cp's OEIS Frontend

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

%I A352328 #15 Jan 05 2025 19:51:42
%S A352328 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,
%T A352328 42,43,44,46,47,48,49,70,71,72,73,75,76,77,78,82,83,84,85,87,88,89,90,
%U A352328 99,100,101,102,104,105,106,107,111,112,113,114,116,117,118
%N A352328 Nonnegative numbers that are the sum of distinct Pell numbers (A000129).
%C A352328 This sequence is the complement of A352323.
%C A352328 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).
%C A352328 Every nonnegative integer is the sum of two (not necessarily distinct) terms of this sequence.
%H A352328 L. Carlitz, Richard Scoville, and V. E. Hoggatt, Jr., <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/10-5/carlitz1.pdf">Pellian Representations</a>, The Fibonacci Quarterly, Vol. 10, No. 5 (1972), pp. 449-488.
%H A352328 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A352328 a(n) = Sum_{k >= 0} b_k * A000129(k+1) where Sum_{k >= 0} b_k * 2^k is the binary expansion of n.
%F A352328 A265744(a(n)) = A000120(n).
%e A352328 For n = 42:
%e A352328 - 42 = 2^5 + 2^3 + 2^1,
%e A352328 - so a(42) = A000129(5+1) + A000129(3+1) + A000129(1+1) = 70 + 12 + 2 = 84.
%t A352328 With[{pell = LinearRecurrence[{2, 1}, {1, 2}, 7]}, Select[Union[Plus @@@ Subsets[pell]], # <= pell[[-1]] &]] (* _Amiram Eldar_, Mar 12 2022 *)
%o A352328 (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) }
%Y A352328 Cf. A000120, A000129, A265744, A352323.
%K A352328 nonn,base
%O A352328 0,3
%A A352328 _Rémy Sigrist_, Mar 12 2022