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A352328 Nonnegative numbers that are the sum of distinct Pell numbers (A000129). 0
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 (list; graph; refs; listen; history; text; internal format)
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
L. Carlitz, Richard Scoville, and V. E. Hoggatt, Jr., Pellian Representations, The Fibonacci Quarterly, Vol. 10, No. 5 (1972), pp. 449-488.
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
Sequence in context: A006431 A285528 A151894 * A028229 A104452 A335073
KEYWORD
nonn,base
AUTHOR
Rémy Sigrist, Mar 12 2022
STATUS
approved

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Last modified March 29 09:32 EDT 2024. Contains 371268 sequences. (Running on oeis4.)