A100764 a(1) = 1, a(2) = 2, a(3) = 3, a(n) = least number not the sum of three or fewer previous terms.

1, 2, 3, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103, 109, 115, 121, 127, 133, 139, 145, 151, 157, 163, 169, 175, 181, 187, 193, 199, 205, 211, 217, 223, 229, 235, 241, 247, 253, 259, 265, 271, 277, 283, 289, 295, 301, 307, 313, 319, 325

Offset: 1

Author

Amarnath Murthy, Nov 25 2004

Keywords

• easy
• nonn

Comments

Generalization: let the first k terms of the sequence be 1,2,...,k, and for n > k, let b(n) be defined as the least positive integer that is not the sum of k or fewer previous terms; then b(n+k) = b(n) + n* k(k+1)/2. b(n) = (n+1)*k*(k+1)/2 + 1. n > k. Here a(n) is for k=3.

Links

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

Crossrefs

Essentially the same as A016921.

Programs

Mathematica

a[1] = 1; a[2] = 2; a[3] = 3; a[n_] := a[n] = (m = 1; l = n - 1; t = Union[ Flatten[ Join[ Table[ a[i], {i, l}], Table[ a[i] + a[j], {i, l}, {j, i + 1, l}], Table[ a[i] + a[j] + a[k], {i, l}, {j, i + 1, l}, {k, j + 1, l}] ]]]; While[ Position[t, m] != {}, m++ ]; m); Table[ a[n], {n, 60}] (* Robert G. Wilson v, Dec 14 2004 *)
LinearRecurrence[{2,-1},{1,2,3,7,13},60] (* Harvey P. Dale, Nov 17 2024 *)

Formula

a(n+4) = a(4) + 6n for n > 4; a(n) = 6n - 17, n >3.

From Chai Wah Wu, Oct 25 2018: (Start)

a(n) = 2*a(n-1) - a(n-2) for n > 5.

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

Extensions

More terms from Robert G. Wilson v, Dec 14 2004