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A173786 Triangle read by rows: T(n,k) = 2^n + 2^k, 0 <= k <= n.

%I A173786 #35 Jun 20 2025 12:41:37
%S A173786 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 A173786 80,96,128,129,130,132,136,144,160,192,256,257,258,260,264,272,288,
%U A173786 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 A173786 Triangle read by rows: T(n,k) = 2^n + 2^k, 0 <= k <= n.
%C A173786 Essentially the same as A048645. - _T. D. Noe_, Mar 28 2011
%H A173786 T. D. Noe, <a href="/A173786/b173786.txt">Rows n = 0..100 of triangle, flattened</a>
%H A173786 Wawrzyniec Bieniawski, Piotr Masierak, Andrzej Tomski, and Szymon Łukaszyk, <a href="https://www.preprints.org/manuscript/202409.1581">Assembly Theory - Formalizing Assembly Spaces and Discovering Patterns and Bounds</a>, Preprints.org (2025).
%F A173786 1 <= A000120(T(n,k)) <= 2.
%F A173786 For n>0, 0<=k<n: T(n,k) = A048645(n+1,k+2) and T(n,n) = A048645(n+2,1).
%F A173786 Row sums give A006589(n).
%F A173786 Central terms give A161168(n).
%F A173786 T(2*n+1,n) = A007582(n+1).
%F A173786 T(2*n+1,n+1) = A028403(n+1).
%F A173786 T(n,k) = A140513(n,k) - A173787(n,k), 0<=k<=n.
%F A173786 T(n,k) = A059268(n+1,k+1) + A173787(n,k), 0<k<=n.
%F A173786 T(n,k) * A173787(n,k) = A173787(2*n,2*k), 0<=k<=n.
%F A173786 T(n,0) = A000051(n).
%F A173786 T(n,1) = A052548(n) for n>0.
%F A173786 T(n,2) = A140504(n) for n>1.
%F A173786 T(n,3) = A175161(n-3) for n>2.
%F A173786 T(n,4) = A175162(n-4) for n>3.
%F A173786 T(n,5) = A175163(n-5) for n>4.
%F A173786 T(n,n-4) = A110287(n-4) for n>3.
%F A173786 T(n,n-3) = A005010(n-3) for n>2.
%F A173786 T(n,n-2) = A020714(n-2) for n>1.
%F A173786 T(n,n-1) = A007283(n-1) for n>0.
%F A173786 T(n,n) = 2*A000079(n).
%e A173786 Triangle begins as:
%e A173786      2;
%e A173786      3,    4;
%e A173786      5,    6,    8;
%e A173786      9,   10,   12,   16;
%e A173786     17,   18,   20,   24,   32;
%e A173786     33,   34,   36,   40,   48,   64;
%e A173786     65,   66,   68,   72,   80,   96,  128;
%e A173786    129,  130,  132,  136,  144,  160,  192,  256;
%e A173786    257,  258,  260,  264,  272,  288,  320,  384,  512;
%e A173786    513,  514,  516,  520,  528,  544,  576,  640,  768, 1024;
%e A173786   1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 1536, 2048;
%t A173786 Flatten[Table[2^n + 2^m, {n,0,10}, {m, 0, n}]] (* _T. D. Noe_, Jun 18 2013 *)
%o A173786 (Magma) [2^n + 2^k: k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 07 2021
%o A173786 (Sage) flatten([[2^n + 2^k for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jul 07 2021
%o A173786 (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
%o A173786 (Python)
%o A173786 from math import isqrt, comb
%o A173786 def A173786(n):
%o A173786     a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
%o A173786     return (1<<a)+(1<<n-comb(a+1,2)) # _Chai Wah Wu_, Jun 20 2025
%Y A173786 Cf. A048645, A118413, A118416.
%Y A173786 Cf. also A087112, A370121.
%K A173786 nonn,tabl,easy
%O A173786 0,1
%A A173786 _Reinhard Zumkeller_, Feb 28 2010
%E A173786 Typo in first comment line fixed by _Reinhard Zumkeller_, Mar 07 2010