A052548 -id:A052548 - cp's OEIS frontend

A348925 Number of 4-sided prudent polygons of area n.

8, 16, 40, 96, 232, 560, 1336, 3176, 7480, 17528, 40776, 94336, 216976, 496432, 1130120, 2560648, 5776304, 12976112, 29036008, 64732992, 143814192, 318455632, 702983256, 1547254832, 3395989288, 7433962168, 16232357608, 35359307144, 76848753032, 166658534216

Offset: 1

Vaclav Kotesovec, following a suggestion from Anthony Guttmann, Nov 04 2021

nonn

Nicholas R. Beaton and Anthony Guttmann, Table of n, a(n) for n = 1..800
Nicholas R. Beaton, Philippe Flajolet, Tim Garoni and Anthony J. Guttmann, Some New Self-avoiding Walk and Polygon Models, Fundamenta Informaticae, vol. 117, 2012, pp. 19-33.
Nicholas R. Beaton, Philippe Flajolet and Anthony J. Guttmann, The Enumeration of Prudent Polygons by Area and its Unusual Asymptotics, arXiv:1011.6195 [math.CO], Nov 29 2010, updated Nov 04 2021.

Cf. A052548, A348838.

A369492 Triangle read by rows. An encoding of compositions of n where the first part is the largest part and the last part is not 1. The number of these compositions (the length of row n) is given by A368279.

Original entry on oeis.org

1, 0, 2, 4, 8, 10, 16, 18, 22, 32, 34, 36, 38, 42, 46, 64, 66, 68, 70, 74, 76, 78, 86, 90, 94, 128, 130, 132, 134, 136, 138, 140, 142, 146, 148, 150, 154, 156, 158, 170, 174, 182, 186, 190, 256, 258, 260, 262, 264, 266, 268, 270, 274, 276, 278, 280, 282, 284, 286

Offset: 0

Peter Luschny, Jan 25 2024

The compositions are in reverse lexicographic order (see A066099).
For n = 0 we get the empty composition, which we encode by 1.
For n = 1 we get the no composition, which we encode by 0.
The definition uses two filter conditions: (a) 'the last part of the composition is not 1' and (b) 'the first part is the largest part'. By changing condition (b) to (b'), 'the parts are nonincreasing', one obtains A002865. Like the current sequence, A002865 has offset 0; the empty composition is nonincreasing (a(0) = 1), and there is no composition of 1, which has a last part that is not 1 (a(1) = 0).

			Encoding the composition as an integer, a binary string, a Dyck path, and a list.
[ n]
[ 0]    1 | 1        |  ()            | [()]
[ 1]    0 | 0        |  .             | []
[ 2]    2 | 10       |  (())          | [2]
[ 3]    4 | 100      |  ((()))        | [3]
[ 4]    8 | 1000     |  (((())))      | [4]
[ 5]   10 | 1010     |  (())(())      | [2, 2]
[ 6]   16 | 10000    |  ((((()))))    | [5]
[ 7]   18 | 10010    |  ((()))(())    | [3, 2]
[ 8]   22 | 10110    |  (())()(())    | [2, 1, 2]
[ 9]   32 | 100000   |  (((((())))))  | [6]
[10]   34 | 100010   |  (((())))(())  | [4, 2]
[11]   36 | 100100   |  ((()))((()))  | [3, 3]
[12]   38 | 100110   |  ((()))()(())  | [3, 1, 2]
[13]   42 | 101010   |  (())(())(())  | [2, 2, 2]
[14]   46 | 101110   |  (())()()(())  | [2, 1, 1, 2]
Sequence seen as table:
  [0]   1;
  [1]   0;
  [2]   2;
  [3]   4;
  [4]   8, 10;
  [5]  16, 18, 22;
  [6]  32, 34, 36, 38, 42, 46;
  [7]  64, 66, 68, 70, 74, 76, 78, 86, 90, 94;
       ...

Peter Luschny, A SageMath notebook for A368279 and A369492, Jan. 2024

Subsequences: A000079, A052548\{3,6}, A369491.

Cf. A066099, A002865, A368279, A368579.

# See the notebook in the links section, that includes a time and space efficient algorithm to generate the compositions. Alternatively, using SageMath's generator:
def pr(bw, w, dw, c):
    print(f"{bw:3d} | {str(w).ljust(7)} | {str(dw).ljust(14)} | {c}")
def Trow(n):
    row, count = [], 0
    for c in reversed(Compositions(n)):
        if c == []:
            count = 1
            pr(1, 1, "()", "[()]")
        elif c == [1]:
            pr(0, 0, ".", "[]")
        elif c[-1] != 1:
            if all(part <= c[0] for part in c):
                w = Words([0, 1])(c.to_code())
                dw = DyckWord(sum([[1]*a + [0]*a for a in c], []))
                bw = int(str(w), 2)
                row.append(bw)
                count += 1
                pr(bw, w, dw, c)
    # print(f"For n = {n} there are {count} composition of type A369492.")
    return row
for n in range(0, 7): Trow(n)

A060802 To weigh from 1 to n, make the heaviest weight as small as possible, under the condition of using fewest pieces of different, single weights; a(n) = weight of the heaviest weight.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6, 7, 8, 6, 6, 6, 7, 7, 8, 8, 9, 9, 10, 11, 12, 13, 14, 15, 16, 10, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 18, 18, 18, 19, 19, 20, 20, 21

Offset: 1

Sen-Peng Eu, Apr 27 2001

nonn

Starting at n = 2^x (x > 2) you get: 3 entries of 2^floor(log_2(x)-2)+2, then 2 entries of each subsequent integer until you reach the halfway point between 2^x and 2^(x+1), then 1 entry of each subsequent integer until you reach 2^(x+1)-1. Proved (see link). - David Consiglio, Jr., Jan 08 2015

			a(20)=7 because every number from 1 to 20 can be obtained from {1,2,4,6,7}.

David Consiglio, Jr., Table of n, a(n) for n = 1..1024
David Consiglio, Jr., Python that quickly provides weights given the proof by Schoenfield
David Consiglio, Jr., Proof for sequence generation - Schoenfield and Consiglio

After the 8th term:
If 2^x <= n <= (2^x)+2 then a(n) = 2 ^ floor(base2log(x)-2)+2 (see A052548)
If (2^x)+2 < n and n+1 < (2^x + 2^x+1)/2 then a(n) and a(n+1) = a(n-1)+1
If (2^x+2^x+1)/2 <= n then a(n) = a(n-1)+1. - David Consiglio, Jr., Jan 08 2015

a(32)-a(1024) from David Consiglio, Jr., Jan 08 2015

A096068 Concatenated in binary representation: largest proper divisor of n and smallest prime factor of n.

Original entry on oeis.org

3, 6, 7, 10, 13, 14, 15, 18, 15, 22, 27, 26, 29, 30, 23, 34, 49, 38, 51, 42, 31, 46, 55, 50, 45, 54, 39, 58, 61, 62, 63, 66, 47, 70, 61, 74, 101, 78, 55, 82, 105, 86, 107, 90, 63, 94, 111, 98, 63, 102, 71, 106, 117, 110, 93, 114, 79, 118, 123, 122, 125, 126, 87, 130, 109

Offset: 1

Reinhard Zumkeller, Jul 21 2004

a(n)=A020639(n)*A000079((A000523(A020639(n))+1))+A020639(n);
a(A000040(n))=A000079((A000523(A000040(n))+1))+A000040(n);
a(A005843(n))=A005843(A005408(n)); a(A000225(n))=A000225(n+1); a(A000079(n))=A052548(n+1).

			n=99: smallest factor = 3->'11' and greatest proper divisor =
99/3 = 33->'100001': '10001'&'11'='10000111'->a(99)=135.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

cbr[n_]:=Module[{sf=FactorInteger[n][[1,1]]},FromDigits[ Join[ IntegerDigits[ n/sf,2],IntegerDigits[sf,2]],2]]; Array[cbr,70] (* Harvey P. Dale, Jul 06 2021 *)

A134351 Binomial transform of [1, 5, -1, 5, -1, 5, ...]. Inverse binomial transform of A134350.

Original entry on oeis.org

1, 6, 10, 18, 34, 66, 130, 258, 514, 1026, 2050, 4098, 8194, 16386, 32770, 65538, 131074, 262146, 524290, 1048578, 2097154, 4194306, 8388610, 16777218, 33554434, 67108866, 134217730, 268435458, 536870914, 1073741826, 2147483650

Offset: 1

Gary W. Adamson, Oct 21 2007

nonn

			a(4) = 18 = (1, 3, 3, 1) dot (1, 5, -1, 5) = (1 + 15 - 3 + 5).

Index entries for linear recurrences with constant coefficients, signature (3, -2).

Cf. A134350.

Essentially the same as A133140, A089985, A052548.

1,seq(2^(n+1)+2,n=1..25); # Emeric Deutsch, Oct 24 2007

a(n) = 2 + 2^(n+1) for n >= 2; a(1)=1. - Emeric Deutsch, Oct 24 2007

O.g.f.: (-1-3*x+6*x^2)/((1-x)*(-1+2*x)). - R. J. Mathar, Apr 02 2008

More terms from Emeric Deutsch, Oct 24 2007

More terms from R. J. Mathar, Apr 02 2008

A139524 Triangle T(n,k) read by rows: the coefficient of [x^k] of the polynomial 2*(x+1)^n + 2^n in row n, column k.

Original entry on oeis.org

3, 4, 2, 6, 4, 2, 10, 6, 6, 2, 18, 8, 12, 8, 2, 34, 10, 20, 20, 10, 2, 66, 12, 30, 40, 30, 12, 2, 130, 14, 42, 70, 70, 42, 14, 2, 258, 16, 56, 112, 140, 112, 56, 16, 2, 514, 18, 72, 168, 252, 252, 168, 72, 18, 2, 1026, 20, 90, 240, 420, 504, 420, 240, 90, 20, 2

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jun 09 2008

			Triangle begins as:
     3;
     4,  2;
     6,  4,  2;
    10,  6,  6,   2;
    18,  8, 12,   8,   2;
    34, 10, 20,  20,  10,   2;
    66, 12, 30,  40,  30,  12,   2;
   130, 14, 42,  70,  70,  42,  14,   2;
   258, 16, 56, 112, 140, 112,  56,  16,  2;
   514, 18, 72, 168, 252, 252, 168,  72, 18,  2;
  1026, 20, 90, 240, 420, 504, 420, 240, 90, 20, 2;

Advanced Number Theory, Harvey Cohn, Dover Books, 1963, Pages 88-89

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007283, A028326, A052548.

A139524:= func< n,k | k eq 0 select 2+2^n else 2*Binomial(n,k) >;
[A139524(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 02 2021

(* First program *)
T[n_, k_]:= SeriesCoefficient[Series[2*(1+x)^n + 2^n, {x, 0, 20}], k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, May 02 2021 *)
(* Second program *)
T[n_, k_]:= T[n, k] = If[k==0, 2 + 2^n, 2*Binomial[n, k]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 02 2021 *)

def A139524(n,k): return 2+2^n if (k==0) else 2*binomial(n,k)
flatten([[A139524(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 02 2021

Sum_{k=0..n} T(n,k) = 3*2^n = A007283(n).
From R. J. Mathar, Sep 12 2013: (Start)
T(n,0) = 2 + 2^n = A052548(n).
T(n,k) = 2*binomial(n,k) = A028326(n,k) if k>0. (End)

A174316 Sequence defined by a(0)=a(1)=a(2)=1, a(3)=2, a(4)=6 and the formula a(n)=2^(n-2)+2 for n>=5.

Original entry on oeis.org

1, 1, 1, 2, 6, 10, 18, 34, 66, 130, 258, 514, 1026, 2050, 4098, 8194, 16386, 32770, 65538, 131074, 262146, 524290, 1048578, 2097154, 4194306, 8388610, 16777218, 33554434, 67108866, 134217730, 268435458, 536870914, 1073741826, 2147483650

Offset: 0

Richard Choulet, Mar 15 2010

			a(5)=2^3+2=10. a(6)=2^4+2=18.

Cf. A052548, A174314.

taylor(((1+z^3+4*z^4-4*z^5)/(1-z))+((8*z^5)/(1-2*z)),z=0,50); v(0):=1:v(1):=1:v(2):=1:v(3):=2:v(4):=6: for n from 5 to 50 do v(n):=2^(n-2)+2:od:seq(v(n),n=0..50);

CoefficientList[Series[(1 + z + z^2 + 2*z^3 + 6*z^4 + ((2*z^5)/(1 - z)) + ((2*z)^5/(4*(1 - 2*z)))), {z, 0, 50}], z] (* Wesley Ivan Hurt, Sep 05 2025 *)

G.f.: (1+x+x^2+2*x^3+6*x^4+((2*x^5)/(1-x))+((2*x)^5/(4*(1-2*x)))).

a(n) = A052548(n-2), n>3. - R. J. Mathar, Feb 29 2016

A179282 Numbers n such that 2^n-2 and 2^n+2 are not squarefree.

Original entry on oeis.org

1, 22, 31, 64, 79, 91, 106, 111, 148, 151, 190, 205, 211, 232, 235, 271, 274, 311, 316, 331, 341, 358, 391, 400, 442, 451, 466, 484, 511, 526, 547, 551, 568, 571, 610, 613, 631, 652, 658, 667, 691, 694, 703, 736, 751, 760, 771, 778, 811, 820, 859, 862, 871, 904

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 22 2011

Most (not all) of the terms are of the form:
x^2-y^2, for x>y>0.
31=16^2-15^2,
64=10^2-6^2=17^2-15^2,
79=40^2-39^2,
91=10^2-3^2=46^2-45^2,
111=20^2-17^2=56^2-55^2,
148=11^2-3^3=38^2-36^2,
151=76^2-75^2,
205=23^2-18^2=103^2-102^2,
211=106^2-105^2.

			2^22-2=2*7^2*127*337, 2^22+2=2*3^2*43*5419.

Amiram Eldar, Table of n, a(n) for n = 1..62
David A. Corneth, More terms (this file might miss terms below 10^5, but listed terms are definetely terms)

Cf. A005117, A000918 (2^n-2), A052548 (2^n+2).

Select[Range@211,!(SquareFreeQ[2^#-2]||SquareFreeQ[2^#+2])&]

isok(n) = !issquarefree(2^n-2) && !issquarefree(2^n+2); \\ Michel Marcus, Oct 04 2019

a(14)-a(30) from D. S. McNeil, Mar 23 2011

a(31)-a(54) from Amiram Eldar, Oct 04 2019

A254148 a(n) = 92^n + 74^n + 38^n + 83^n + 29^n + 65^n + 56^n + 47^n + 10^n + 10.

Original entry on oeis.org

55, 220, 1210, 7942, 57838, 450670, 3682030, 31153342, 270739678, 2403012910, 21693441550, 198578979742, 1838853136318, 17193665419150, 162090976108270, 1538867288166142, 14698448516729758, 141129617123665390, 1361277292619082190

Offset: 0

Luciano Ancora, Jan 28 2015

This is the sequence of tenth terms of "second partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..999
Luciano Ancora, Demonstration of formulas
Index entries for linear recurrences with constant coefficients, signature (55,-1320,18150,-157773,902055,-3416930,8409500,-12753576,10628640,-3628800).

Cf. A052548, A254028, A254030, A254031, A254144, A254145, A254146, A254147.

vector(30, n, n--; 9*2^n + 7*4^n + 3*8^n + 8*3^n + 2*9^n + 6*5^n + 5*6^n + 4*7^n + 10^n + 10) \\ Colin Barker, Jan 28 2015

G.f.: -(80627040*x^9 -184920912*x^8 +175484892*x^7 -91478420*x^6 +29111445*x^5 -5902743*x^4 +766458*x^3 -61710*x^2 +2805*x -55) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)*(7*x -1)*(8*x -1)*(9*x -1)*(10*x -1)). - Colin Barker, Jan 28 2015

A334164 a(n) is the number of ON-cells in the completed n-th level of a triangular wedge in the hexagonal grid of A151723 (i.e., after 2^k >= n generations of the automaton in A151723 have been computed).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 4, 8, 2, 10, 6, 10, 4, 14, 8, 16, 2, 18, 10, 16, 8, 20, 12, 22, 6, 26, 14, 22, 8, 30, 16, 32, 2, 34, 18, 28, 16, 34, 18, 32, 14, 40, 22, 34, 16, 42, 24, 44, 10, 50, 26, 40, 20, 48, 28, 50, 14, 58, 30, 46, 16, 62, 32, 64

Offset: 1

Hartmut F. W. Hoft, Apr 17 2020

nonn

Conjecture 1: Except for a(2^n + 1) = 2, n >= 1, for odd-numbered completed levels a lower bound of the ratio of ON-cells to the length of the level is (2^n + 2)/(3*2^(n+1) + 1) with limit 1/6, determined by the subsequence of levels starting with: 13, 25, 49, 97, 193, 385, 769, 1537, 3073, ..., and the associated ON-cell counts: 4, 6, 10, 18, 34, 66, 130, 258, 514, ..., as listed in the second column of each of the two triangles below.
The ON-cell counts for the indices in each row define a line of slope 1/2.  The formula for the indices of levels in row k >= 2 is  L(k, i) = 1 + Sum_{j = 0, ..., i} 2^(k-j), 0 <= i <= k - 2, and the formula for the associated numbers of ON-cells is C(k, i) = 2 + Sum_{j = 1..i} 2^(k-1-j), 0 <= i <= k - 2:
Index of the level: L(k, i) number of ON-cells: C(k, i)
k\i  0   1   2   3   4   5   6    k/i  0   1   2   3   4   5   6
2:   5                             2:  2
3:   9  13                         3:  2   4
4:  17  25  29                     4:  2   6   8
5:  33  49  57  61                 5:  2  10  14  16
6:  65  97 113 121 125             6:  2  18  26  30  32
7: 129 193 225 241 249 253         7:  2  34  50  58  62  64
8: 257 385 449 481 497 505 509     8:  2  66  98 114 122 126 128
...
The pairs ( L(k, i), C(k, i) ), for 0 <= i <= k-2, define a line of slope 1/2 for each k >= 3.
For triangle L(k, i): column 0 is A000051(n), n >= 2; column 1 is A181565(n), n >= 3; column 2 is A083686(n), n >= 2; columns 3 is A195744(n), n >= 1; column 4 is A206371(n), n >= 2; column 5 is A196657(n), n >= 1; the bounding diagonal is A036563(n), n >= 3.
For triangle C(k, i): column 1 is A052548(n), n >= 1; column 2 is A164094(n), n >= 1.
Conjecture 2: In an even-numbered completed level 2*n the fraction of ON-cells is bounded below by (23 * 2^n - 24)/(2^(n+5) - 36) with limit 23/32, determined by the subsequence of levels starting with: 28, 92, 220, 476, 988, 2012, 4060, ... .
There are 16 numbers less than 1000 that do not occur as the number of ON-cells in a completed level through level 16384: 136, 164, 330, 334, 402, 444, 526, 570, 598, 604, 614, 714, 740, 822, 832, 878.
Sequence A334169 of even-numbered completed levels in which all cells are ON-cells is a subsequence of this sequence.

Hartmut F. W. Hoft, Diagram of ON-cell counts

Cf. A000051, A036563, A052548, A083686, A151723, A164094, A181565, A195744, A196657, A206371, A334169.

(* a169781[] and support functions are defined in A169781 and create the list nTriangle *)
a334164[n_] := Module[{k, levels={}}, a169781[n]; For[k=1, k<=n, k++, AppendTo[levels, Count[nTriangle[[k]], 1] - 2]]; levels]/;(n>=3 && IntegerQ[Log[2,n]])
a334164[64] (* sequence data *)

cp's OEIS Frontend

A348925 Number of 4-sided prudent polygons of area n.

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A369492 Triangle read by rows. An encoding of compositions of n where the first part is the largest part and the last part is not 1. The number of these compositions (the length of row n) is given by A368279.

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SageMath

A060802 To weigh from 1 to n, make the heaviest weight as small as possible, under the condition of using fewest pieces of different, single weights; a(n) = weight of the heaviest weight.

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A096068 Concatenated in binary representation: largest proper divisor of n and smallest prime factor of n.

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Mathematica

A134351 Binomial transform of [1, 5, -1, 5, -1, 5, ...]. Inverse binomial transform of A134350.

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Maple

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A139524 Triangle T(n,k) read by rows: the coefficient of [x^k] of the polynomial 2*(x+1)^n + 2^n in row n, column k.

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A174316 Sequence defined by a(0)=a(1)=a(2)=1, a(3)=2, a(4)=6 and the formula a(n)=2^(n-2)+2 for n>=5.

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A179282 Numbers n such that 2^n-2 and 2^n+2 are not squarefree.

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PARI

A254148 a(n) = 92^n + 74^n + 38^n + 83^n + 29^n + 65^n + 56^n + 47^n + 10^n + 10.