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A054223
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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 k values.
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3
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4, 4, 6, 11, 9, 26, 16, 25, 110, 36, 49, 64, 335, 81, 276, 100, 649, 121, 144, 169, 196, 225, 670, 2024, 256, 1166, 289, 517, 324, 3522, 361, 3068, 4071, 400, 2485, 441, 484, 6137, 529, 1534, 6816, 576, 625, 676, 729, 784, 841, 900, 961, 15851, 16199, 12099
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OFFSET
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0,1
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COMMENTS
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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; binomial(8+2,3)=120=binomial(6+2,3)+4^3;
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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[k]; AppendTo[s, {i, k}]], {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, ", k) ; 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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