A165665 -id:A165665 - cp's OEIS frontend

A060867 a(n) = (2^n - 1)^2.

1, 9, 49, 225, 961, 3969, 16129, 65025, 261121, 1046529, 4190209, 16769025, 67092481, 268402689, 1073676289, 4294836225, 17179607041, 68718952449, 274876858369, 1099509530625, 4398042316801, 17592177655809, 70368727400449, 281474943156225, 1125899839733761

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), May 04 2001

Number of n X n matrices over GF(2) with rank 1.
Let M_2(n) be the 2 X 2 matrix M_2(n)(i,j)=i^n+j^n; then a(n)=-det(M_2(n)). - Benoit Cloitre, Apr 21 2002
Number of distinct lines through the origin in the n-dimensional lattice of side length 3. A001047 gives lines in the n-dimensional lattice of side length 2, A049691 gives lines in the 2-dimensional lattice of side length n. - Joshua Zucker, Nov 19 2003
a(n) is also the number of n-tuples with each entry chosen from the subsets of {1,2} such that the intersection of all n entries is empty. See example. This may be shown by exhibiting a bijection to a set whose cardinality is obviously (2^n-1)^2, namely the set of all pairs with each entry chosen from the 2^n-1 proper subsets of {1,..,n}, i.e., for both entries {1,..,n} is forbidden. The bijection is given by (X_1,..,X_n) |-> (Y_1,Y_2) where for each j in {1,2} and each i in {1,..,n}, i is in Y_j if and only if j is in X_i. For example, a(2)=9, because the nine pairs of subsets of {1,2} with empty intersection are: ({},{}), ({},{1}), ({},{2}), ({},{1,2}), ({1},{}), ({2},{}), ({1,2},{}), ({1},{2}), ({2},{1}). - Peter C. Heinig (algorithms(AT)gmx.de), Apr 13 2007
Partial sums of A165665. - J. M. Bergot, Dec 06 2014
Except for a(1)=4, the number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 737", based on the 5-celled von Neumann neighborhood. - Robert Price, May 23 2016
Apparently (with offset 0) also the number of active cells at state 2^n-1 of the automaton defined by "Rule 7". - Robert Price, Apr 12 2016
a(n) is the difference x-y where positive integer x has binary form of n leading ones followed by n zeros and nonnegative integer y has binary form of n leading zeros followed by n ones. For example, a(4) = (1111000-00001111)(base 2) = 240-15 = 225 = 15^2. The result follows readily by noting y=2^n-1 and x=2^(2*n)-1-y. Therefore x-y=2^(2*n)-2^(n+1)+1=(2^n-1)^2. - Dennis P. Walsh, Sep 19 2016
Also the number of dominating sets in the n-barbell graph. - Eric W. Weisstein, Jun 29 2017
For n > 1, also the number of connected dominating sets in the complete bipartite graph K_n,n. - Eric W. Weisstein, Jun 29 2017

			a(2) = 9 because there are 10 (the second element in sequence A060704) singular 2 X 2 matrices over GF(2), that have rank <= 1 of which only the zero matrix has rank zero so a(2) = 10 - 1 = 9.

Richard P. Stanley, Enumerative Combinatorics: Volume 1: Wadsworth & Brooks: 1986: p. 11.

Harry J. Smith, Table of n, a(n) for n = 1..200
Michael Baake, Franz Gähler and Uwe Grimm, Examples of substitution systems and their factors, Journal of Integer Sequences, Vol. 16 (2013), Article 13.2.14; arXiv preprint, arXiv:1211.5466 [math.DS], 2012-2013. - From _N. J. A. Sloane_, Jan 03 2013
Franck Ramaharo, A one-variable bracket polynomial for some Turk's head knots, arXiv:1807.05256 [math.CO], 2018.
Eric Weisstein's World of Mathematics, Near-Square Prime.
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A000225, A060704, A065443, A165665 (first differences)

seq((Stirling2(n+1, 2))^2, n=1..25); # Zerinvary Lajos, Dec 20 2006

(2^Range[30] - 1)^2 (* Harvey P. Dale, Sep 15 2013 *)
LinearRecurrence[{7, -14, 8}, {1, 9, 49}, 30] (* Harvey P. Dale, Sep 15 2013 *)
Table[(2^n - 1)^2, {n, 30}] (* Eric W. Weisstein, Jun 29 2017 *)

a(n) = (2^n - 1)^2; \\ Michel Marcus, Mar 11 2016

[stirling_number2(n+1,2)^2 for n in range(1,25)] # Zerinvary Lajos, Mar 14 2009

a(n) = (2^n - 1)^2 = A000225(n)^2.
a(n) = sum_{j=1..n} sum_{k=1..n} binomial(n+j,n-k). - Yalcin Aktar, Dec 28 2011
G.f.: x*(1+2*x)/((1-x)(1-2*x)(1-4*x)). a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3). - Colin Barker, Feb 03 2012
E.g.f.: (1 - 2*exp(x) + exp(3*x))*exp(x). - Ilya Gutkovskiy, May 23 2016
Sum_{n>=1} 1/a(n) = A065443. - Amiram Eldar, Nov 12 2020

Description changed to formula by Eric W. Weisstein, Jun 29 2017

A227074 A triangle formed like Pascal's triangle, but with 4^n on the borders instead of 1.

Original entry on oeis.org

1, 4, 4, 16, 8, 16, 64, 24, 24, 64, 256, 88, 48, 88, 256, 1024, 344, 136, 136, 344, 1024, 4096, 1368, 480, 272, 480, 1368, 4096, 16384, 5464, 1848, 752, 752, 1848, 5464, 16384, 65536, 21848, 7312, 2600, 1504, 2600, 7312, 21848, 65536, 262144, 87384, 29160

Offset: 0

T. D. Noe, Aug 06 2013

All rows except the zeroth are divisible by 4. Is there a closed-form formula for these numbers, like for binomial coefficients?

			Triangle begins:
  1,
  4, 4,
  16, 8, 16,
  64, 24, 24, 64,
  256, 88, 48, 88, 256,
  1024, 344, 136, 136, 344, 1024,
  4096, 1368, 480, 272, 480, 1368, 4096,
  16384, 5464, 1848, 752, 752, 1848, 5464, 16384,
  65536, 21848, 7312, 2600, 1504, 2600, 7312, 21848, 65536

T. D. Noe, Rows n = 0..50 of triangle, flattened

Cf. A007318 (Pascal's triangle), A228053 ((-1)^n on the borders).
Cf. A051601 (n on the borders), A137688 (2^n on borders).
Cf. A165665 (row sums: 3*4^n - 2*2^n), A227075 (3^n edges), A227076 (5^n edges).

t = {}; Do[r = {}; Do[If[k == 0 || k == n, m = 4^n, m = t[[n, k]] + t[[n, k + 1]]]; r = AppendTo[r, m], {k, 0, n}]; AppendTo[t, r], {n, 0, 10}]; t = Flatten[t]

A166065 Triangle, read by rows, given by [0,1,1,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 2, 0, 2, 2, 0, 4, 2, 2, 0, 8, 4, 2, 2, 0, 16, 8, 4, 2, 2, 0, 32, 16, 8, 4, 2, 2, 0, 64, 32, 16, 8, 4, 2, 2, 0, 128, 64, 32, 16, 8, 4, 2, 2, 0, 256, 128, 64, 32, 16, 8, 4, 2, 2, 0, 512, 256, 128, 64, 32, 16, 8, 4, 2, 2, 0, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 2, 0, 2048, 1024

Offset: 0

Philippe Deléham, Oct 05 2009

			Triangle begins :
1,
0,2,
0,2,2,
0,4,2,2,
0,8,4,2,2,
0,16,8,4,2,2,
0,32,16,8,4,2,2,
0,64,32,16,8,4,2,2,
0,128,64,32,16,8,4,2,2,
0,256,128,64,32,16,8,4,2,2,
0,512,256,128,64,32,16,8,4,2,2,

Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A084247(n), A000007(n), A000079(n), A001787(n+1), A166060(n), A165665(n), A083585(n) for x= -1, 0, 1, 2, 3, 4, 5 respectively. Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A040000(n), A000079(n), A095121(n), A047851(n), A047853(n), A047855(n) for x = 0, 1, 2, 3, 4, 5 respectively.

G.f.: (1-2*x+x*y)/((-1+2*x)*(x*y-1)). - R. J. Mathar, Aug 11 2015

A256255 Triangle read by rows: T(n,k) = 6*k + 1, n>=0, 0<=k<=(2^n-1).

Original entry on oeis.org

1, 1, 7, 1, 7, 13, 19, 1, 7, 13, 19, 25, 31, 37, 43, 1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103, 109, 115, 121, 127, 133, 139, 145, 151, 157, 163, 169, 175, 181, 187, 1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103

Offset: 0

Omar E. Pol, Apr 30 2015

Row n lists the first 2^n terms of A016921, n >= 0.
Row sums give A165665.
Right border gives A048488.
The sum of all terms of the first k rows gives A060867(k).
The product of the terms of the third row is equal to the Hardy-Ramanujan number: 1 * 7 * 13 * 19 = 1729.

			Triangle begins:
1;
1,7;
1,7,13,19;
1,7,13,19,25,31,37,43;
1,7,13,19,25,31,37,43,49,55,61,67,73,79,85,91;
...
Illustration of initial terms in the fourth quadrant of the square grid:
------------------------------------------------------------------------
n   a(n)             Compact diagram
------------------------------------------------------------------------
.            _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
0    1      |_|_  |_ _ _  |_ _ _ _ _ _ _  |
1    1      | |_| |_ _  | |_ _ _ _ _ _  | |
2    7      |_ _ _|_  | | |_ _ _ _ _  | | |
3    1      | | | |_| | | |_ _ _ _  | | | |
4    7      | | |_ _ _| | |_ _ _  | | | | |
5   13      | |_ _ _ _ _| |_ _  | | | | | |
6   19      |_ _ _ _ _ _ _|_  | | | | | | |
7    1      | | | | | | | |_| | | | | | | |
8    7      | | | | | | |_ _ _| | | | | | |
9   13      | | | | | |_ _ _ _ _| | | | | |
10  19      | | | | |_ _ _ _ _ _ _| | | | |
11  25      | | | |_ _ _ _ _ _ _ _ _| | | |
12  31      | | |_ _ _ _ _ _ _ _ _ _ _| | |
13  37      | |_ _ _ _ _ _ _ _ _ _ _ _ _| |
14  43      |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
a(n) is also the number of cells in the n-th region of the diagram.
For other diagrams of the same family see A241717 and A256258.

Paolo Xausa, Table of n, a(n) for n = 0..16382 (rows 0..13 of the triangle, flattened)

Cf. A000225, A001235, A016921, A048488, A060867, A165665, A241717, A256258.

With[{rows=7},Array[Range[1,6*2^#,6]&,rows,0]] (* Paolo Xausa, Sep 26 2023 *)

cp's OEIS Frontend

A060867 a(n) = (2^n - 1)^2.

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A227074 A triangle formed like Pascal's triangle, but with 4^n on the borders instead of 1.

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A166065 Triangle, read by rows, given by [0,1,1,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A256255 Triangle read by rows: T(n,k) = 6*k + 1, n>=0, 0<=k<=(2^n-1).

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