A254708 -id:A254708 - cp's OEIS frontend

A254707 Expansion of (1 + 2x^2) / ((1 - x^2)^2 (1 - x^3) * (1 - x^4)) in powers of x.

1, 0, 4, 1, 8, 4, 15, 8, 25, 15, 38, 25, 55, 38, 77, 55, 103, 77, 135, 103, 173, 135, 217, 173, 268, 217, 327, 268, 393, 327, 468, 393, 552, 468, 645, 552, 748, 645, 862, 748, 986, 862, 1122, 986, 1270, 1122, 1430, 1270, 1603, 1430, 1790, 1603, 1990, 1790

Offset: 0

Michael Somos, Feb 06 2015

The number of quadruples of integers [x, u, v, w] which satisfy x > u > v > w >=0, n+7 = x+u, u+v != x+w, and x+u+v+w is even.

			G.f. = 1 + 4*x^2 + x^3 + 8*x^4 + 4*x^5 + 15*x^6 + 8*x^7 + 25*x^8 + ...

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

Cf. A006578, A254594, A254708.

a[ n_] := Quotient[ n^3 + If[ OddQ[n], 8 n^2 + 9 n + 18, 17 n^2 + 84 n + 148], 96];
a[ n_] := Module[{m = n}, SeriesCoefficient[ If[ n < 0, m = -9 - n; -2 - x^2, 1 + 2 x^2] / ((1 - x^2)^2 (1 - x^3) (1 - x^4)), {x, 0, m}]];
a[ n_] := Length @ FindInstance[ {x > u, u > v, v > w, w >= 0, x + u == n + 7, u + v != x + w, x + u + v + w == 2 k}, {x, u, v, w, k}, Integers, 10^9];

{a(n) = (n^3 + if( n%2, 8*n^2 + 9*n + 18, 17*n^2 + 84*n + 148)) \ 96};

{a(n) = polcoeff( if( n<0, n = -9-n; -2 - x^2, 1 + 2*x^2) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) + x * O(x^n), n)};

G.f.: (1 + 2*x^2) / (1 - 2*x^2 - x^3 + 2*x^5 + 2*x^6 - x^8 - 2*x^9 + x^11).
0 = a(n) + a(n+1) - a(n+2) - 2*a(n+3) - 2*a(n+4) + 2*a(n+6) + 2*a(n+7) + a(n+8) - a(n+9) - a(n+10) + 3 for all n in Z.
a(n+3) - a(n) = 0 if n even else A006578((n+5)/2) for all n in Z.
a(n+2) = 2*A254594(n) + A254594(n+2) for all n in Z.
a(n) = -A254708(-9 - n) for all n in Z.

A254745 Chebyshev polynomials of the second kind, U(n,x)^2, evaluated at x = sqrt(3)/2.

Original entry on oeis.org

1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4

Offset: 0

Michael Somos, Feb 07 2015

Period 6: repeat [1, 3, 4, 3, 1, 0].

			G.f. = 1 + 3*x + 4*x^2 + 3*x^3 + x^4 + x^6 + 3*x^7 + 4*x^8 + 3*x^9 + ...

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

Cf. A078070, A131027, A254744, A254612, A254707, A254708, A254875.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)/((1-x)*(1-x+x^2)))); // G. C. Greubel, Aug 03 2018

a[ n_] := {3, 4, 3, 1, 0, 1}[[Mod[n, 6, 1]]];
a[ n_] := ChebyshevU[ n, Sqrt[3] / 2]^2;
CoefficientList[Series[(1 + x) / ((1 - x) (1 - x + x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Jul 14 2017 *)

PARI
```
{a(n) = [1, 3, 4, 3, 1, 0][n%6 + 1]};
```

{a(n) = simplify( polchebyshev( n, 2, quadgen(12) / 2)^2)};

Euler transform of length 6 sequence [3, -2, -1, 0, 0, 1].
G.f.: (1 + x) / ((1 - x) * (1 - x + x^2)) = (1 - x^2)^2 * (1 - x^3) / ((1 - x)^3 * (1 - x^6)).
a(n) = a(-2-n) = a(n+6) for all n in Z.
a(n) = (-1)^n*A078070(n) = A131027(n-1) for all n in Z.
a(n) = (n+1)*(Sum_{k=0..n} (-1)^k/(k+1)*binomial(n+k+1,2*k+1)) for n >= 0. - Werner Schulte, Jul 10 2017
Sum_{n>=0} a(n)/(n+1)*x^(n+1) = log(1-x+x^2)-2*log(1-x) for -1 < x < 1. - Werner Schulte, Jul 10 2017
a(n) = sqrt(3)*sin(Pi*n/3) - cos(Pi*n/3) + 2. - Peter Luschny, Jul 16 2017
a(n) = 2 + 2*cos(Pi/3*(n+4)) for n >= 0. - Werner Schulte, Jul 18 2017

A251260 Expansion of (1 + 2x + x^2 + x^3) / ((1 - x^2)^2 (1 - x^3) * (1 - x^4)) in powers of x.

Original entry on oeis.org

1, 2, 3, 6, 8, 13, 16, 24, 28, 40, 45, 61, 68, 89, 97, 124, 134, 167, 179, 219, 233, 281, 297, 353, 372, 437, 458, 533, 557, 642, 669, 765, 795, 903, 936, 1056, 1093, 1226, 1266, 1413, 1457, 1618, 1666, 1842, 1894, 2086, 2142, 2350, 2411, 2636, 2701, 2944

Offset: 0

Michael Somos, Mar 20 2015

			G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 8*x^4 + 13*x^5 + 16*x^6 + 24*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (0,2,1,0,-2,-2,0,1,2,0,-1).

Cf. A008763, A165188, A254594, A254708, A254875.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 + 2*x+x^2+x^3)/((1-x^2)^2*(1-x^3)*(1-x^4)))); // G. C. Greubel, Aug 03 2018

a[ n_] := Quotient[ 5 n^3 + If[ OddQ[n], 66 n^2 + 249 n, 57 n^2 + 204 n] + 288, 288];
a[ n_] := Length @ FindInstance[ {x > u, u > v, v > w, w >= 0, x + u == n + 6, (u + v < x + w && k == 0) || (u + v > x + w && x + u + v + w == 2 k + 1)}, {x, u, v, w, k}, Integers, 10^9];
LinearRecurrence[{0,2,1,0,-2,-2,0,1,2,0,-1},{1,2,3,6,8,13,16,24,28,40,45},60] (* Harvey P. Dale, Jul 16 2025 *)

{a(n) = (5*n^3 + if( n%2, 66*n^2 + 249*n, 57*n^2 + 204*n) + 288) \ 288};

{a(n) = polcoeff( if( n<0, n = -8-n; -(1 + x + 2*x^2 + x^3), 1 + 2*x + x^2 + x^3) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) + x * O(x^n), n)};

a(n) = A165188(n+1) + A254708(n-1) = A254594(n-1) + A008763(n+4) for all n in Z.
0 = a(n) - 2*a(n+2) - a(n+3) + 2*a(n+5) + 2*a(n+6) - a(n+8) - 2*a(n+9) + a(n+11) for all n in Z.
a(2*n) = A254875(n) for all n in Z.
G.f.: (1 + 2*x + x^2 + x^3) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)).

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A254707 Expansion of (1 + 2*x^2) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.

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A254745 Chebyshev polynomials of the second kind, U(n,x)^2, evaluated at x = sqrt(3)/2.

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A251260 Expansion of (1 + 2*x + x^2 + x^3) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.

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A254707 Expansion of (1 + 2x^2) / ((1 - x^2)^2 (1 - x^3) * (1 - x^4)) in powers of x.

A251260 Expansion of (1 + 2x + x^2 + x^3) / ((1 - x^2)^2 (1 - x^3) * (1 - x^4)) in powers of x.