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
A140504 a(n) = 2^n + 4.
5, 6, 8, 12, 20, 36, 68, 132, 260, 516, 1028, 2052, 4100, 8196, 16388, 32772, 65540, 131076, 262148, 524292, 1048580, 2097156, 4194308, 8388612, 16777220, 33554436, 67108868, 134217732, 268435460, 536870916, 1073741828
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (3,-2).
Crossrefs
Programs
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Magma
[2^n +4: n in [0..30]]; // G. C. Greubel, Jul 08 2021
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Mathematica
Table[2^n + 4, {n, 0, 30}] (* Stefan Steinerberger, Aug 04 2008 *) LinearRecurrence[{3,-2}, {5,6}, 40] (* Harvey P. Dale, May 26 2018 *) -
PARI
a(n)=2^n+4 \\ Charles R Greathouse IV, Dec 21 2011
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Sage
[2^n + 4 for n in range(0,31)] # Zerinvary Lajos, May 31 2009
Formula
G.f.: (5 - 9*x)/((1 - x)*(1 - 2*x)). - Jaume Oliver Lafont, Aug 30 2009
a(n) = 2*a(n-1) - 4 with a(0) = 5. - Vincenzo Librandi, Nov 24 2009
From Reinhard Zumkeller, Feb 28 2010: (Start)
a(n) = A173786(n,2) for n > 1.
From G. C. Greubel, Jul 08 2021: (Start)
a(n) = m*(2^(n-2) + 1), with m = 4.
E.g.f.: exp(2*x) + 4*exp(x). (End)
Extensions
More terms from Stefan Steinerberger, Aug 04 2008
A175161 a(n) = 8*(2^n + 1).
16, 24, 40, 72, 136, 264, 520, 1032, 2056, 4104, 8200, 16392, 32776, 65544, 131080, 262152, 524296, 1048584, 2097160, 4194312, 8388616, 16777224, 33554440, 67108872, 134217736, 268435464, 536870920, 1073741832, 2147483656, 4294967304, 8589934600, 17179869192
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (3,-2).
Crossrefs
Programs
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Magma
I:=[16,24]; [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jul 08 2021
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Mathematica
8*(2^Range[0, 40] + 1) (* G. C. Greubel, Jul 08 2021 *) LinearRecurrence[{3,-2},{16,24},40] (* Harvey P. Dale, Feb 10 2022 *)
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Sage
[8*(2^n +1) for n in (0..40)] # G. C. Greubel, Jul 08 2021
Formula
a(n) = A173786(n+3, 3).
a(n) = 3*a(n-1) -2*a(n-2) with a(0)=16, a(1)=24. - Vincenzo Librandi, Dec 28 2010
G.f.: 8*(2 - 3*x)/((1-x)*(1-2*x)). - Chai Wah Wu, Jun 20 2020
a(n) = 8 * A000051(n). - Alois P. Heinz, Jun 20 2020
E.g.f.: 8*(exp(2*x) + exp(x)). - G. C. Greubel, Jul 08 2021
A175162 a(n) = 16*(2^n + 1).
32, 48, 80, 144, 272, 528, 1040, 2064, 4112, 8208, 16400, 32784, 65552, 131088, 262160, 524304, 1048592, 2097168, 4194320, 8388624, 16777232, 33554448, 67108880, 134217744, 268435472, 536870928, 1073741840, 2147483664, 4294967312, 8589934608, 17179869200
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (3,-2).
Crossrefs
Programs
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Magma
I:=[32,48]; [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jul 08 2021
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Mathematica
16*(2^Range[0,30] +1) (* or *) LinearRecurrence[{3,-2},{32,48},30] (* Harvey P. Dale, Jun 08 2017 *) -
Sage
[16*(2^n +1) for n in (0..40)] # G. C. Greubel, Jul 08 2021
Formula
a(n) = A173786(n+4, 4).
a(n) = 3*a(n-1) - 2*a(n-2), a(0)=32, a(1)=48. - Vincenzo Librandi, Dec 28 2010
From G. C. Greubel, Jul 08 2021: (Start)
G.f.: 16*(2 - 3*x)/((1-x)*(1-2*x)).
E.g.f.: 16*(exp(2*x) + exp(x)). (End)
Comments