A006127 -id:A006127 - cp's OEIS frontend

A091264 Matrix defined by a(n,k) = 2^n + (k-1), read by antidiagonals.

0, 1, 1, 3, 2, 2, 7, 4, 3, 3, 15, 8, 5, 4, 4, 31, 16, 9, 6, 5, 5, 63, 32, 17, 10, 7, 6, 6, 127, 64, 33, 18, 11, 8, 7, 7, 255, 128, 65, 34, 19, 12, 9, 8, 8, 511, 256, 129, 66, 35, 20, 13, 10, 9, 9, 1023, 512, 257, 130, 67, 36, 21, 14, 11, 10, 10, 2047, 1024, 513, 258, 131, 68, 37, 22

Offset: 0

Ross La Haye, Feb 23 2004

			{0};
{1,1};
{3,2,2};
{7,4,3,3};
{15,8,5,4,4};
{31,16,9,6,5,5};
{63,32,17,10,7,6,6};
a(5,3) = 34 because 2^5 + (3-1) = 34.

Rows: a(0, k) = A001477(k), a(1, k) = A000027(k+1) etc. etc. Columns: a(n, 0) = A000225(n). a(n, 1) = A000079(n). a(n, 2) = A000051(n). a(n, 3) = A052548(n). a(n, 4) = A062709(n). Diagonals: a(n, n+3) = A052968(n+1). a(n, n+2) = A005126(n). a(n, n+1) = A006127(n). a(n, n) = A052944(n). a(n, n-1) = A083706(n-1). Also note that the sums of the antidiagonals = the partial sums of the main diagonal, i.e., a(n, n).

Flatten[ Table[ Table[ a[i, n - i], {i, n, 0, -1}], {n, 0, 11}]] (* both from Robert G. Wilson v, Feb 26 2004 *)
Table[a[n, k], {n, 0, 10}, {k, 0, 10}] // TableForm (* to view the table *)

For k > 0, a(n, k)= a(n, k-1) + 1.

a(n, k) = 2^n + (k-1).

More terms from Robert G. Wilson v, Feb 23 2004

A104795 Triangle T(n,k) = C(n,k)+1 for k

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 5, 7, 5, 1, 2, 6, 11, 11, 6, 1, 2, 7, 16, 21, 16, 7, 1, 2, 8, 22, 36, 36, 22, 8, 1, 2, 9, 29, 57, 71, 57, 29, 9, 1, 2, 10, 37, 85, 127, 127, 85, 37, 10, 1, 2, 11, 46, 121, 211, 253, 211, 121, 46, 11, 1, 2, 12, 56, 166, 331, 463, 463, 331, 166, 56, 12, 1

Offset: 0

Gary W. Adamson, Mar 26 2005

			First few rows of the triangle are:
1;
2, 1;
2, 3, 1;
2, 4, 4, 1;
2, 5, 7, 5, 1;
2, 6, 11, 11, 6, 1;
2, 7, 16, 21, 16, 7, 1;
2, 8, 22, 36, 36, 22, 8, 1;
...

Row sums are in A006127. Cf. A007318.

Edited by Ralf Stephan, Apr 05 2009

a(28) = 28 replaced by 2, 8 by Georg Fischer, Apr 09 2022

A131415 (A007318 * A000012) + (A000012 * A007318) - A007318.

Original entry on oeis.org

1, 3, 1, 6, 4, 1, 11, 10, 5, 1, 20, 21, 15, 6, 1, 37, 41, 36, 21, 7, 1, 70, 78, 77, 57, 28, 8, 1, 135, 148, 155, 134, 85, 36, 9, 1, 264, 283, 303, 289, 219, 121, 45, 10, 1, 521, 547, 586, 592, 508, 340, 166, 55, 11, 1

Offset: 0

Gary W. Adamson, Jul 08 2007

Left column = A006127: (1, 3, 6, 11, 20, 37,...). Row sums = A047859: (1, 4, 11, 27, 63,...).

			First few rows of the triangle are:
1;
3, 1;
6, 4, 1;
11, 10, 5, 1;
20, 21, 15, 6, 1;
37, 41, 36, 21, 7, 1;
...

Cf. A007318, A000012, A006127, A047859.

(A007318 * A000012) + (A000012 * A007318) - A007318 as infinite lower triangular matrices.

A137810 a(n) = 2^(2^n+n) - 1.

Original entry on oeis.org

1, 7, 63, 2047, 1048575, 137438953471, 1180591620717411303423, 43556142965880123323311949751266331066367, 29642774844752946028434172162224104410437116074403984394101141506025761187823615

Offset: 0

Ant King, Feb 12 2008

An integer is simultaneously a Mersenne number and a Woodall number if and only if it is a member of this sequence. Hence this sequence is the intersection of A000225 and A003261.

			The fourth integer which is both a Mersenne number and a Woodall number is 2047. Hence a(3)=2047 (as the offset is zero).

Wilfrid Keller, New Cullen Primes, Mathematics of Computation, Vol. 64, No. 212 (Ocober 1995), pp. 1733-1741.

Cf. A000225, A003261, A006127.

Mathematica
```
2^(2^#+#)-1 &/@Range[0,8]
```

a(n) = 2^(2^n+n)-1 = A000225(2^n+n) = A003261(2^n).

A201824 G.f.: Sum_{n>=0} log( 1/sqrt(1-2^n*x) )^n / n!.

Original entry on oeis.org

1, 1, 3, 20, 330, 15504, 2324784, 1198774720, 2214919483920, 14955617450039552, 372282884729800002816, 34307640086657221926620160, 11737947382912650038702322439680, 14950677150224267346380689021913026560, 71100479076279984636914230616119201295462400

Offset: 0

Paul D. Hanna, Dec 05 2011

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 20*x^3 + 330*x^4 + 15504*x^5 +...
where
A(x) = 1 + log(1/sqrt(1-2*x)) + log(1/sqrt(1-4*x))^2/2! + log(1/sqrt(1-8*x))^3/3! + log(1/sqrt(1-16*x))^4/4! +...

Cf. A002416, A006127, A060690.

PARI
```
{a(n)=binomial(2^(n-1)+n-1, n)}
```

{a(n)=polcoef(sum(m=0,n+1,log(1/sqrt(1-2^m*x +x^2*O(x^n)))^m/m!),n)}

a(n) = binomial(2^(n-1) + n - 1, n).

a(n) = A006127(n-1)*A060690(n-1)/n for n>0. - Hugo Pfoertner, Jul 19 2024

A340228 a(n) is the sum of the lengths of all the segments used to draw a rectangle of height 2^(n-1) and width n divided into 2^(n-1) rectangles of unit height, in turn, divided into rectangles of unit height and lengths corresponding to the parts of the compositions of n.

Original entry on oeis.org

4, 11, 27, 64, 149, 342, 775, 1736, 3849, 8458, 18443, 39948, 86029, 184334, 393231, 835600, 1769489, 3735570, 7864339, 16515092, 34603029, 72351766, 150994967, 314572824, 654311449, 1358954522, 2818572315, 5838471196, 12079595549, 24964497438, 51539607583, 106300440608

Offset: 1

Stefano Spezia, Jan 01 2021

			Illustrations for n = 1..4:
      _           _ _
     |_|         |_ _|
                 |_|_|
  a(1) = 4     a(2) = 11
    _ _ _       _ _ _ _
   |_ _ _|     |_ _ _ _|
   |_ _|_|     |_ _ _|_|
   |_|_ _|     |_|_ _ _|
   |_|_|_|     |_ _|_ _|
               |_ _|_|_|
               |_|_ _|_|
               |_|_|_ _|
               |_|_|_|_|
  a(3) = 27    a(4) = 64

Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).
Index entries for sequences related to compositions.

Cf. A000079, A001792, A006127, A011782, A045623, A053220, A188626, A215149, A228525.

Cf. A338969.

LinearRecurrence[{6,-13,12,-4},{4,11,27,64},32]

O.g.f.: x*(4 - 13*x + 13*x^2 - 3*x^3)/(1 - 3*x + 2*x^2)^2.
E.g.f.: (exp(2*x)*(3 + 6*x) + 4*x*exp(x) - 3)/4.
a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4) for n > 4.
a(n) = n + 3*(n + 1)*2^(n-2).
a(n) = A001792(n) + A188626(n).
a(n) = A045623(n) + A215149(n).
a(n) = A006127(n) + A053220(n).

cp's OEIS Frontend

A091264 Matrix defined by a(n,k) = 2^n + (k-1), read by antidiagonals.

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A104795 Triangle T(n,k) = C(n,k)+1 for k

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A131415 (A007318 * A000012) + (A000012 * A007318) - A007318.

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A137810 a(n) = 2^(2^n+n) - 1.

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A201824 G.f.: Sum_{n>=0} log( 1/sqrt(1-2^n*x) )^n / n!.

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A340228 a(n) is the sum of the lengths of all the segments used to draw a rectangle of height 2^(n-1) and width n divided into 2^(n-1) rectangles of unit height, in turn, divided into rectangles of unit height and lengths corresponding to the parts of the compositions of n.

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