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 A302648 a(n) is the largest integer bn such that there exists a set of n integers b1=0,b2,...,bn such that their pairwise sums cover all integers between 0 and 2*bn. 1
 0, 1, 2, 4, 6, 8, 10, 13, 16, 20, 22, 27, 32, 36, 40, 46, 52, 58, 64, 70, 76, 82, 90, 98, 106, 114, 122, 131 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS All known terms satisfy a(n) = (A123509(n)-1)/2 except for n=11, where A123509(11) = 47 but the corresponding item in this array is 22. Enumerating all sequences corresponding to A123509(11)=47, we found the solutions are (0 1 2 3 7 11 15 19 21 22 24) and (0 1 2 5 7 11 15 19 21 22 24); and none of them satisfy this problem. For most solutions of the problem, the differences between adjacent items are symmetric, and one of the differences is repeated multiple times in the middle of the difference array. E.g. for n=23 we have 0 1 3 4 6 10 13 15 21 29 37 45 53 61 69 75 77 80 84 86 87 89 90 with differences {1 2 1 2 4 3 2 6 8 8 8 8 8 8 6 2 3 4 2 1 2 1}. Using this property, we find that a(29) >= 140, a(30) >= 149, a(31) >= 158, a(32) >= 168, a(33) >= 178, a(34) >= 188. LINKS FORMULA a(n) <= (A123509(n)-1)/2. EXAMPLE 3: 0 1 2 4: 0 1 3 4 5: 0 1 3 5 6 6: 0 1 3 5 7 8 7: 0 1 2 5 8 9 10 8: 0 1 3 4 9 10 12 13 9: 0 1 2 5 8 11 14 15 16 10:0 1 3 4 9 11 16 17 19 20 11:0 1 3 4 6 11 13 18 19 21 22 12:0 1 3 5 6 13 14 21 22 24 26 27 13:0 1 3 4 9 11 16 21 23 28 29 31 32 14:0 1 3 4 9 11 16 20 25 27 32 33 35 36 15:0 1 3 4 5 8 14 20 26 32 35 36 37 39 40 16:0 1 3 4 5 8 14 20 26 32 38 41 42 43 45 46 17:0 1 3 4 5 8 14 20 26 32 38 44 47 48 49 51 52 18:0 1 3 4 5 8 14 20 26 32 38 44 50 53 54 55 57 58 19:0 1 3 4 5 8 14 20 26 32 38 44 50 56 59 60 61 63 64 20:0 1 3 4 5 8 14 20 26 32 38 44 50 56 62 65 66 67 69 70 21:0 1 3 4 5 8 14 20 26 32 38 44 50 56 62 68 71 72 73 75 76 22:0 1 3 4 6 10 13 15 21 29 37 45 53 61 67 69 72 76 78 79 81 82 23:0 1 3 4 6 10 13 15 21 29 37 45 53 61 69 75 77 80 84 86 87 89 90 24:0 1 3 4 6 10 13 15 21 29 37 45 53 61 69 77 83 85 88 92 94 95 97 98 25:0 1 3 4 6 10 13 15 21 29 37 45 53 61 69 77 85 91 93 96 100 102 103 105 106 26:0 1 3 4 6 10 13 15 21 29 37 45 53 61 69 77 85 93 99 101 104 108 110 111 113 114 PROG (C) /*C code to generate first part of the sets change K to larger value to generate more sets*/ #include #include #include #ifndef K #define K 8 #endif #ifndef R #define R 1 #endif #define UPBOUND 40960 unsigned short data[K+R]; unsigned short sumbuf[UPBOUND]; unsigned short diffbuf[UPBOUND]; unsigned short modbuf[K]; int rcount; int level; int next_sum, next_diff; int cur_best=10000000; void output() {     int i, j;     int b=data[level-1]+K;     int tindex=1;     for(i=b; i=data[j]&&(h-data[j])%K==0){                   min_index=(h-data[j])/K;               }            }            if(min_index<0)return;            if(min_index>tindex)tindex=min_index;        }     }     for(i=0; itindex)tindex=min_index;        }     }     if(K*(level-1)-data[level-1]<=cur_best){        cur_best=K*(level-1)-data[level-1];        printf("%d, >=%d | ", K*(level-1)-data[level-1], tindex);        for(i=0; i0){         if(rcount>=R)return 0;         rcount++;     }     modbuf[r]++;     for(i=0; i0){        rcount--;     }     sumbuf[x+x]--; diffbuf[0]--;     for(i=0; i=K&&data[level-1]+K<=next_sum){         output();     }     for(i=startv; i<=next_sum&&i<=K-1+data[level-1]; ++i){         if(push(i)){             search(i+1);             pop();         }     } } int main() {     data[0]=0; data[1]=1;     sumbuf[0]=sumbuf[1]=sumbuf[2]=1; rcount=0;     diffbuf[0]=2; diffbuf[1]=1; next_diff=2;     next_sum = 3;     level=2;     search( 2); } CROSSREFS Cf. A123509. Sequence in context: A294023 A096182 A186347 * A269746 A056827 A024172 Adjacent sequences:  A302645 A302646 A302647 * A302649 A302650 A302651 KEYWORD nonn,more AUTHOR Zhao Hui Du, Apr 11 2018 STATUS approved

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Last modified February 22 11:06 EST 2020. Contains 332135 sequences. (Running on oeis4.)