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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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%I #15 Jan 15 2019 02:48:21

%S 4,4,6,11,9,26,16,25,110,36,49,64,335,81,276,100,649,121,144,169,196,

%T 225,670,2024,256,1166,289,517,324,3522,361,3068,4071,400,2485,441,

%U 484,6137,529,1534,6816,576,625,676,729,784,841,900,961,15851,16199,12099

%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 k values.

%C i values are A054221 and j values are A054222.

%H Bert Dobbelaere, <a href="/A054223/b054223.txt">Table of n, a(n) for n = 0..271</a>

%e binomial(7+2,3)=84=binomial(4+2,3)+4^3; binomial(8+2,3)=120=binomial(6+2,3)+4^3;

%t 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 *)

%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,",k) ; fflush(stdout) ; } } } } } /* _R. J. Mathar_, Nov 10 2006 */

%Y Cf. A054221, A054222.

%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