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 A069999 Number of possible dimensions for commutators of n X n matrices; it is independent of the field. Or, given a partition P = (p_1, p_2, ..., p_m) of n with p_1 >= p_2 >= ... >= p_m, let S(P) = sum_j (2j-1)p_j; then a(n) = number of integers that are an S(P) for some partition. 8
 1, 1, 2, 3, 5, 7, 9, 13, 18, 21, 27, 34, 39, 46, 54, 61, 72, 83, 92, 106, 118, 130, 145, 162, 176, 193, 209, 226, 246, 265, 284, 308, 330, 352, 375, 402, 426, 453, 480, 508, 538, 570, 598, 631, 661, 694, 730, 765, 800, 835, 872, 911, 951, 992, 1030, 1071, 1115 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Or, given such a partition P of n, let T(P) = sum_i p_i^2; then a(n) = number of integers that are a T(P) for some P. While T(P) need not equal S(P) for a given partition, the two sets of integers are equal. Or, expand the infinite product prod_k 1/(1-x^{k^2}y^k) as a power series; then a(n) = number of terms of the form x^my^n having a nonzero coefficient. The least m for which there are distinct partitions x(1)+...+x(k) of n for which the sums of squares {x(i)^2} are not distinct is 6. [Clark Kimberling, Mar 06 2012] a(n) is also the number of possible counts of intersection points of n lines in the plane, no three concurrent. This is because n lines, grouped into pencils of size a_1,...,a_k, meet in P=Sum_{i x+i^2, b(n-i, min(n-i, i)))[]}) end: a:= n-> nops(b(n\$2)): seq(a(n), n=0..56); # Alois P. Heinz, Jun 02 2022 MATHEMATICA p[n_, k_] := (IntegerPartitions[n]^2)[[k]]; s[n_, k_] := Sum[p[n, k][[i]], {i, 1, Length[p[n, k]]}]; t = Table[s[n, k], {n, 1, 20}, {k, 1, Length[IntegerPartitions[n]]}]; Table[Length[Union[t[[n]]]], {n, 1, 20}] (* Clark Kimberling, Mar 06 2012 *) PROG (PARI) a069999(N)= \\ terms up to a(N), b-file format { my( V = vector(N) ); V[1] = 'x; print(1, " ", 1 ); for (j=2, N, my( t = x^(j*j) ); for (a=1, j-1, my( b = j - a ); if ( a > b, break() ); t += V[a] * V[b]; ); t = Pol( apply( x->x!=0, Vec(t) ) ); print(j, " ", vecsum( Vec(t) ) ); V[j] = t; ); } \\ Joerg Arndt, Apr 19 2019 CROSSREFS Cf. A111212, A354468, A354800. Sequence in context: A200672 A341497 A332686 * A271661 A035563 A240063 Adjacent sequences: A069996 A069997 A069998 * A070000 A070001 A070002 KEYWORD easy,nonn,nice AUTHOR Jim Kuzmanovich (kuz(AT)wfu.edu), Apr 26 2002 EXTENSIONS More terms from Robert Gerbicz, Aug 27 2002 a(0)=1 prepended by Alois P. Heinz, Jun 02 2022 STATUS approved

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Last modified February 7 06:02 EST 2023. Contains 360112 sequences. (Running on oeis4.)