A375317 -id:A375317 - cp's OEIS frontend

A375315 Expansion of (1 + x)/(1 - x^2*(1 + x)^3).

1, 1, 1, 4, 7, 11, 23, 45, 81, 154, 296, 555, 1046, 1986, 3753, 7085, 13404, 25348, 47904, 90568, 171245, 323728, 612009, 1157071, 2187496, 4135527, 7818464, 14781237, 27944604, 52830706, 99879234, 188826693, 356986401, 674901117, 1275934888, 2412219633, 4560424135

Offset: 0

Seiichi Manyama, Aug 12 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (0,1,3,3,1).

Cf. A093040, A375316.

Cf. A116090, A375317.

my(N=40, x='x+O('x^N)); Vec((1+x)/(1-x^2*(1+x)^3))

a(n) = sum(k=0, n\2, binomial(3*k+1, n-2*k));

a(n) = a(n-2) + 3*a(n-3) + 3*a(n-4) + a(n-5).
a(n) = Sum_{k=0..floor(n/2)} binomial(3*k+1,n-2*k).
a(n) = A116090(n) + A116090(n-1).

A375364 Expansion of 1/( (1 + x) * (1 - x^2*(1 + x)^3) ).

Original entry on oeis.org

1, -1, 2, 1, 3, 4, 12, 17, 35, 67, 127, 234, 451, 850, 1602, 3031, 5740, 10837, 20490, 38751, 73253, 138471, 261814, 494972, 935738, 1769079, 3344568, 6323022, 11953992, 22599700, 42725842, 80775309, 152709941, 288705926, 545813095, 1031887517, 1950836006

Offset: 0

Seiichi Manyama, Aug 13 2024

sign

Index entries for linear recurrences with constant coefficients, signature (-1,1,4,6,4,1).

Cf. A375315, A375317, A375365.

my(N=40, x='x+O('x^N)); Vec(1/((1+x)*(1-x^2*(1+x)^3)))

a(n) = sum(k=0, n\2, binomial(3*k-1, n-2*k));

a(n) = -a(n-1) + a(n-2) + 4*a(n-3) + 6*a(n-4) + 4*a(n-5) + a(n-6).
a(n) = Sum_{k=0..floor(n/2)} binomial(3*k-1,n-2*k).
a(n) = A375365(n) + A375365(n-1).

A375365 Expansion of 1/( (1 + x)^2 * (1 - x^2*(1 + x)^3) ).

Original entry on oeis.org

1, -2, 4, -3, 6, -2, 14, 3, 32, 35, 92, 142, 309, 541, 1061, 1970, 3770, 7067, 13423, 25328, 47925, 90546, 171268, 323704, 612034, 1157045, 2187523, 4135499, 7818493, 14781207, 27944635, 52830674, 99879267, 188826659, 356986436, 674901081, 1275934925, 2412219595

Offset: 0

Seiichi Manyama, Aug 13 2024

sign

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,0,5,10,10,5,1).

Cf. A375315, A375317, A375364.

CoefficientList[Series[1/((1+x)^2(1-x^2(1+x)^3)),{x,0,40}],x] (* or *) LinearRecurrence[{-2,0,5,10,10,5,1},{1,-2,4,-3,6,-2,14},40] (* Harvey P. Dale, Aug 07 2025 *)

my(N=40, x='x+O('x^N)); Vec(1/((1+x)^2*(1-x^2*(1+x)^3)))

a(n) = sum(k=0, n\2, binomial(3*k-2, n-2*k));

a(n) = -2*a(n-1) + 5*a(n-3) + 10*a(n-4) + 10*a(n-5) + 5*a(n-6) + a(n-7).

a(n) = Sum_{k=0..floor(n/2)} binomial(3*k-2,n-2*k).

A375321 Expansion of (1 + x)^2/(1 - x^3*(1 + x)^3).

Original entry on oeis.org

1, 2, 1, 1, 5, 10, 11, 13, 29, 57, 81, 111, 194, 352, 554, 827, 1348, 2303, 3739, 5843, 9382, 15519, 25317, 40431, 64933, 105863, 172321, 277696, 447272, 725140, 1177181, 1903186, 3072365, 4972113, 8057421, 13038606, 21075947, 34094041, 55199573, 89336141

Offset: 0

Seiichi Manyama, Aug 12 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (0,0,1,3,3,1).

Cf. A115055, A375319.

Cf. A375317,

my(N=40, x='x+O('x^N)); Vec((1+x)^2/(1-x^3*(1+x)^3))

a(n) = sum(k=0, n\3, binomial(3*k+2, n-3*k));

a(n) = a(n-3) + 3*a(n-4) + 3*a(n-5) + a(n-6).
a(n) = Sum_{k=0..floor(n/3)} binomial(3*k+2,n-3*k).
a(n) = A375319(n) + A375319(n-1).

cp's OEIS Frontend

A375315 Expansion of (1 + x)/(1 - x^2*(1 + x)^3).

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A375364 Expansion of 1/( (1 + x) * (1 - x^2*(1 + x)^3) ).

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A375365 Expansion of 1/( (1 + x)^2 * (1 - x^2*(1 + x)^3) ).

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A375321 Expansion of (1 + x)^2/(1 - x^3*(1 + x)^3).

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