A001792 -id:A001792 - cp's OEIS frontend

A228367 n-th element of the ruler function plus the highest power of 2 dividing n.

2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 21, 2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 38, 2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 21, 2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 71, 2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 21, 2, 4, 2, 7

Offset: 1

Omar E. Pol, Aug 22 2013

a(n) is also the length of the n-th pair of orthogonal line segments in a diagram of compositions, see example.
a(n) is also the largest part plus the number of parts of the n-th region of the mentioned diagram (if the axes both "x" and "y" are included in the diagram).
a(n) is also the number of toothpicks added at n-th stage to the structure of A228366. Essentially the first differences of A228366.
The equivalent sequence for partitions is A207779.

			Illustration of initial terms (n = 1..16) using a diagram of compositions in which A001511(n) is the length of the horizontal line segment in row n and A006519(n) is the length of the vertical line segment ending in row n. Hence a(n) is the length of the n-th pair of orthogonal line segments. Also counting both the x-axis and the y-axis we have that A001511(n) is also the largest part of the n-th region of the diagram and A006519(n) is also the number of parts of the n-th region of the diagram, see below.
---------------------------------------------------------
.                Diagram of
n   A001511(n)  compositions   A006519(n)    a(n)
---------------------------------------------------------
1       1        _| | | | |        1          2
2       2        _ _| | | |        2          4
3       1        _|   | | |        1          2
4       3        _ _ _| | |        4          7
5       1        _| |   | |        1          2
6       2        _ _|   | |        2          4
7       1        _|     | |        1          2
8       4        _ _ _ _| |        8         12
9       1        _| | |   |        1          2
10      2        _ _| |   |        2          4
11      1        _|   |   |        1          2
12      3        _ _ _|   |        4          7
13      1        _| |     |        1          2
14      2        _ _|     |        2          4
15      1        _|       |        1          2
16      5        _ _ _ _ _|       16         21
...
If written as an irregular triangle the sequence begins:
  2;
  4;
  2, 7;
  2, 4, 2, 12;
  2, 4, 2, 7, 2, 4, 2, 21;
  2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 38;
  ...
Row lengths is A011782. Right border gives A005126.
Counting both the x-axis and the y-axis we have that A038712(n) is the area (or the number of cells) of the n-th region of the diagram. Note that adding only the x-axis to the diagram we have a tree. - _Omar E. Pol_, Nov 07 2018

Antti Karttunen, Table of n, a(n) for n = 1..16383
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A001511, A001792, A005126, A006519, A011782, A038712, A139250, A139251, A207779, A228366.

Array[1 + # + 2^# &[IntegerExponent[#, 2]] &, 84] (* Michael De Vlieger, Nov 06 2018 *)

A228367(n) = (1 + valuation(n,2) + 2^valuation(n,2)); \\ Antti Karttunen, Nov 06 2018

def A228367(n): return (m:=n&-n)+m.bit_length() # Chai Wah Wu, Jul 14 2022

a(n) = A001511(n) + A006519(n).

A053539 a(n) = n * 8^(n-1).

Original entry on oeis.org

0, 1, 16, 192, 2048, 20480, 196608, 1835008, 16777216, 150994944, 1342177280, 11811160064, 103079215104, 893353197568, 7696581394432, 65970697666560, 562949953421312, 4785074604081152, 40532396646334464, 342273571680157696, 2882303761517117440, 24211351596743786496

Offset: 0

Barry E. Williams, Jan 15 2000

The Szeged index of the hypercube Q_n (see the Ashrafi et al. reference, p. 45, last line). - Emeric Deutsch, Aug 06 2014

For n > 3, 2*a(n) is the number of spanning trees in a superprism on 2*n vertices (see Bogdanowicz). - Stefano Spezia, May 05 2024

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
A. R. Ashrafi, B. Manoochehrian, and H. Yousefi-Azari, On Szeged polynomial of a graph, Bull. Iranian Math. Soc., 33, 2007, 37-46. - _Emeric Deutsch_, Aug 06 2014
Zbigniew R. Bogdanowicz, The number of spanning trees in a superprism, Discrete Math. Lett. 13 (2024) 66-73. See Theorem 3.1.
Frank Ellermann, Illustration of binomial transforms.
Index entries for linear recurrences with constant coefficients, signature (16,-64).

Binomial transform of A027473.

Cf. A001787, A053464, A053469, A053540.

List([0..20], n-> n*8^(n-1)); # G. C. Greubel, May 16 2019

[n*8^(n-1): n in [0..20]]; // Vincenzo Librandi, Feb 09 2011

a := proc(n) option remember; if n<2 then n else 16*a(n-1)-64*a(n-2) end if end proc: seq(a(n), n = 0 .. 20); # Emeric Deutsch, Aug 06 2014

Table[n 8^(n-1),{n,0,20}] (* or *) LinearRecurrence[{16,-64},{0,1},20] (* Harvey P. Dale, Feb 01 2017 *)

a(n) = n*8^(n-1); \\ Joerg Arndt, Aug 07 2014

[n*8^(n-1) for n in (0..20)] # G. C. Greubel, May 16 2019

a(n) = 16*a(n-1) - 64*a(n-2), with a(0)=0, a(1)=1. - Emeric Deutsch, Aug 06 2014
From G. C. Greubel, May 16 2019: (Start)
G.f.: x/(1-8*x)^2.
E.g.f.: x*exp(8*x). (End)
From Amiram Eldar, Oct 28 2020: (Start)
Sum_{n>=1} 1/a(n) = 8*log(8/7).
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(9/8). (End)

Offset corrected and name edited by Emeric Deutsch, Aug 06 2014

A059450 Triangle read by rows: T(n,k) = Sum_{j=0..k-1} T(n,j) + Sum_{j=1..n-k} T(n-j,k), with T(0,0)=1 and T(n,k) = 0 for k > n.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 4, 8, 17, 29, 8, 20, 50, 107, 185, 16, 48, 136, 336, 721, 1257, 32, 112, 352, 968, 2370, 5091, 8925, 64, 256, 880, 2640, 7116, 17304, 37185, 65445, 128, 576, 2144, 6928, 20168, 53596, 129650, 278635, 491825, 256, 1280, 5120, 17664, 54880

Offset: 0

N. J. A. Sloane, Sep 16 2003

G.f. A(x,y) satisfies 0 = -(1-x)^2 + (1-x)(1-4x+3xy)A + 2x(1-2x-2y+3xy)A^2. G.f.: (1-x)(-(1-4x+3xy) + sqrt((1-xy)(1-9xy)))/(4x(1-2x-2y+3xy)) = 2(1-x)/(1-4x+3xy+sqrt((1-xy)(1-9xy))). - Michael Somos, Mar 06 2004
T(n,k) = number of below-diagonal lattice paths from (0,0) to (n,k) consisting of steps (k,0) (k=1,2,...) and (0,k) (k=1,2,...). Example: T(2,1)=3 because we have (1,0)(1,0)(0,1), (2,0)(0,1) and (1,0)(0,1)(1,0). - Emeric Deutsch, Mar 19 2004
T(n,k) is odd if and only if (n,k) = (0,0), k = n > 0, or k + 1 = n > 0. - Peter Kagey, Apr 20 2020

			1;
1,  1;
2,  3,  5;
4,  8, 17,  29;
8, 20, 50, 107, 185;

Wen-jin Woan, Diagonal lattice paths, Congressus Numerantium, 151, 2001, 173-178.

Peter Kagey, Table of n, a(n) for n = 0..10011 (141 rows flattened, first 50 rows from G. C. Greubel)
C. Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003), 13-28.

Columns include A000079, A001792 (I guess), A086866, A059231. Rows sums give A086871.

A059231(n) = T(n, n).

l := 1:a[0,0] := 1:b[l] := 1:T := (n,k)->sum(a[n,j],j=0..k-1)+sum(a[n-j,k],j=1..n-k): for n from 1 to 15 do for k from 0 to n do a[n,k] := T(n,k):l := l+1:b[l] := a[n,k]: od:od:seq(b[w],w=1..l); # Sascha Kurz
# alternative
A059450 := proc(n,k)
    option remember;
    local j ;
    if k =0 and n= 0 then
        1;
    elif k > n or k < 0 then
        0 ;
    else
        add( procname(n,j),j=0..k-1) + add(procname(n-j,k),j=1..n-k) ;
    end if;
end proc:
seq(seq(A059450(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Mar 25 2024

t[0, 0] = 1; t[n_, k_] /; k > n = 0; t[n_, k_] := t[n, k] = Sum[t[n, j], {j, 0, k-1}] + Sum[t[n-j, k], {j, 1, n-k}]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 08 2014 *)

T(n,k)=if(k<0||k>n,0,polcoeff(polcoeff(2*(1-x)/((1-4*x+3*x*y)+sqrt((1-x*y)*(1-9*x*y)+x^2*O(x^n))),n),k)) /* Michael Somos, Mar 06 2004 */

T(n,k)=local(A,t);if(k<0||k>n,0,A=matrix(n+1,n+1);A[1,1]=1;for(m=1,n,t=0;for(j=0,m,t+=(A[m+1,j+1]=t+sum(i=1,m-j,A[m-i+1,j+1]))));A[n+1,k+1]) /* Michael Somos, Mar 06 2004 */

T(n,k)=if(k<0||k>n,0,(n==0)+sum(j=0,k-1,T(n,j))+sum(j=1,n-k,T(n-j,k))) /* Michael Somos, Mar 06 2004 */

More terms from Ray Chandler, Sep 17 2003

A066637 Total number of elements in all factorizations of n with all factors > 1.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 6, 3, 3, 1, 8, 1, 3, 3, 12, 1, 8, 1, 8, 3, 3, 1, 17, 3, 3, 6, 8, 1, 10, 1, 20, 3, 3, 3, 22, 1, 3, 3, 17, 1, 10, 1, 8, 8, 3, 1, 34, 3, 8, 3, 8, 1, 17, 3, 17, 3, 3, 1, 27, 1, 3, 8, 35, 3, 10, 1, 8, 3, 10, 1, 46, 1, 3, 8, 8, 3, 10, 1, 34, 12, 3, 1, 27, 3, 3, 3, 17, 1, 27, 3, 8, 3, 3, 3

Offset: 1

Amarnath Murthy, Dec 28 2001

nonn

From Gus Wiseman, Apr 18 2021: (Start)
Number of ways to choose a factor index or position in a factorization of n. The version selecting a factor value is A339564. For example, the factorizations of n = 2, 4, 8, 12, 16, 24, 30 with a selected position (in parentheses) are:
  ((2))  ((4))    ((8))      ((12))     ((16))       ((24))       ((30))
         ((2)*2)  ((2)*4)    ((2)*6)    ((2)*8)      ((3)*8)      ((5)*6)
         (2*(2))  (2*(4))    (2*(6))    (2*(8))      (3*(8))      (5*(6))
                  ((2)*2*2)  ((3)*4)    ((4)*4)      ((4)*6)      ((2)*15)
                  (2*(2)*2)  (3*(4))    (4*(4))      (4*(6))      (2*(15))
                  (2*2*(2))  ((2)*2*3)  ((2)*2*4)    ((2)*12)     ((3)*10)
                             (2*(2)*3)  (2*(2)*4)    (2*(12))     (3*(10))
                             (2*2*(3))  (2*2*(4))    ((2)*2*6)    ((2)*3*5)
                                        ((2)*2*2*2)  (2*(2)*6)    (2*(3)*5)
                                        (2*(2)*2*2)  (2*2*(6))    (2*3*(5))
                                        (2*2*(2)*2)  ((2)*3*4)
                                        (2*2*2*(2))  (2*(3)*4)
                                                     (2*3*(4))
                                                     ((2)*2*2*3)
                                                     (2*(2)*2*3)
                                                     (2*2*(2)*3)
                                                     (2*2*2*(3))
(End)

			a(12) = 8: there are 4 factorizations of 12: (12), (6*2), (4*3), (3*2*2) having 1, 2, 2, 3 elements respectively, a total of 8.

Amarnath Murthy, Generalization of Partition function, Introducing Smarandache Factor partitions, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.
Amarnath Murthy, Length and extent of Smarandache Factor partitions, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)

The version for normal multisets is A001787.
The version for compositions is A001792.
The version for partitions is A006128 (strict: A015723).
Choosing a value instead of position gives A339564.
A000070 counts partitions with a selected part.
A001055 counts factorizations.
A002033 and A074206 count ordered factorizations.
A067824 counts strict chains of divisors starting with n.
A336875 counts compositions with a selected part.
Cf. A000005, A000041, A045778, A050336, A066186, A162247, A264401, A281116, A292504, A292886, A322794.

# Return a list of lists which are factorizations (product representations)
# of n. Within each sublist, the factors are sorted. A minimum factor in
# each element of sublists returned can be specified with 'mincomp'.
# If mincomp=2, the number of sublists contained in the list returned is A001055(n).
# Example:
# n=8 and mincomp=2 return [[2,2,2],[4,8],[8]]
listProdRep := proc(n,mincomp)
    local dvs,resul,f,i,j,rli,tmp ;
    resul := [] ;
    # list returned is empty if n < mincomp
    if n >= mincomp then
        if n = 1 then
            RETURN([1]) ;
        else
            # compute the divisors, and take each divisor
            # as a head element (minimum element) of one of the
            # sublists. Example: for n=8 use {1,2,4,8}, and consider
            # (for mincomp=2) sublists [2,...], [4,...] and [8].
            dvs := numtheory[divisors](n) ;
            for i from 1 to nops(dvs) do
                # select the head element 'f' from the divisors
                f := op(i,dvs) ;
                # if this is already the maximum divisor n
                # itself, this head element is the last in
                # the sublist
                if f =n and f >= mincomp then
                    resul := [op(resul),[f]] ;
                elif f >= mincomp then
                    # if this is not the maximum element
                    # n itself, produce all factorizations
                    # of the remaining factor recursively.
                    rli := procname(n/f,f) ;
                    # Prepend all the results produced
                    # from the recursion with the head
                    # element for the result.
                    for j from 1 to nops(rli) do
                        tmp := [f,op(op(j,rli))] ;
                        resul := [op(resul),tmp] ;
                    od ;
                fi ;
            od ;
        fi ;
    fi ;
    resul ;
end:
A066637 := proc(n)
    local f,d;
    a := 0 ;
    for d in listProdRep(n,2) do
        a := a+nops(d) ;
    end do:
    a ;
end proc: # R. J. Mathar, Jul 11 2013
# second Maple program:
with(numtheory):
b:= proc(n, k) option remember; `if`(n>k, 0, [1$2])+
      `if`(isprime(n), 0, (p-> p+[0, p[1]])(add(
      `if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n})))
    end:
a:= n-> `if`(n<2, 0, b(n$2)[2]):
seq(a(n), n=1..120); # Alois P. Heinz, Feb 12 2019

g[1, r_] := g[1, r]={1, 0}; g[n_, r_] := g[n, r]=Module[{ds, i, val}, ds=Select[Divisors[n], 1<#<=r&]; val={0, 0}+Sum[g[n/ds[[i]], ds[[i]]], {i, 1, Length[ds]}]; val+{0, val[[1]]}]; a[n_] := g[n, n][[2]]; a/@Range[95] (* g[n, r] = {c, f}, where c is the number of factorizations of n with factors <= r and f is the total number of factors in them. - Dean Hickerson, Oct 28 2002 *)
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];Table[Sum[Length[fac],{fac,facs[n]}],{n,50}] (* Gus Wiseman, Apr 18 2021 *)

A079859 a(n) = n*2^(n-4).

Original entry on oeis.org

4, 10, 24, 56, 128, 288, 640, 1408, 3072, 6656, 14336, 30720, 65536, 139264, 294912, 622592, 1310720, 2752512, 5767168, 12058624, 25165824, 52428800, 109051904, 226492416, 469762048, 973078528, 2013265920, 4160749568, 8589934592, 17716740096, 36507222016

Offset: 4

Silvia Heubach (sheubac(AT)calstatela.edu), Jan 11 2003

a(n) = the number of occurrences of 3s in the palindromic compositions of m = 2*n-1 = the number of occurrences of 4s in the palindromic compositions of k = 2*n.
This sequence is part of a family of sequences, namely R(n,k), the number of ks in palindromic compositions of n. See also A057711, A001792, A078836, A079861, A079862, A079863. General formula: R(n,k)=2^(floor(n/2) - k) * (2 + floor(n/2) - k) if n and k have different parity and R(n,k)=2^(floor(n/2) - k) * (2 + floor(n/2) - k + 2^(floor((k+1)/2 - 1)) otherwise, for n >= 2k.
Number of 2 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (10;0) and (01;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1Sergey Kitaev, Nov 11 2004
a(n) appears to be the coefficient of Pi^n in the closed-form expression for the expected value of X^n, where X is the area of a spherical triangle formed by three random points on the unit sphere. (The n*2^(n-4) formula applies when n=2,3 as well, and produces fractional coefficients.) - Drake Thomas, Jan 24 2021

			a(4)=4 since the palindromic compositions of 7 that contain a 3 are 2+3+2, 1+1+3+1+1 and 3+1+3, for a total of 4 3s. The palindromic compositions of 8 that contain a 4 are 2+4+2, 1+1+4+1+1 and 4+4.

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
Phyllis Chinn, Ralph Grimaldi and Silvia Heubach, The frequency of summands of a particular size in Palindromic Compositions, Ars Combin., Vol. 69 (2003), pp. 65-78.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 4 (2004), A21, 20pp.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, University of Kentucky Research Reports (2004).
Math StackExchange, The distribution of areas of a random triangle on the sphere.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A057711, A001792, A078836, A079861, A079862.

Main diagonal of A049089.

[n*2^(n-4) : n in [4..40]]; // Vincenzo Librandi, Sep 22 2011

Mathematica
```
Table[i*2^(i - 4), {i, 4, 50}]
```

Vec(-2*x^4*(3*x-2)/(2*x-1)^2 + O(x^50)) \\ Colin Barker, Sep 29 2015

a(n) = n*2^(n-4);
vector(40, n, a(n+3)) \\ Altug Alkan, Sep 29 2015

O.g.f.: 2*x^4*(2-3*x)/(1-2*x)^2. a(n) = 2*A045623(n-3). - R. J. Mathar, Jun 13 2008
a(n) = 4*a(n-1) - 4*a(n-2) for n>5. - Colin Barker, Sep 29 2015
From Amiram Eldar, Jan 12 2021: (Start)
Sum_{n>=4} 1/a(n) = 16*log(2) - 32/3.
Sum_{n>=4} (-1)^n/a(n) = 20/3 - 16*log(3/2). (End)
E.g.f.: x*(exp(2*x) - 1 - 2*x - 2*x^2)/8. - Stefano Spezia, Apr 06 2021

A079861 a(n) is the number of occurrences of 7's in the palindromic compositions of 2n-1, or also, the number of occurrences of 8's in the palindromic compositions of 2n.

Original entry on oeis.org

10, 22, 48, 104, 224, 480, 1024, 2176, 4608, 9728, 20480, 43008, 90112, 188416, 393216, 819200, 1703936, 3538944, 7340032, 15204352, 31457280, 65011712, 134217728, 276824064, 570425344, 1174405120, 2415919104, 4966055936

Offset: 8

Silvia Heubach (sheubac(AT)calstatela.edu), Jan 11 2003

This sequence is part of a family of sequences, namely R(n,k), the number of k's in palindromic compositions of n. See also A057711, A001792, A078836, A079861, A079862. General formula: R(n,k) = 2^(floor(n/2) - k) * (2 + floor(n/2) - k) if n and k have different parity and R(n,k) = 2^(floor(n/2) - k) * (2 + floor(n/2) - k + 2^(floor((k+1)/2 - 1)) otherwise, for n >= 2*k.

			a(8)=10 since the palindromic compositions of 15 that contain a 7 are 7+1+7, 4+7+4, 1+3+7+3+1, 3+1+7+1+3, 2+2+7+2+2, 1+1+1+1+7+1+1+1+1, 1+1+2+7+2+1+1, 1+2+1+7+1+2+1 and 2+1+1+7+1+1+2, for a total of 10 7's.

Vincenzo Librandi, Table of n, a(n) for n = 8..1000
Phyllis Chinn, Ralph Grimaldi and Silvia Heubach, The Frequency of Summands of a Particular Size in Palindromic Compositions, Ars Combin., Vol. 69 (2003), pp. 65-78.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A057711, A001792, A079859, A061256, A079862, A079863, A111297.

[(2+n)*2^(n-8) : n in [8..40]]; // Vincenzo Librandi, Sep 22 2011

Table[(2 + i)*2^(i - 8), {i, 8, 50}]
LinearRecurrence[{4,-4},{10,22},50] (* Harvey P. Dale, Jun 04 2025 *)

Vec(-2*x^8*(9*x-5)/(2*x-1)^2 + O(x^100)) \\ Colin Barker, Dec 16 2014

a(n) = (2+n)*2^(n-8).
a(n) = 2*A111297(n-6). - Colin Barker, Dec 16 2014
a(n) = 4*a(n-1) - 4*a(n-2). - Colin Barker, Dec 16 2014
G.f.: -2*x^8*(9*x-5) / (2*x-1)^2. - Colin Barker, Dec 16 2014
From Amiram Eldar, Jan 13 2021: (Start)
Sum_{n>=8} 1/a(n) = 1024*log(2) - 447047/630.
Sum_{n>=8} (-1)^n/a(n) = 261617/630 - 1024*log(3/2). (End)

A079862 a(i) = the number of occurrences of 9's in the palindromic compositions of n=2i-1 = the number of occurrences of 10's in the palindromic compositions of n=2i.

Original entry on oeis.org

18, 38, 80, 168, 352, 736, 1536, 3200, 6656, 13824, 28672, 59392, 122880, 253952, 524288, 1081344, 2228224, 4587520, 9437184, 19398656, 39845888, 81788928, 167772160, 343932928, 704643072, 1442840576, 2952790016, 6039797760, 12348030976, 25232932864

Offset: 10

Silvia Heubach (sheubac(AT)calstatela.edu), Jan 11 2003

This sequence is part of a family of sequences, namely R(n,k), the number of ks in palindromic compositions of n. See also A057711, A001792, A078836, A079861, A079862. General formula: R(n,k)=2^(floor(n/2) - k) * (2 + floor(n/2) - k) if n and k have different parity and R(n,k)=2^(floor(n/2) - k) * (2 + floor(n/2) - k + 2^(floor((k+1)/2 - 1)) otherwise, for n >= 2k.

			a(10) = 18 since the palindromic compositions of 19 that contain a 9 are 9+1+9 and the 16 compositions of the form c+9+(reverse of c), where c represents a composition of 5.

Colin Barker, Table of n, a(n) for n = 10..1000
P. Chinn, R. Grimaldi and S. Heubach, The frequency of summands of a particular size in Palindromic Compositions, Ars Combin. 69 (2003), 65-78.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A057711, A001792, A079859, A078836, A079861, A079863.

Mathematica
```
Table[(8 + i)*2^(i - 10), {i, 10, 50}]
```

Vec(-2*x^10*(17*x-9)/(2*x-1)^2 + O(x^100)) \\ Colin Barker, Sep 29 2015

a(n) = (n+8)*2^(n-10).
From Colin Barker, Sep 29 2015: (Start)
a(n) = 2*A159697(n-10).
a(n) = 4*a(n-1) - 4*a(n-2) for n>11.
G.f.: -2*x^10*(17*x-9) / (2*x-1)^2.
(End)

A097910 Number of parts in all compositions of n into distinct parts.

Original entry on oeis.org

1, 1, 5, 5, 9, 27, 31, 49, 71, 185, 207, 339, 457, 685, 1421, 1745, 2577, 3615, 5143, 6877, 13439, 15965, 23823, 31983, 45553, 59425, 83549, 139013, 173769, 244803, 330391, 452257, 597935, 810929, 1052559, 1692723, 2074321, 2890333, 3783821, 5178041, 6658377

Offset: 1

Vladeta Jovovic, Sep 04 2004

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Cf. A001792, A008289, A015723, A032020, A072574, A336875.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(n>i*(i+1)/2, [][], zip((x, y)->x+y, [b(n, i-1)],
      `if`(i>n, [], [0, b(n-i, i-1)]), 0)[]))
    end:
a:= n-> (l-> add(i*l[i+1]*i!, i=1..nops(l)-1))([b(n$2)]):
seq(a(n), n=1..50);  # Alois P. Heinz, Nov 20 2012
# second Maple program:
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 10 2020

Drop[ CoefficientList[ Series[ Sum[ k*k!*x^((k^2 + k)/2)/Product[1 - x^j, {j, 1, k}], {k, 1, 45}], {x, 0, 40}], x], 1] (* Robert G. Wilson v, Sep 08 2004 *)

G.f.: Sum(k >= 0; k*k! x^((k^2+k)/2) / Prod(1<=j<=k; 1-x^j)).

a(n) = Sum_{k=1..floor((sqrt(8*n+1)-1)/2)} k! * k * A008289(n,k). - Alois P. Heinz, Aug 10 2020

More terms from Robert G. Wilson v and John W. Layman, Sep 08 2004

A159694 a(n) = 2*a(n-1) + 2^(n-1), for n > 0, with a(0)=6.

Original entry on oeis.org

6, 13, 28, 60, 128, 272, 576, 1216, 2560, 5376, 11264, 23552, 49152, 102400, 212992, 442368, 917504, 1900544, 3932160, 8126464, 16777216, 34603008, 71303168, 146800640, 301989888, 620756992, 1275068416, 2617245696, 5368709120

Offset: 0

Philippe Deléham, Apr 20 2009

Diagonal of triangles A062111, A152920.

			a(0) = 6,
a(1) = 2* 6 + 1 =  13,
a(2) = 2*13 + 2 =  28,
a(3) = 2*28 + 4 =  60,
a(4) = 2*60 + 8 = 128, ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Milan Janjić and Boris Petković, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Milan Janjić and Boris Petković, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq., Vol. 17 (2014), Article 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A000079, A001787, A001792, A034007, A045623.
Cf. A045891, A062111, A111297, A152920.
Seventh row of triangle A062111. - Klaus Brockhaus, Sep 27 2009

[(12+n)*2^(n-1): n in [0..30]]; // G. C. Greubel, Sep 27 2022

Table[(6 + n/2)*2^n, {n, 0, 30}] (* Amiram Eldar, Jan 19 2021 *)

[(12+n)*2^(n-1) for n in range(30)] # G. C. Greubel, Sep 27 2022

a(n) = Sum_{k=0..n} (k+6)*binomial(n,k).
From Klaus Brockhaus, Sep 27 2009: (Start)
a(n) = (6 + n/2)*2^n.
G.f.: (6 - 11*x)/(1-2*x)^2. (End)
From Amiram Eldar, Jan 19 2021: (Start)
Sum_{n>=0} 1/a(n) = 8192*log(2) - 3934820/693.
Sum_{n>=0} (-1)^n/a(n) = 11509636/3465 - 8192*log(3/2). (End)
E.g.f.: (6 + x)*exp(2*x). - G. C. Greubel, Sep 27 2022

A188553 T(n,k) = Number of n X k binary arrays without the pattern 0 1 diagonally, vertically, antidiagonally or horizontally.

Original entry on oeis.org

2, 3, 3, 4, 5, 4, 5, 8, 7, 5, 6, 12, 12, 9, 6, 7, 17, 20, 16, 11, 7, 8, 23, 32, 28, 20, 13, 8, 9, 30, 49, 48, 36, 24, 15, 9, 10, 38, 72, 80, 64, 44, 28, 17, 10, 11, 47, 102, 129, 112, 80, 52, 32, 19, 11, 12, 57, 140, 201, 192, 144, 96, 60, 36, 21, 12, 13, 68, 187, 303, 321, 256, 176

Offset: 1

R. H. Hardin, Apr 04 2011

From Miquel A. Fiol, Feb 06 2024: (Start)
Also, T(n,k) is the number of words of length k, x(1)x(2)...x(k), on the alphabet {0,1,...,n}, such that, for i=2,...,k, x(i)=either x(i-1) or x(i)=x(i-1)-1.
For the bijection between arrays and sequences, notice that the i-th column consists of 1's and then 0's, and there are x(i)=0 to n of 1's.
Such a bijection implies that all the empirical/conjectured formulas in A188554, A188555, A188556, A188557, A188558, and A188559 become correct.
(End)

			Table starts
..2..3..4..5...6...7...8...9...10...11...12....13....14....15....16.....17
..3..5..8.12..17..23..30..38...47...57...68....80....93...107...122....138
..4..7.12.20..32..49..72.102..140..187..244...312...392...485...592....714
..5..9.16.28..48..80.129.201..303..443..630...874..1186..1578..2063...2655
..6.11.20.36..64.112.192.321..522..825.1268..1898..2772..3958..5536...7599
..7.13.24.44..80.144.256.448..769.1291.2116..3384..5282..8054.12012..17548
..8.15.28.52..96.176.320.576.1024.1793.3084..5200..8584.13866.21920..33932
..9.17.32.60.112.208.384.704.1280.2304.4097..7181.12381.20965.34831..56751
.10.19.36.68.128.240.448.832.1536.2816.5120..9217.16398.28779.49744..84575
.11.21.40.76.144.272.512.960.1792.3328.6144.11264.20481.36879.65658.115402
Some solutions for 5 X 3:
  1 1 1   1 0 0   0 0 0   1 1 1   1 1 1   1 1 1   1 1 1
  1 1 1   0 0 0   0 0 0   1 1 1   1 1 1   1 1 1   1 1 1
  1 1 1   0 0 0   0 0 0   1 1 1   1 0 0   1 1 0   1 1 1
  1 1 1   0 0 0   0 0 0   1 1 0   0 0 0   1 0 0   1 1 1
  1 1 1   0 0 0   0 0 0   1 0 0   0 0 0   0 0 0   1 1 0
Some solutions for T(5,3): By taking the sums of the columns in the above arrays we get 555, 100, 000, 543, 322, 432, 554. - _Miquel A. Fiol_, Feb 04 2024

R. H. Hardin, Table of n, a(n) for n = 1..9934

Diagonal is A045623.
Column 4 is A086570.
Rows 2..8 are A022856(n+4), A188554, A188555, A188556, A188557, A188558, A188559.
Upper diagonals T(n,n+i) for i=1..8 give: A001792, A001787(n+1), A000337(n+1), A045618, A045889, A034009, A055250, A055251.
Lower diagonals T(n+i,n) for i=1..7 give: A045891(n+1), A034007(n+2), A111297(n+1), A159694(n-1), A159695(n-1), A159696(n-1), A159697(n-1).
Antidiagonal sums give A065220(n+5).

T:= (n,k)-> `if`(k<=n+1, (2*n+3-k)*2^(k-2), (n+1-k)*binomial(k-1, n) * add(binomial(n, j-1)/(k-j)*T(n, j)*(-1)^(n-j), j=1..n+1)): seq(seq(T(n, 1+d-n), n=1..d), d=1..15); #Alois P. Heinz in the Sequence Fans Mailing List, Apr 04 2011 [We do not permit programs based on conjectures, but this program is now justified by Fiol's comment. - N. J. A. Sloane, Mar 09 2024]

Empirical: T(n,k) = (n+1)*2^(k-1) + (1-k)*2^(k-2) for k < n+3, and then the entire row n is a polynomial of degree n in k.
From Miquel A. Fiol, Feb 06 2024: (Start)
The above empirical formula is correct.
It can be proved that T(n,k) satisfies the recurrence
  T(n,k) = Sum_{r=1..n+1} (-1)^(r+1)*binomial(n+1,r)*T(n,k-r)
with initial values
  T(n,k) = Sum_{r=0..k-1} (n+1-r)*binomial(k-1,r) for k = 1..n+1. (End)

cp's OEIS Frontend

A228367 n-th element of the ruler function plus the highest power of 2 dividing n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A053539 a(n) = n * 8^(n-1).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A059450 Triangle read by rows: T(n,k) = Sum_{j=0..k-1} T(n,j) + Sum_{j=1..n-k} T(n-j,k), with T(0,0)=1 and T(n,k) = 0 for k > n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Extensions

A066637 Total number of elements in all factorizations of n with all factors > 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

A079859 a(n) = n*2^(n-4).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A079861 a(n) is the number of occurrences of 7's in the palindromic compositions of 2*n-1, or also, the number of occurrences of 8's in the palindromic compositions of 2*n.

Views

Author

Keywords

Comments

Examples

Links

A079861 a(n) is the number of occurrences of 7's in the palindromic compositions of 2n-1, or also, the number of occurrences of 8's in the palindromic compositions of 2n.

A079862 a(i) = the number of occurrences of 9's in the palindromic compositions of n=2i-1 = the number of occurrences of 10's in the palindromic compositions of n=2i.