A054223 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.

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

Offset

0,1

Comments

i values are A054221 and j values are A054222.

Links

Bert Dobbelaere, Table of n, a(n) for n = 0..271

Example

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

Mathematica

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 *)

Prog

(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 */

Crossrefs

Cf. A054221, A054222.

Sequence in context: A213383 A213425 A213426 * A160643 A078321 A195954

Adjacent sequences: A054220 A054221 A054222 * A054224 A054225 A054226

Keyword

nice,nonn

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Feb 04 2000

Extensions

More terms from R. J. Mathar, Nov 10 2006

Status

approved