, 6, 9, 16, 25, ...).

			First few rows of the triangle:
  1;
  1,  1;
  1,  1,  1;
  1,  3,  1,  1;
  1,  1,  5,  1,  1;
  1,  5,  1,  7,  1,  1,
  1,  1, 11,  1,  9,  1,  1;
  1,  7,  1, 19,  1, 11,  1,  1;
  ...

Cf. A049310, A062114, A131325, A131326, A131327.

2*A049310 - A000012(signed + - + - ... by columns).

A193664 Q-residue of A049310 (triangle of coefficients of Fibonacci polynomials), where Q=Pascal's triangle. (See Comments.)

Original entry on oeis.org

0, 1, 1, 6, 11, 68, 177, 1215, 4059, 30733, 124408, 1027972, 4862600, 43450761, 234283662, 2247091674, 13563976285, 138780931929, 925063455844, 10044476018973, 73144254450840, 839146997933059, 6618306039456419

Offset: 0

Clark Kimberling, Aug 02 2011

nonn

The definition of Q-residue is given at A193649.

Cf. A049310, A193649, A000110.

f[n_, x_] := Fibonacci[n, x];
q[n_, k_] := Coefficient[(x + 1)^n, x, k]; (* Pascal's triangle *)
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}];
p[n_, k_] := Coefficient[f[n, x], x, k];
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 22}]    (* A193664 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* A000110 *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]

A248163 Chebyshev's S polynomials (A049310) evaluated at 34/3 and multiplied by powers of 3 (A000244).

Original entry on oeis.org

1, 34, 1147, 38692, 1305205, 44028742, 1485230383, 50101574344, 1690086454249, 57012025275370, 1923198081274339, 64875626535849196, 2188462519487403613, 73823845023749080078, 2490314568132082090135, 84006280711277049343888, 2833800713070230938880977, 95593167717986358477858226

Offset: 0

Wolfdieter Lang, Nov 07 2014

This sequence appears in the solution for the curvature sequence of the touching circle and chord example given in A249457. See also the pair A249862(n) and a(n-1), with a(-1) = 0, for which details are given in A249862.

G. C. Greubel, Table of n, a(n) for n = 0..650
Index entries for linear recurrences with constant coefficients, signature (34,-9).

Cf. A000244, A049310, A249457, A249862.

I:=[1, 34]; [n le 2 select I[n] else 34*Self(n-1) - 9*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 08 2014

CoefficientList[Series[1/(1-34 x +(3 x)^2), {x,0,40}], x] (* Vincenzo Librandi, Nov 08 2014 *)
Table[3^n*ChebyshevU[n,17/3], {n,0,40}] (* G. C. Greubel, May 31 2025 *)

a(n) = 3^n*polchebyshev(n, 2, 17/3); \\ Michel Marcus, May 31 2025

def A248163(n): return 3^n*chebyshev_U(n,17/3)
print([A248163(n) for n in range(41)]) # G. C. Greubel, May 31 2025

a(n) = 3^n*S(n, 34/3) with Chebyshev's S polynomial (for S see the coefficient triangle A049310).
O.g.f.: 1/(1 - 34*x + 9*x^2).
a(n) = 34*a(n-1) - 9*a(n-2), a(-1) = 0, a(0) = 1 .
E.g.f.: exp(17*x)*(140*cosh(2*sqrt(70)*x) + 17*sqrt(70)*sinh(2*sqrt(70)*x))/140. - Stefano Spezia, Mar 24 2023

a(16)-a(17) from Stefano Spezia, Mar 24 2023

A249863 Chebyshev S polynomial (A049310) evaluated at x = 26/7 and multiplied by powers of 7 (A000420).

Original entry on oeis.org

1, 26, 627, 15028, 360005, 8623758, 206577463, 4948449896, 118537401609, 2839498396930, 68018625641339, 1629348845225244, 39030157319430733, 934945996889162102, 22396118210466108735, 536486719624549884112

Offset: 0

Wolfdieter Lang, Nov 09 2014

This sequence appears in the solution of the curvature sequence of the touching circle and chord example given by Kival Ngaokrajang in A249458. See also the pair A249864(n) and a(n-1), with a(-1) = 0, for which details are given in A249864.

Index entries for linear recurrences with constant coefficients, signature (26,-49)
Index entries for sequences related to Chebyshev polynomials

Cf. A000420, A049310, A249864, A248163.

I:=[1,26]; [n le 2 select I[n] else 26*Self(n-1)-49*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Nov 09 2014

LinearRecurrence[{26,-49},{1,26},20] (* Harvey P. Dale, Jun 30 2017 *)

a(n) = 7^n*S(n, 26/7) with Chebyshev's S polynomial (for S see the coefficient triangle A049310).
O.g.f.: 1/(1 - 26*x + (7*x)^2).
a(n) = 26*a(n-1) - 49*a(n-2), a(-1) = 0, a(0) = 1 .

A131375 A007318 + A046854 - A049310.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 2, 3, 4, 1, 1, 6, 6, 5, 1, 2, 5, 13, 10, 6, 1, 1, 9, 15, 24, 15, 7, 1, 2, 7, 27, 35, 40, 21, 8, 1, 1, 12, 28, 66, 70, 62, 28, 9, 1, 2, 9, 46, 84, 141, 126, 91, 36, 10, 1

Offset: 0

Gary W. Adamson, Jul 04 2007

Row sums = A117591: (1, 3, 5, 10, 19, 37, 72,...).

			First few rows of the triangle are:
1;
2, 1;
1, 3, 1;
2, 3, 4, 1;
1, 6, 6, 5, 1;
2, 5, 13, 10, 6, 1;
1, 9, 15, 24, 15, 7, 1;
...

Cf. A046854, A007318, A049310, A117591.

A007318 + A046854 - A049310 as infinite lower triangular matrices.

A135222 Triangle A049310 + A000012 - I, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 4, 1, 1, 1, 4, 1, 5, 1, 1, 2, 1, 7, 1, 6, 1, 1, 1, 5, 1, 11, 1, 7, 1, 1, 2, 1, 11, 1, 16, 1, 8, 1, 1, 1, 6, 1, 21, 1, 22, 1, 9, 1, 1, 2, 1, 16, 1, 36, 1, 29, 1, 10, 1, 1, 1, 7, 1, 36, 1, 57, 1, 37, 1, 11, 1, 1, 2, 1, 22, 1, 71, 1, 85, 1, 46, 1, 12, 1, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

Row sums = A081659: (1, 2 4, 6, 9, 13, 19, 28, ...).

			First few rows of the triangle:
  1;
  1, 1;
  2, 1,  1;
  1, 3,  1,  1;
  2, 1,  4,  1,  1;
  1, 4,  1,  5,  1, 1;
  2, 1,  7,  1,  6, 1, 1;
  1, 5,  1, 11,  1, 7, 1, 1;
  2, 1, 11,  1, 16, 1, 8, 1, 1;
...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A049310, A081659.

T:= proc(n, k) option remember;
      if k=n then 1
    else 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*Pi/2) )
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..15); # G. C. Greubel, Nov 20 2019

T[n_, k_]:= T[n, k]= If[k==n, 1, 1 + Abs[Simplify[((1+(-1)^(n-k))/2)* Binomial[(n+k)/2, (n-k)/2]*Cos[(n-k)*Pi/2]]] ]; Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

@CachedFunction
def T(n, k):
    if (k==n): return 1
    else: return 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*pi/2) )
[[T(n, k) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Nov 20 2019

T(n,k) = A049310(n,k) + A000012(n,k) - Identity matrix, as infinite lower triangular matrices.

T(n,k) = 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*Pi/2) ), with T(n,n) = 1. - G. C. Greubel, Nov 20 2019

More terms added and offset changed by G. C. Greubel, Nov 20 2019

cp's OEIS Frontend

A131327 Triangle |4*|A049310(n,k)| - 3| read by rows, 0<=k<=n.

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A131777 5*A065941 - 4*A049310.

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A131778 6*A065941 - 5*A049310.

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A285872 a(n) is the number of zeros of the Chebyshev S(n, x) polynomial (A049310) in the open interval (-sqrt(3), +sqrt(3)).

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A131324 2*A049310 - A000012(signed).

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A193664 Q-residue of A049310 (triangle of coefficients of Fibonacci polynomials), where Q=Pascal's triangle. (See Comments.)

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A248163 Chebyshev's S polynomials (A049310) evaluated at 34/3 and multiplied by powers of 3 (A000244).

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A249863 Chebyshev S polynomial (A049310) evaluated at x = 26/7 and multiplied by powers of 7 (A000420).

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A131777 5A065941 - 4A049310.

A131778 6A065941 - 5A049310.