A369849 -id:A369849 - cp's OEIS frontend

A369850 Number of compositions of 5*n-2 into parts 4 and 5.

0, 1, 3, 6, 10, 16, 29, 64, 156, 376, 860, 1873, 3965, 8347, 17751, 38303, 83488, 182497, 397978, 864339, 1871236, 4046070, 8751965, 18952107, 41080980, 89090436, 193201660, 418857277, 907802332, 1967174726, 4262706876, 9237582992, 20020241409, 43391549313

Offset: 1

Seiichi Manyama, Feb 03 2024

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-4,1).

Cf. A099131, A368475, A369849, A369851.

Cf. A017827.

LinearRecurrence[{5, -10, 10, -4, 1}, {0, 1, 3, 6, 10}, 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n\4, binomial(n+k, n-2-4*k));

a(n) = A017827(5*n-2).
a(n) = Sum_{k=0..floor(n/4)} binomial(n+k,n-2-4*k).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 4*a(n-4) + a(n-5).
G.f.: x^2*(1-x)^2/((1-x)^5 - x^4).

A373933 Number of compositions of 7*n-1 into parts 6 and 7.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 17, 54, 175, 506, 1299, 3017, 6465, 13021, 25142, 47651, 91104, 180254, 374077, 810381, 1800140, 4019204, 8888489, 19322901, 41223071, 86520282, 179574728, 370946309, 767426451, 1597653852, 3354537225, 7101005320, 15118658953

Offset: 1

Seiichi Manyama, Jun 23 2024

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-6,1).

Cf. A369809, A373912, A373934, A373935, A373936, A373937.

Cf. A017847, A369849.

a(n) = sum(k=0, n\6, binomial(n+k, n-1-6*k));

a(n) = A017847(7*n-1).
a(n) = Sum_{k=0..floor(n/6)} binomial(n+k,n-1-6*k).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 6*a(n-6) + a(n-7).
G.f.: x*(1-x)^5/((1-x)^7 - x^6).
a(n) = A373934(n+1)-A373934(n). - R. J. Mathar, Jun 24 2024

A369851 Number of compositions of 5*n-3 into parts 4 and 5.

Original entry on oeis.org

0, 0, 1, 4, 10, 20, 36, 65, 129, 285, 661, 1521, 3394, 7359, 15706, 33457, 71760, 155248, 337745, 735723, 1600062, 3471298, 7517368, 16269333, 35221440, 76302420, 165392856, 358594516, 777451793, 1685254125, 3652428851, 7915135727, 17152718719, 37172960128

Offset: 1

Seiichi Manyama, Feb 03 2024

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-4,1).

Cf. A099131, A368475, A369849, A369850.

Cf. A017827.

LinearRecurrence[{5, -10, 10, -4, 1}, {0, 0, 1, 4, 10}, 50] (* Paolo Xausa, Mar 15 2024 *)

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

a(n) = A017827(5*n-3).
a(n) = Sum_{k=0..floor(n/4)} binomial(n+k,n-3-4*k).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 4*a(n-4) + a(n-5).
G.f.: x^3*(1-x)/((1-x)^5 - x^4).

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A369850 Number of compositions of 5*n-2 into parts 4 and 5.

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A373933 Number of compositions of 7*n-1 into parts 6 and 7.

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A369851 Number of compositions of 5*n-3 into parts 4 and 5.

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