A036563 -id:A036563 - cp's OEIS frontend

A141725 a(n) = 4^(n+1) - 3.

1, 13, 61, 253, 1021, 4093, 16381, 65533, 262141, 1048573, 4194301, 16777213, 67108861, 268435453, 1073741821, 4294967293, 17179869181, 68719476733, 274877906941, 1099511627773, 4398046511101, 17592186044413, 70368744177661

Offset: 0

Paul Curtz, Sep 13 2008

Inverse binomial transform yields A003946 with A003946(1)=4 deleted. - R. J. Mathar, Sep 13 2008
Starting with n=1, binary numbers of the form 1X01 where X is an odd number of 1's. - Brad Clardy, Mar 22 2011
Column 4 of A193871. - Omar E. Pol, Aug 22 2011
The Sierpinski square curve is a representation of this sequence, where a(n) is the number squares filled by the Sierpinski (space-filling) square curve. The square footprint expands at a rate of (2^n-1)^2 (A000225)^2. The number of nodes per iteration grows at a rate of (4^n-1)/3 (A002450). See illustration in links. - John Elias, Jul 25 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
John Elias, Illustration of Initial Terms: Sierpinski Square Curve
Mattia Fregola, Elementary Cellular Automata Rule 1 generating OEIS sequence A277799, A058896, A141725, A002450
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A001045, A036563.

List([0..25], n -> 4^(n+1)-3); # Muniru A Asiru, Feb 20 2018

[4^(n+1)-3: n in [0..30]]; // Vincenzo Librandi, Aug 08 2011

a:= n-> 4^(n+1)-3: seq(a(n), n=0..25); # Muniru A Asiru, Feb 20 2018

4^(Range[2,25]-1)-3 (* or *) LinearRecurrence[{5,-4},{1,13},25] (* or *) NestList[4#+9&,1,25] (* Harvey P. Dale, Sep 25 2011 *)

a(n)=4^(n+1)-3 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 10*A001045(2*n) + A001045(2*n+1).
a(n) = 4*a(n-1) + 9 for n > 0, a(0) = 1.
a(n) = A036563(2*n+2).
G.f.: (1 + 8*x)/((1 - x)*(1 - 4*x)). - R. J. Mathar, Sep 13 2008
a(n) = 4^n - 3, with offset 1. - Omar E. Pol, Aug 22 2011
a(n) = 5*a(n-1) - 4*a(n-2) for n > 1, a(0) = 1, a(1) = 13. - Harvey P. Dale, Sep 25 2011
E.g.f.: exp(4*x) - 3*exp(x). - Elmo R. Oliveira, Nov 15 2023

Edited by N. J. A. Sloane, Sep 13 2008

More terms from R. J. Mathar, Sep 13 2008

A329644 Möbius transform of A323244, the deficiency of A156552(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 4, -1, 3, 1, 5, 1, 14, 0, 0, 1, 9, 1, 12, -5, 16, 1, 8, -5, 44, 4, 5, 1, 2, 1, 24, 12, 80, -4, -4, 1, 254, -14, 0, 1, 22, 1, 47, 7, 224, 1, 24, -13, 19, 6, 83, 1, 12, -21, 44, -14, 746, 1, 14, 1, 1360, 20, -8, 8, 9, 1, 131, 252, 24, 1, 12, 1, 3836, 13, 149, -12, 71, 1, 56, -16, 5456, 1, -21, -74, 12248, -350, -40, 1

Offset: 1

Antti Karttunen, Nov 21 2019

sign

The first eleven zeros occur at n = 1, 15, 16, 40, 96, 119, 120, 160, 893, 2464, 6731. There are 3091 negative terms among the first 10000 terms.
Applying this function to the divisors of the first four terms of A324201 reveals the following pattern:
------------------------------------------------------------------------------------
A324201(n) divisors                             a(n) applied       Sum of positive
                                                 to each:          terms, A329610
        9: [1, 3, 9]                         -> [0, 1, -1]                     1
      125: [1, 5, 25, 125]                   -> [0, 1, -5, 4]                  5
   161051: [1, 11, 121, 1331, 14641, 161051] -> [0, 1, -29, 4, -240, 264]    269
410338673: [1, 17, 289, 4913, 83521, 1419857, 24137569, 410338673]
                             -> [0, 1, -125, 4, -1008, 1032, -5048, 5144]   6181
The positive and negative terms seem to alternate, and the fourth term (from case n=125 onward) is always 4. See also array A329637.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A008683, A036563, A156552, A297112, A297113, A323243, A323244, A324543, A329610, A329637, A329638, A329639, A329645, A329646.

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A323244(n) = if(1==n, 0, my(k=A156552(n)); (2*k)-sigma(k));
A329644(n) = sumdiv(n,d,moebius(n/d)*A323244(d));

a(n) = Sum_{d|n} A008683(n/d) * A323244(d).
a(n) = Sum_{d|n} A008683(n/d) * (2*A156552(d) - A323243(d)).
a(1) = 0; for n > 1, a(n) = 2*A297112(n) - A324543(n) = 2^A297113(n) - A324543(n).
a(n) = A329642(n) - A329643(n).
For all n >= 1, a(A000040(n)^2) = A323244(A000040(n)^2)-1 = -A036563(n).
For all primes p, a(p^3) = A323244(p^3) - A323244(p^2) = 4.

A263017 n is the a(n)-th positive integer having its binary weight.

Original entry on oeis.org

1, 2, 1, 3, 2, 3, 1, 4, 4, 5, 2, 6, 3, 4, 1, 5, 7, 8, 5, 9, 6, 7, 2, 10, 8, 9, 3, 10, 4, 5, 1, 6, 11, 12, 11, 13, 12, 13, 6, 14, 14, 15, 7, 16, 8, 9, 2, 15, 17, 18, 10, 19, 11, 12, 3, 20, 13, 14, 4, 15, 5, 6, 1, 7, 16, 17, 21, 18, 22, 23, 16, 19, 24, 25, 17

Offset: 1

Paul Tek, Oct 07 2015

Binary weight is given by A000120.
a(2^k) = k+1 for any k>=0.
a(2^k-1) = 1 for any k>0.
a(A057168(k)) = a(k)+1 for any k>0.
a(A036563(k+1)) = k for any k>0.
Ordinal transform of A000120. - Alois P. Heinz, Dec 23 2018

			The numbers with binary weight 3 are: 7, 11, 13, 14, 19, ...
Hence: a(7)=1, a(11)=2, a(13)=3, a(14)=4, a(19)=5, ...
And more generally: a(A014311(k))=k for any k>0.

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, PERL program for this sequence

One more than A068076.
Cf. A000120, A036563, A057168, A254524, A286478, A286552, A286554, A286558.
Cf. A263109, A263110.

import Data.IntMap (empty, findWithDefault, insert)
a263017 n = a263017_list !! (n-1)
a263017_list = f 1 empty where
   f x m = y : f (x + 1) (insert h (y + 1) m) where
           y = findWithDefault 1 h m
           h = a000120 x
-- Reinhard Zumkeller, Oct 09 2015

a:= proc() option remember; local a, b, t; b, a:=
      proc() 0 end, proc(n) option remember; a(n-1);
        t:= add(i, i=convert(n, base, 2)); b(t):= b(t)+1
      end; a(0):=0; a
    end():
seq(a(n), n=1..120);  # Alois P. Heinz, Dec 23 2018

Perl
```
# See Links section.
```

from math import comb
def A263017(n):
    c, k = 1, 0
    for i,j in enumerate(bin(n)[-1:1:-1]):
        if j == '1':
            k += 1
            c += comb(i,k)
    return c # Chai Wah Wu, Mar 01 2023

a(n) = 1 + A068076(n). - Antti Karttunen, May 22 2017

A081118 Triangle of first n numbers per row having exactly n 1's in binary representation.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 23, 27, 29, 31, 47, 55, 59, 61, 63, 95, 111, 119, 123, 125, 127, 191, 223, 239, 247, 251, 253, 255, 383, 447, 479, 495, 503, 507, 509, 511, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1023, 1535, 1791, 1919, 1983, 2015, 2031, 2039, 2043

Offset: 1

Reinhard Zumkeller, Mar 06 2003

T(n,n) = A036563(n+1) = 2^(n+1) - 3.

Numbers of the form 2^t - 2^k - 1, 1 <= k < t.

			Triangle begins:
.......... 1 ......... ................ 1
........ 3...5 ....... .............. 11 101
...... 7..11..13 ..... .......... 111 1011 1101
... 15..23..27..29 ... ...... 1111 10111 11011 11101
. 31..47..55..59..61 . . 11111 101111 110111 111011 111101.

Reinhard Zumkeller, Rows n=1..150 of triangle, flattened

Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691, A038461, A038462, A038463.
Cf. A181741 (primes), A208083, subsequence of A089633.
Cf. A131094.

a081118 n k = a081118_tabl !! (n-1) !! (k-1)
a081118_row n = a081118_tabl !! (n-1)
a081118_tabl  = iterate
   (\row -> (map ((+ 1) . (* 2)) row) ++ [4 * (head row) + 1]) [1]
a081118_list = concat a081118_tabl
-- Reinhard Zumkeller, Feb 23 2012

Table[2^(n+1)-2^(n-k+1)-1,{n,10},{k,n}]//Flatten (* Harvey P. Dale, Apr 09 2020 *)

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

a(n) = (2^A002260(n)-1)*2^A004736(n)-1; a(n)=(2^i-1)*2^j-1, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Apr 04 2013

A152187 a(n) = 3a(n-1) + 5a(n-2), with a(0)=1, a(1)=5.

Original entry on oeis.org

1, 5, 20, 85, 355, 1490, 6245, 26185, 109780, 460265, 1929695, 8090410, 33919705, 142211165, 596232020, 2499751885, 10480415755, 43940006690, 184222098845, 772366329985, 3238209484180, 13576460102465, 56920427728295

Offset: 0

Philippe Deléham, Nov 28 2008

Unsigned version of A152185.
From Johannes W. Meijer, Aug 01 2010: (Start)
The a(n) represent the number of n-move routes of a fairy chess piece starting in a given side square (m = 2, 4, 6 and 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.
The sequence above corresponds to 24 red king vectors, i.e., A[5] vectors, with decimal values 27, 30, 51, 54, 57, 60, 90, 114, 120, 147, 150, 153, 156, 177, 180, 210, 216, 240, 282, 306, 312, 402, 408 and 432. These vectors lead for the corner squares to A015523 and for the central square to A179606.
This sequence belongs to a family of sequences with g.f. (1+2*x)/(1 - 3*x - k*x^2). Red king sequences that are members of this family are A007483 (k=2), A108981 (k=4), A152187 (k=5; this sequence), A154964 (k=6), A179602 (k=7) and A179598 (k=8). We observe that there is no red king sequence for k=3. Other members of this family are A036563 (k=-2), A054486 (k=-1), A084244 (k=0), A108300 (k=1) and A000351 (k=10).
Inverse binomial transform of A015449 (without the first leading 1).
(End)

Index entries for linear recurrences with constant coefficients, signature (3, 5).

Cf. A015523, A072263, A072264, A179606, A197189.

LinearRecurrence[{3,5},{1,5},40] (* Harvey P. Dale, May 03 2013 *)

G.f.: (1+2*x)/(1 - 3*x - 5*x^2).
Lim_{k->infinity} a(n+k)/a(k) = (A072263(n) + A015523(n)*sqrt(29))/2. - Johannes W. Meijer, Aug 01 2010
G.f.: G(0)*(1+2*x)/(2-3*x), where G(k) = 1 + 1/(1 - x*(29*k-9)/(x*(29*k+20) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013

A185346 a(n) = 2^n - 9.

Original entry on oeis.org

-8, -7, -5, -1, 7, 23, 55, 119, 247, 503, 1015, 2039, 4087, 8183, 16375, 32759, 65527, 131063, 262135, 524279, 1048567, 2097143, 4194295, 8388599, 16777207, 33554423, 67108855, 134217719, 268435447, 536870903, 1073741815, 2147483639, 4294967287, 8589934583

Offset: 0

Andreas Rieber, Dec 04 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000225, A036563, A172252 (essentially the same).

A185346:=n->2^n-9: seq(A185346(n), n=0..50); # Wesley Ivan Hurt, Jun 28 2017

Table[2^n - 9, {n, 0, 40}] (* T. D. Noe, Dec 04 2012 *)

a(n)=2^n-9 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: ( -8+17*x ) / ( (2*x-1)*(x-1) ). - R. J. Mathar, Dec 17 2012
E.g.f.: exp(2*x) - 9*exp(x). - G. C. Greubel, Jun 28 2017
From Elmo R. Oliveira, Nov 08 2023: (Start)
a(n) = 2*a(n-1) + 9 with a(0) = -8.
a(n) = 3*a(n-1) - 2*a(n-2) for n > 1. (End)

A272644 Triangle read by rows: T(n,m) = Sum_{i=0..m} Stirling2(m+1,i+1)(-1)^(m-i)i^(n-m)*i!, for n >= 2, m = 1..n-1.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 13, 13, 1, 1, 29, 73, 29, 1, 1, 61, 301, 301, 61, 1, 1, 125, 1081, 2069, 1081, 125, 1, 1, 253, 3613, 11581, 11581, 3613, 253, 1, 1, 509, 11593, 57749, 95401, 57749, 11593, 509, 1, 1, 1021, 36301, 268381, 673261, 673261, 268381, 36301, 1021, 1

Offset: 2

N. J. A. Sloane, May 07 2016

Gives number of bitriangular permutations. Could be prefixed with row 0 containing a single 1. - N. J. A. Sloane, Jan 10 2018

			Triangle begins:
n\m  [1]     [2]     [3]     [4]     [5]     [6]     [7]     [8]
[2]  1;
[3]  1,      1;
[4]  1,      5,      1;
[5]  1,     13,     13,      1;
[6]  1,     29,     73,     29,      1;
[7]  1,     61,    301,    301,     61,      1;
[8]  1,    125,   1081,   2069,   1081,    125,      1;
[9]  1,    253,   3613,  11581,  11581,   3613,    253,      1;
...

Gheorghe Coserea, Rows n = 2..101, flattened
F. Alayont and N. Krzywonos, Rook Polynomials in Three and Higher Dimensions, 2012.
Beáta Bényi, A Bijection for the Boolean Numbers of Ferrers Graphs, Graphs and Combinatorics (2022) Vol. 38, No. 10.
Beata Bényi and Peter Hajnal, Combinatorial properties of poly-Bernoulli relatives, arXiv preprint arXiv:1602.08684 [math.CO], 2016. See D_{n,k}.
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259-268. The array is on page 267.
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. [Annotated scanned copy]
D. E. Knuth, Parades and poly-Bernoulli bijections, Mar 31 2024. See (16.2).
D. E. Knuth, Notes on four arrays of numbers arising from the enumeration of CRC constraints and min-and-max-closed constraints, May 06 2024. Mentions this sequence.
J. Riordan, Letter to N. J. A. Sloane, Dec. 1976.

Column 2 is A036563.
Largest term in each row gives A272645.
Second diagonal from the right is 2^i - 3.
Third diagonal from the right edge is A006230.
T(2n,n) gives A048144.
For row sums see A297195.
Cf. A008277, A001469, A371761.

A272644 := proc(n,m)
    add(combinat[stirling2](m+1,i+1)*(-1)^(m-i)*i^(n-m)*i!,i=0..m) ;
end proc:
seq(seq(A272644(n,m),m=1..n-1),n=2..10) ; # R. J. Mathar, Mar 04 2018

Table[Sum[StirlingS2[m + 1, i + 1] (-1)^(m - i) i^(n - m) i!, {i, 0, m} ], {n, 11}, {m, n - 1}] /. {} -> {0} // Flatten  (* Michael De Vlieger, May 19 2016 *)

A(n,m) = sum(i=0, m, stirling(m+1, i+1, 2) * (-1)^((m-i)%2) * i^(n - m) * i!);
concat(vector(10, n, vector(n, m, A(n+1, m))))  \\ Gheorghe Coserea, May 16 2016

T(n,m) = Sum_{i=0..m} Stirling2(m+1, i+1)*(-1)^(m-i)*i^(n-m)*i!, for n>=2, m=1..n-1, where Stirling2(n,k) is defined by A008277.

A001469(n+1) = Sum_{m=1..2*n-1} (-1)^(m-1)*T(2*n,m). - Gheorghe Coserea, May 18 2016

More terms from Gheorghe Coserea, May 16 2016

A048483 Array read by antidiagonals: T(k,n) = (k+1)2^n - k.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 7, 4, 1, 16, 15, 10, 5, 1, 32, 31, 22, 13, 6, 1, 64, 63, 46, 29, 16, 7, 1, 128, 127, 94, 61, 36, 19, 8, 1, 256, 255, 190, 125, 76, 43, 22, 9, 1, 512, 511, 382, 253, 156, 91, 50, 25, 10, 1, 1024, 1023, 766, 509, 316, 187

Offset: 0

Clark Kimberling

n-th difference of (T(k,n),T(k,n-1),...,T(k,0)) is k+1, for n=1,2,3,...; k=0,1,2,...

			1 2 4 8 16 32 ...
1 3 7 15 31 63 ...
1 4 10 22 46 94 ...
1 5 13 29 61 125 ...
1 6 16 36 76 156 ...

Rows are A000079 (k=0), A000225 (k=1), A033484 (k=2), A036563 (k=3), A048487 (k=4), A048488 (k=5), A048489 (k=6), A048490 (k=7), A048491 (k=8).

Main diagonal is A048493. Cf. A048494.

G.f.: (1-x+kx)/[(1-x)(1-2x)]. E.g.f.: (k+1)*exp(2x) - k*exp(x).

Recurrences: T(k, n) = 2T(k, n-1)+k = T(k-1, n)+2^n-1, T(k, 0) = 1.

Edited by Ralf Stephan, Feb 05 2004

A067576 Array T(i,j) read by downward antidiagonals, where T(i,j) is the j-th term whose binary expansion has i 1's.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 6, 11, 15, 16, 9, 13, 23, 31, 32, 10, 14, 27, 47, 63, 64, 12, 19, 29, 55, 95, 127, 128, 17, 21, 30, 59, 111, 191, 255, 256, 18, 22, 39, 61, 119, 223, 383, 511, 512, 20, 25, 43, 62, 123, 239, 447, 767, 1023, 1024, 24, 26, 45, 79, 125, 247, 479, 895, 1535, 2047

Offset: 1

Robert G. Wilson v, Jan 30 2002

This is a permutation of the positive integers; the inverse permutation is A356419. - Jianing Song, Aug 06 2022

			Array begins:
        j=1  j=2  j=3  j=4  j=5  j=6
  i=1:    1,   2,   4,   8,  16,  32, ...
  i=2:    3,   5,   6,   9,  10,  12, ...
  i=3:    7,  11,  13,  14,  19,  21, ...
  i=4:   15,  23,  27,  29,  30,  39, ...
  i=5:   31,  47,  55,  59,  61,  62, ...
  i=6:   63,  95, 111, 119, 123, 125, ...

Ivan Neretin, Table of n, a(n) for n = 1..8001 (126 antidiagonals)
Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A356419.
T(n,n) gives A036563(n+1).
The antidiagonals are read in the opposite direction from those in A066884.
Antidiagonal sums give A361074.
Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691.

a = {}; Do[ a = Append[a, Last[ Take[ Select[ Range[2^13], Count[ IntegerDigits[ #, 2], 1] == j & ], i - j]]], {i, 2, 12}, {j, 1, i - 1} ]; a

A123203 a(n) = 2^(n+1) - 3*n.

Original entry on oeis.org

1, 2, 7, 20, 49, 110, 235, 488, 997, 2018, 4063, 8156, 16345, 32726, 65491, 131024, 262093, 524234, 1048519, 2097092, 4194241, 8388542, 16777147, 33554360, 67108789, 134217650, 268435375, 536870828, 1073741737, 2147483558

Offset: 1

Gary W. Adamson, Jun 13 2007

An elephant sequence, see A175654. For the corner squares just one A[5] vector, with decimal value 186, leads to this sequence. For the central square this vector leads to the companion sequence A036563. - Johannes W. Meijer, Aug 15 2010

			a(4) = 20, row sums of 4th row of triangle A131062: (1, 9, 9, 1).
a(4) = 20 = (1, 3, 3, 1) dot (1, 1, 4, 4) = (1 + 3 + 12 + 4).

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Joseph Breen and Emma Copeland, Non-orientable Nurikabe, arXiv:2506.12612 [math.CO], 2025. See pp. 1, 4.
Tamas Lengyel, On p-adic properties of the Stirling numbers of the first kind, Journal of Number Theory, 148 (2015) 73-94.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000079, A000295, A109128, A131060, A131061, A131063, A131064, A131065, A131066.

[2^(n+1) -3*n: n in [1..40]]; // G. C. Greubel, Sep 14 2024

Table[2^(n+1) - 3*n, {n,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *)
LinearRecurrence[{4,-5,2},{1,2,7},40] (* Harvey P. Dale, Mar 30 2024 *)

def A123203(n): return 2^(n+1) -3*n
[A123203(n) for n in range(1,41)] # G. C. Greubel, Sep 14 2024

Binomial transform of [1, 1, 4, 4, 4, ...].
Equals row sums of triangle A131061.
From Johannes W. Meijer, Aug 15 2010; corrected by Colin Barker, Jul 28 2012: (Start)
a(n) = 2^(1+n) - 3*n.
a(n) = 3*A000295(n-1) + A000079(n-1).
(End)
G.f.: x*(1 - 2*x + 4*x^2)/((1-x)^2*(1-2*x)). - Colin Barker, Jul 28 2012
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Colin Barker, Jul 29 2012
E.g.f.: 2*exp(2*x) - 3*x*exp(x) - 2. - G. C. Greubel, Sep 14 2024

More terms from Vladimir Joseph Stephan Orlovsky, Nov 15 2008

Title changed by G. C. Greubel, Sep 14 2024

cp's OEIS Frontend

A141725 a(n) = 4^(n+1) - 3.

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A329644 Möbius transform of A323244, the deficiency of A156552(n).

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A263017 n is the a(n)-th positive integer having its binary weight.

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Haskell

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Perl

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Formula

A081118 Triangle of first n numbers per row having exactly n 1's in binary representation.

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Haskell

Mathematica

Formula

A152187 a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=1, a(1)=5.

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Mathematica

Formula

A185346 a(n) = 2^n - 9.

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Maple

Mathematica

PARI

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A272644 Triangle read by rows: T(n,m) = Sum_{i=0..m} Stirling2(m+1,i+1)*(-1)^(m-i)*i^(n-m)*i!, for n >= 2, m = 1..n-1.

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A152187 a(n) = 3a(n-1) + 5a(n-2), with a(0)=1, a(1)=5.

A272644 Triangle read by rows: T(n,m) = Sum_{i=0..m} Stirling2(m+1,i+1)(-1)^(m-i)i^(n-m)*i!, for n >= 2, m = 1..n-1.