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%I #20 Jan 28 2022 01:29:16
%S 4,6,2,17,25,22,62,123,23,214,341,510,246,727,932,998,573,1329,1726,
%T 2195,2742,3373,3515,2516,4094,4155,4911,5006,5830,1746,6857,5352,
%U 4057,7998,8273,9259,10646,1331,12165,12239,884,13822,15623,17574,19681
%N Consider all integer triples (i,j,k), j,k>0, with binomial(i+2, 3) = binomial(j+2, 3) + k^3, ordered by increasing i; sequence gives j values.
%C i values are A054221 and k values are A054223.
%C Up to a(59), which is as far as computed, there is only one unique pair (j,k) associated with each i. - _R. J. Mathar_, Nov 10 2006
%H Bert Dobbelaere, <a href="/A054222/b054222.txt">Table of n, a(n) for n = 0..271</a>
%e binomial(7+2, 3) = 84 = binomial(4+2, 3) + 4^3, so 4 is a term;
%e binomial(8+2, 3) = 120 = binomial(6+2, 3) + 4^3, so 6 is a term.
%t max = 20000; s = {}; Do[k = ((i*(i+1)*(i+2) - j*(j+1)*(j+2))/6)^(1/3); If[IntegerQ[k], Print[j]; AppendTo[s, {i, j}]], {j, 1, max}, {i, j+1, max}]; Sort[s, #1[[1]] < #2[[1]] &][[All, 2]] (* _Jean-François Alcover_, Oct 12 2011 *)
%o (C) #include <stdio.h> #include <limits.h> #include <math.h> unsigned A000578inv(unsigned long long n) { unsigned long long n3 = (unsigned long long)cbrt((double)n) ; for(unsigned long long k= n3-1 ; k <= n3+1 ; k++) if ( k*k*k == n) return k ; return 0 ; } int main(int argc, char *argv[]) { const unsigned long long imax = cbrt((double)ULLONG_MAX)-2. ; for(unsigned i=1; i<imax; i++) { unsigned long long i3 = i*(unsigned long long)(i+1)*(unsigned long long)(i+2) ; for(unsigned j=1 ; j < i ; j++) { unsigned long long k3 = i3- j*(unsigned long long)(j+1)*(unsigned long long)(j+2) ; if( k3 % 6 == 0) { unsigned k=A000578inv(k3/6) ; if ( k ) { printf("%d,",j) ; fflush(stdout) ; } } } } } /* _R. J. Mathar_, Nov 10 2006 */
%Y Cf. A054221, A054223.
%K nice,nonn
%O 0,1
%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Feb 04 2000
%E More terms from _R. J. Mathar_, Nov 10 2006
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