A173786 - OEIS

%I #22 Mar 09 2024 08:18:52

%S 2,3,4,5,6,8,9,10,12,16,17,18,20,24,32,33,34,36,40,48,64,65,66,68,72,

%T 80,96,128,129,130,132,136,144,160,192,256,257,258,260,264,272,288,

%U 320,384,512,513,514,516,520,528,544,576,640,768,1024,1025,1026,1028,1032,1040,1056,1088,1152,1280,1536,2048

%N Triangle read by rows: T(n,k) = 2^n + 2^k, 0 <= k <= n.

%C Essentially the same as A048645. - _T. D. Noe_, Mar 28 2011

%H T. D. Noe, <a href="/A173786/b173786.txt">Rows n = 0..100 of triangle, flattened</a>

%F 1 <= A000120(T(n,k)) <= 2.

%F For n>0, 0<=k<n: T(n,k) = A048645(n+1,k+2) and T(n,n) = A048645(n+2,1).

%F Row sums give A006589(n).

%F Central terms give A161168(n).

%F T(2*n+1,n) = A007582(n+1).

%F T(2*n+1,n+1) = A028403(n+1).

%F T(n,k) = A140513(n,k) - A173787(n,k), 0<=k<=n.

%F T(n,k) = A059268(n+1,k+1) + A173787(n,k), 0<k<=n.

%F T(n,k) * A173787(n,k) = A173787(2*n,2*k), 0<=k<=n.

%F T(n,0) = A000051(n).

%F T(n,1) = A052548(n) for n>0.

%F T(n,2) = A140504(n) for n>1.

%F T(n,3) = A175161(n-3) for n>2.

%F T(n,4) = A175162(n-4) for n>3.

%F T(n,5) = A175163(n-5) for n>4.

%F T(n,n-4) = A110287(n-4) for n>3.

%F T(n,n-3) = A005010(n-3) for n>2.

%F T(n,n-2) = A020714(n-2) for n>1.

%F T(n,n-1) = A007283(n-1) for n>0.

%F T(n,n) = 2*A000079(n).

%e Triangle begins as:

%e      2;

%e      3,    4;

%e      5,    6,    8;

%e      9,   10,   12,   16;

%e     17,   18,   20,   24,   32;

%e     33,   34,   36,   40,   48,   64;

%e     65,   66,   68,   72,   80,   96,  128;

%e    129,  130,  132,  136,  144,  160,  192,  256;

%e    257,  258,  260,  264,  272,  288,  320,  384,  512;

%e    513,  514,  516,  520,  528,  544,  576,  640,  768, 1024;

%e   1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 1536, 2048;

%t Flatten[Table[2^n + 2^m, {n,0,10}, {m, 0, n}]] (* _T. D. Noe_, Jun 18 2013 *)

%o (Magma) [2^n + 2^k: k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 07 2021

%o (Sage) flatten([[2^n + 2^k for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jul 07 2021

%o (PARI) A173786(n) = { my(c = (sqrtint(8*n + 1) - 1) \ 2); 1 << c + 1 << (n - binomial(c + 1, 2)); }; \\ _Antti Karttunen_, Feb 29 2024, after _David A. Corneth_'s PARI-program in A048645

%Y Cf. A048645, A118413, A118416.

%Y Cf. also A087112, A370121.

%K nonn,tabl,easy

%O 0,1

%A _Reinhard Zumkeller_, Feb 28 2010

%E Typo in first comment line fixed by _Reinhard Zumkeller_, Mar 07 2010