A375316 -id:A375316 - 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).

A375320 Expansion of (1 + x)/(1 - x^3*(1 + x)^4).

Original entry on oeis.org

1, 1, 0, 1, 5, 10, 11, 14, 37, 85, 139, 204, 371, 768, 1432, 2398, 4117, 7685, 14422, 25744, 45037, 80888, 148408, 269402, 480873, 859580, 1554254, 2817518, 5074004, 9103828, 16384908, 29588793, 53381548, 96068570, 172864927, 311535484, 561770980, 1012168575

Offset: 0

Seiichi Manyama, Aug 12 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (0,0,1,4,6,4,1).

Cf. A375316, A375318.

my(N=40, x='x+O('x^N)); Vec((1+x)/(1-x^3*(1+x)^4))

a(n) = sum(k=0, n\3, binomial(4*k+1, n-3*k));

a(n) = a(n-3) + 4*a(n-4) + 6*a(n-5) + 4*a(n-6) + a(n-7).
a(n) = Sum_{k=0..floor(n/3)} binomial(4*k+1,n-3*k).
a(n) = A375318(n) + A375318(n-1).

cp's OEIS Frontend

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

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A375320 Expansion of (1 + x)/(1 - x^3*(1 + x)^4).

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