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A372557 Numbers k such that the least number of Jacobsthal numbers that add up to k, A372555(k), is less than the number needed with the greedy algorithm, A265745(k).

%I A372557 #16 Dec 20 2024 17:42:33
%S A372557 63,84,148,169,191,212,234,255,276,297,319,340,404,425,489,510,532,
%T A372557 553,575,596,617,638,660,681,703,724,746,767,788,809,831,852,874,895,
%U A372557 917,937,938,959,980,1002,1022,1023,1044,1065,1087,1108,1129,1150,1172,1193,1215,1236,1258,1278,1279,1300,1321,1343,1363,1364,1428
%N A372557 Numbers k such that the least number of Jacobsthal numbers that add up to k, A372555(k), is less than the number needed with the greedy algorithm, A265745(k).
%H A372557 Antti Karttunen, <a href="/A372557/b372557.txt">Table of n, a(n) for n = 1..22785</a>
%e A372557 63 = 21+21+21 has A372555(63)=3 for its optimal, non-greedy solution, and A265745(63) = 5 for its greedy solution 63 = 43+11+5+3+1, therefore 63 is included in this sequence. (From _Yuriko Suwa_'s Jul 11 2021 comment in A265745.)
%e A372557 84 = 21+21+21+21 has A372555(84)=4 for its optimal, non-greedy solution, and A265745(84) = 6 for its greedy solution 84 = 43+21+11+5+3+1, therefore 84 is included in this sequence.
%e A372557 169 = 85+21+21+21+21 has A372555(169)=5 for its optimal, non-greedy solution, and A265745(169) = 7 for its greedy solution 169 = 85+43+21+11+5+3+1, therefore 169 is included in this sequence.
%o A372557 (PARI)
%o A372557 \\ For A372555, use the program given under that entry.
%o A372557 A130249(n) = (#binary(3*n+1)-1);
%o A372557 A001045(n) = (2^n - (-1)^n) / 3;
%o A372557 A265745(n) = { my(s=0); while(n,s++; n -= A001045(A130249(n))); (s); };
%o A372557 isA372557(k) = (A372555(k)<A265745(k));
%Y A372557 Cf. A001045, A265745, A372555.
%Y A372557 Cf. A372558 (subsequence).
%K A372557 nonn
%O A372557 1,1
%A A372557 _Antti Karttunen_, May 07 2024