A057945 -id:A057945 - cp's OEIS frontend

A354763 a(n) is the minimum number of square tiles needed for constructing a figure whose corresponding graph has n cycles.

0, 4, 7, 6, 9, 12, 8, 11, 14, 13, 10, 13, 16, 15, 18, 12, 15, 18, 17, 20, 23, 14, 17, 20, 19, 22, 25, 21, 16, 19, 22, 21, 24, 27, 23, 26, 18, 21, 24, 23, 26, 29, 25, 28, 31, 20, 23, 26, 25, 28, 31, 27, 30, 33, 32, 22, 25, 28, 27, 30, 33, 29, 32, 35, 34, 31, 24

Offset: 0

Stefano Spezia, Jun 06 2022

The square tiles are connected only at corners.

Stefano Spezia, Illustrations for n = 1..10

Cf. A000217, A057945, A163300, A354762.

r[n_]:=First[IntegerPartitions[n,All,Table[k(k+1)/2,{k,Sqrt[1+8n]}]]]; (* A354762 *)
Join[{0}, Table[1+Sum[Sqrt[1+8Part[r[n],i]],{i,Length[r[n]]}],{n,66}]]

a(n) = 1 + Sum_{i=1..A057945(n)} sqrt(1 + 8*A354762(n, i)) for n > 0.

a(A000217(n)) = A163300(n+1).

A354762 Irregular triangle read by rows in which the row n lists the partition of n into the minimum number of triangular parts.

Original entry on oeis.org

0, 1, 1, 1, 3, 3, 1, 3, 1, 1, 6, 6, 1, 6, 1, 1, 6, 3, 10, 10, 1, 10, 1, 1, 10, 3, 10, 3, 1, 15, 15, 1, 15, 1, 1, 15, 3, 15, 3, 1, 15, 3, 1, 1, 21, 21, 1, 21, 1, 1, 21, 3, 21, 3, 1, 21, 3, 1, 1, 21, 6, 28, 28, 1, 28, 1, 1, 28, 3, 28, 3, 1, 28, 3, 1, 1, 28, 6, 28, 6, 1

Offset: 0

Stefano Spezia, Jun 06 2022

The representation of the partitions (for fixed n) is as (weakly) decreasing list of the parts.

			The irregular triangle begins:
     0;
     1;
     1, 1;
     3;
     3, 1;
     3, 1, 1;
     6;
     6, 1;
     6, 1, 1;
     6, 3;
    10;
    10, 1;
    10, 1, 1;
    10, 3;
    10, 3, 1;
    15;
    15, 1;
    15, 1, 1;
    15, 3;
    15, 3, 1;
    15, 3, 1, 1;
    ...

Cf. A000041, A000217, A007294.
Cf. A001477 (row sums), A057944 (1st column), A057945 (row lengths).
Cf. A354763.

Flatten[Join[{0}, Table[First[IntegerPartitions[n, All, Table[k(k+1)/2, {k, (Sqrt[1+8n]-1)/2}]]], {n, 35}]]]

cp's OEIS Frontend

A354763 a(n) is the minimum number of square tiles needed for constructing a figure whose corresponding graph has n cycles.

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