A352328 Nonnegative numbers that are the sum of distinct Pell numbers (A000129).

0, 1, 2, 3, 5, 6, 7, 8, 12, 13, 14, 15, 17, 18, 19, 20, 29, 30, 31, 32, 34, 35, 36, 37, 41, 42, 43, 44, 46, 47, 48, 49, 70, 71, 72, 73, 75, 76, 77, 78, 82, 83, 84, 85, 87, 88, 89, 90, 99, 100, 101, 102, 104, 105, 106, 107, 111, 112, 113, 114, 116, 117, 118

Offset: 0

Author

Rémy Sigrist, Mar 12 2022

Keywords

• nonn
• base

Comments

This sequence is the complement of A352323.

Although this is a list, it has offset 0 for mathematical reasons: indeed, so, the binary expansion of n encodes the positive Pell numbers summing to a(n).

Every nonnegative integer is the sum of two (not necessarily distinct) terms of this sequence.

Examples

			For n = 42:
- 42 = 2^5 + 2^3 + 2^1,
- so a(42) = A000129(5+1) + A000129(3+1) + A000129(1+1) = 70 + 12 + 2 = 84.
		

Links

• L. Carlitz, Richard Scoville, and V. E. Hoggatt, Jr., Pellian Representations, The Fibonacci Quarterly, Vol. 10, No. 5 (1972), pp. 449-488.
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000120, A000129, A265744, A352323.

Programs

Mathematica

With[{pell = LinearRecurrence[{2, 1}, {1, 2}, 7]}, Select[Union[Plus @@@ Subsets[pell]], # <= pell[[-1]] &]] (* Amiram Eldar, Mar 12 2022 *)

PARI

a(n) = { my (v=0, k); while (n, n-=2^k=valuation(n, 2); v+=([2, 1; 1, 0]^(k+1))[2, 1]); return (v) }

Formula

a(n) = Sum_{k >= 0} b_k * A000129(k+1) where Sum_{k >= 0} b_k * 2^k is the binary expansion of n.

A265744(a(n)) = A000120(n).