A173786 Triangle read by rows: T(n,k) = 2^n + 2^k, 0 <= k <= n.
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, 80, 96, 128, 129, 130, 132, 136, 144, 160, 192, 256, 257, 258, 260, 264, 272, 288, 320, 384, 512, 513, 514, 516, 520, 528, 544, 576, 640, 768, 1024, 1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 1536, 2048
Offset: 0
Examples
Triangle begins as:
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, 80, 96, 128;
129, 130, 132, 136, 144, 160, 192, 256;
257, 258, 260, 264, 272, 288, 320, 384, 512;
513, 514, 516, 520, 528, 544, 576, 640, 768, 1024;
1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 1536, 2048;
Links
- T. D. Noe, Rows n = 0..100 of triangle, flattened
- Wawrzyniec Bieniawski, Piotr Masierak, Andrzej Tomski, and Szymon Łukaszyk, Assembly Theory - Formalizing Assembly Spaces and Discovering Patterns and Bounds, Preprints.org (2025).
Programs
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Magma
[2^n + 2^k: k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 07 2021
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Mathematica
Flatten[Table[2^n + 2^m, {n,0,10}, {m, 0, n}]] (* T. D. Noe, Jun 18 2013 *) -
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
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Python
from math import isqrt, comb def A173786(n): a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)) return (1<
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Sage
flatten([[2^n + 2^k for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 07 2021
Formula
1 <= A000120(T(n,k)) <= 2.
For n>0, 0<=kA048645(n+1,k+2) and T(n,n) = A048645(n+2,1).
Row sums give A006589(n).
Central terms give A161168(n).
T(2*n+1,n) = A007582(n+1).
T(2*n+1,n+1) = A028403(n+1).
T(n,k) = A059268(n+1,k+1) + A173787(n,k), 0
T(n,0) = A000051(n).
T(n,1) = A052548(n) for n>0.
T(n,2) = A140504(n) for n>1.
T(n,3) = A175161(n-3) for n>2.
T(n,4) = A175162(n-4) for n>3.
T(n,5) = A175163(n-5) for n>4.
T(n,n-4) = A110287(n-4) for n>3.
T(n,n-3) = A005010(n-3) for n>2.
T(n,n-2) = A020714(n-2) for n>1.
T(n,n-1) = A007283(n-1) for n>0.
T(n,n) = 2*A000079(n).
Extensions
Typo in first comment line fixed by Reinhard Zumkeller, Mar 07 2010
Comments