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

0,3

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.

Links

Table of n, a(n) for n=0..62.

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

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).

Example

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.

Mathematica

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

Prog

(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) }

Crossrefs

Cf. A000120, A000129, A265744, A352323.

Sequence in context: A006431 A285528 A151894 * A028229 A104452 A335073

Adjacent sequences: A352325 A352326 A352327 * A352329 A352330 A352331

Keyword

nonn,base,changed

Author

Rémy Sigrist, Mar 12 2022

Status

approved