A351873 Number of subsets of {1,2,...,n} such that no two elements 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

Author

Michael A. Allen, Feb 22 2022

Keywords

• easy
• nonn

Examples

			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}.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,1,2).
• Index entries for subsets of {1,..,n} with disallowed differences

Crossrefs

See A375981 for other sequences related to restricted combinations.

Column k=12 of A376033.

Programs

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 *)

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).