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A352328
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Nonnegative numbers that are the sum of distinct Pell numbers (A000129).
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0
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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L. Carlitz, Richard Scoville, and V. E. Hoggatt, Jr., Pellian Representations, The Fibonacci Quarterly, Vol. 10, No. 5 (1972), pp. 449-488.
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FORMULA
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a(n) = Sum_{k >= 0} b_k * A000129(k+1) where Sum_{k >= 0} b_k * 2^k is the binary expansion of n.
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EXAMPLE
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For n = 42:
- 42 = 2^5 + 2^3 + 2^1,
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MATHEMATICA
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With[{pell = LinearRecurrence[{2, 1}, {1, 2}, 7]}, Select[Union[Plus @@@ Subsets[pell]], # <= pell[[-1]] &]] (* Amiram Eldar, Mar 12 2022 *)
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PROG
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(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) }
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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