A351873 Number of subsets of {1,2,...,n} whose elements do not differ by 3 or 4.

1, 2, 4, 8, 12, 16, 21, 29, 45, 73, 117, 178, 260, 376, 552, 832, 1273, 1945, 2937, 4385, 6521, 9730, 14612, 22040, 33252, 50032, 75053, 112437, 168549, 253065, 380429, 572018, 859572, 1290664, 1937152, 2907744, 4366321, 6558769, 9853041, 14800001, 22226225

Offset

0,2

Links

Table of n, a(n) for n=0..40.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,1,2).

Formula

a(n) = a(n-1) + a(n-5) + a(n-6) + 2*a(n-7) + delta(n,0) + delta(n,1) + 2*delta(n,2) + 4*delta(n,3) + 4*delta(n,4) + 3*delta(n,5) + 2*delta(n,6).

G.f.: (1 + x + 2*x^2 + 4*x^3 + 4*x^4 + 3*x^5 + 2*x^6)/(1 - x - x^5 - x^6 - 2*x^7).

Example

When n = 5, the 16 subsets are {}, {1}, {2}, {3}, {4}, {5}, {1,2}, {1,3}, {2,3}, {2,4}, {3,4}, {3,5}, {4,5}, {1,2,3}, {2,3,4}, and {3,4,5}.

Mathematica

CoefficientList[Series[(1 + x + 2x^2 + 4x^3 + 4x^4 + 3x^5 + 2x^6)/(1 - x - x^5 - x^6 - 2*x^7), {x, 0, 35}], x]
LinearRecurrence[{1, 0, 0, 0, 1, 1, 2}, {1, 2, 4, 8, 12, 16, 21}, 50] (* Harvey P. Dale, Mar 01 2023 *)

Crossrefs

Other sequences giving numbers of restricted combinations: A000045, A006498, A006500, A031923, A000930, A130137, A263710, A079972, A224809, A224810, A224815, A003269, A317669, A351874, A177485, A121832, A224808, A003520, A224811, A005708, A224812, A005709, A224813, A005710.

Sequence in context: A256403 A308013 A260090 * A256941 A324174 A047836

Adjacent sequences: A351870 A351871 A351872 * A351874 A351875 A351876

Keyword

easy,nonn

Author

Michael A. Allen, Feb 22 2022

Status

approved