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A054222
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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.
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3
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4, 6, 2, 17, 25, 22, 62, 123, 23, 214, 341, 510, 246, 727, 932, 998, 573, 1329, 1726, 2195, 2742, 3373, 3515, 2516, 4094, 4155, 4911, 5006, 5830, 1746, 6857, 5352, 4057, 7998, 8273, 9259, 10646, 1331, 12165, 12239, 884, 13822, 15623, 17574, 19681
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OFFSET
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0,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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binomial(7+2, 3) = 84 = binomial(4+2, 3) + 4^3, so 4 is a term;
binomial(8+2, 3) = 120 = binomial(6+2, 3) + 4^3, so 6 is a term.
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nice,nonn
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AUTHOR
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Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Feb 04 2000
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EXTENSIONS
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STATUS
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approved
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