A070770 - OEIS

%I #21 Dec 11 2024 15:42:29

%S 0,1,2,3,2,3,4,4,5,6,3,4,5,5,6,7,6,7,8,9,4,5,6,6,7,8,7,8,9,10,8,9,10,

%T 11,12,5,6,7,7,8,9,8,9,10,11,9,10,11,12,13,10,11,12,13,14,15,6,7,8,8,

%U 9,10,9,10,11,12,10,11,12,13,14,11,12,13,14,15,16,12,13,14,15,16,17,18,7

%N b + c + d where b >= c >= d >= 0 ordered by b then c then d.

%H Robert Israel, <a href="/A070770/b070770.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = A056556(n) + A056557(n) + A056558(n).

%e Triangle begins:

%e   0,

%e   ;

%e   1;

%e   2, 3;

%e   ;

%e   2;

%e   3, 4;

%e   4, 5, 6;

%e   ;

%e   3;

%e   4, 5,

%e   5, 6, 7;

%e   6, 7, 8, 9;

%e   ;

%e   4;

%e   5, 6;

%e   6, 7,  8;

%e   7, 8,  9, 10;

%e   8, 9, 10, 11, 12;

%e   ;

%e   ...

%p seq(seq(seq(b+c+d,d=0..c),c=0..b),b=0..10); # _Robert Israel_, Jun 21 2018

%o (PARI) for(x=0,5,for(y=0,x,for(z=0,y,print1(x+y+z", ")))) \\ _Charles R Greathouse IV_, Sep 17 2015

%o (Python)

%o from math import isqrt, comb

%o from sympy import integer_nthroot

%o def A070770(n): return (m:=integer_nthroot(6*(n+1),3)[0])+(a:=n>=comb(m+2,3))+(k:=isqrt(b:=(c:=n+1-comb(m+a+1,3))<<1))-((b<<2)<=(k<<2)*(k+1)+1)+c-2-comb(k+(b>k*(k+1)),2) # _Chai Wah Wu_, Dec 11 2024

%Y Cf. A001477, A051162, A070771, A070772 for similar sequences with different numbers of terms summed.

%K nonn,look,tabf

%O 0,3

%A _Henry Bottomley_, May 06 2002