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%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
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