A006127 -id:A006127 - cp's OEIS frontend

A188917 Where powers of 2 occur in the union of squares and powers of 2.

1, 2, 3, 4, 6, 8, 11, 15, 20, 27, 37, 51, 70, 97, 135, 189, 264, 371, 521, 734, 1034, 1459, 2059, 2908, 4108, 5805, 8205, 11599, 16398, 23185, 32783, 46356, 65552, 92698, 131089, 185381, 262162, 370746, 524307, 741475, 1048596, 1482931, 2097173, 2965842, 4194326, 5931664, 8388631

Offset: 0

Reinhard Zumkeller, Apr 14 2011

A188915(a(n)) = A000079(n); A188915(A188916(n)) = A000290(n).

Robert Israel, Table of n, a(n) for n = 0..6637

Cf. A000079, A000290, A017910, A036987.

Cf. A188915, A188916, A209229, A006127 (subsequence).

a188917 n = a188917_list !! n
a188917_list = filter ((== 1) . a209229. a188915) [0..]
-- Reinhard Zumkeller, May 19 2015

seq(floor((n+1)/2) + floor(2^(n/2)), n=0..100); # Robert Israel, Jun 13 2019

Table[Floor[(n+1)/2] + Floor[2^(n/2)], {n, 0, 50}] (* Paolo Xausa, Oct 01 2024 *)

from math import isqrt
def A188917(n): return (n+1>>1)+isqrt(1<Chai Wah Wu, Oct 01 2024

a(n) = floor((n+1)/2) + floor(2^(n/2)). - Robert Israel, Jun 13 2019

A073113 a(n) = 2^(2^n + n).

Original entry on oeis.org

2, 8, 64, 2048, 1048576, 137438953472, 1180591620717411303424, 43556142965880123323311949751266331066368

Offset: 0

Joe Mathes (oldschoolchaz(AT)hotmail.com), Aug 19 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 0..11

Cf. A000079, A001146, A006127.

a[n_] := 2^(2^n + n); Array[a, 8, 0] (* Amiram Eldar, Aug 14 2022 *)

a(n) = 2^A006127(n) = A000079(n)*A001146(n). - Amiram Eldar, Aug 14 2022

A134466 See A134457.

Original entry on oeis.org

0, 1, 1, 2, 6, 6, 11, 22, 44, 92, 92, 157, 311, 622, 1239, 2478, 4956, 9912, 19832, 19832, 36217, 72058, 144061, 288122, 576123, 1152239, 2304478, 4608943

Offset: 1

David W. Wilson, Dec 17 2007

nonn

Note that a(n) = a(n-1) for n = [1,] 3, 6, 11, 20,... = A006127(2^k + k) (conjectured); this corresponds to the case where the string c (see A134457) satisfies c(n)="0".c(n-1) (pre-pended "0"). The next string c(n+1) is obtained by inserting "0" after the first "1" and adding 1 (except for c(4) where both of these operations are equivalent and only one must be performed). - M. F. Hasler, Dec 17 2007

A162574 Primes of the form 2^x+x+y+2^y, with x and y integers of any sign.

Original entry on oeis.org

2, 7, 17, 23, 31, 43, 71, 73, 107, 541, 2129, 4111, 4243, 9239, 18457, 32789, 32803, 65563, 65687, 73757, 135197, 147487, 264221, 589859, 786469, 1048633, 1049117, 4194329, 4194847, 4227109, 8388637, 8392739, 8405029, 8650793, 16908329

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 06 2009

nonn

			a(1) = A006127(1)+A006127(1) =2. a(2) = A006127(1)+A006127(3)=7.
a(3) = A006127(4)+A006127(3) =17. a(4) = A006127(5)+A006127(2) = 23.

Cf. A162573, A006127.

f[x_,y_]:=2^x+x+y+2^y; lst={};Do[Do[p=f[x,y];If[PrimeQ[p],AppendTo[lst, p]],{y,-5!,6!}],{x,-5!,6!}];Take[Union[lst],5! ]

Examples added by R. J. Mathar, Sep 17 2009

A198061 Array read by antidiagonals, m>=0, n>=0, A(m,n) = sum{k=0..n} sum{j=0..m} sum{i=0..m} (-1)^(j+i)C(i,j)n^j*k^(m-j).

Original entry on oeis.org

1, 0, 2, 0, 2, 3, 0, 2, 6, 4, 0, 2, 11, 12, 5, 0, 2, 20, 32, 20, 6, 0, 2, 37, 84, 70, 30, 7, 0, 2, 70, 224, 240, 130, 42, 8, 0, 2, 135, 612, 834, 550, 217, 56, 9, 0, 2, 264, 1712, 2968, 2354, 1092, 336, 72, 10, 0, 2, 521, 4884, 10826, 10310, 5551, 1960, 492

Offset: 0

Peter Luschny, Nov 02 2011

			m\n  [0] [1]  [2]   [3]    [4]     [5]    [6]
----------------------------------------------
[0]   1   2    3     4      5       6       7    A000027
[1]   0   2    6    12     20      30      42    A002378
[2]   0   2   11    32     70     130     217    A033994
[3]   0   2   20    84    240     550    1092    A098077
[4]   0   2   37   224    834    2354    5551
[5]   0   2   70   612   2968   10310   28854

Cf. A198060.

A198061 := proc(m, n) local i,j,k,pow;
pow := (a,b) -> if a=0 and b=0 then 1 else a^b fi;
add(add(add((-1)^(j+i)*binomial(i,j)*pow(n,j)*pow(k,m-j),i=0..m),j=0..m),k=0..n) end:
for m from 0 to 8 do lprint(seq(A198061(m,n), n=0..6)) od;

Unprotect[Power]; 0^0 = 1; Protect[Power]; a[m_, n_] :=  Sum[(-1)^(j+i)*Binomial[i, j]*n^j*k^(m-j) , {i, 0, m}, {j, 0, m}, {k, 0, n}]; Table[a[m-n, n], {m, 0, 10}, {n, 0, m}] // Flatten (* Jean-François Alcover, Jul 26 2013 *)

A198061(n,2) = A006127(n+1)

A024038 a(n) = 4^n - n^2.

Original entry on oeis.org

1, 3, 12, 55, 240, 999, 4060, 16335, 65472, 262063, 1048476, 4194183, 16777072, 67108695, 268435260, 1073741599, 4294967040, 17179868895, 68719476412, 274877906583, 1099511627376, 4398046510663, 17592186043932

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy]
Index entries for linear recurrences with constant coefficients, signature (7,-15,13,-4).

Cf. A000290, A000302, A000325, A006127.

[ 4^n-n^2: n in [0..30] ]; // Vincenzo Librandi, Dec 25 2010

Table[4^n-n^2,{n,0,30}] (* or *) LinearRecurrence[{7,-15,13,-4},{1,3,12,55},30] (* Harvey P. Dale, Sep 14 2013 *)

[4^n-n^2 for n in range(31)] # G. C. Greubel, Aug 18 2023

a(n) = A000325(n)*A006127(n). - Reinhard Zumkeller, Apr 10 2010
G.f.: (1 - 4*x + 6*x^2 + 3*x^3)/((1 - x)^3*(1 - 4*x)). - Colin Barker, May 29 2012
E.g.f.: exp(4*x) - x*(1 + x)*exp(x). - G. C. Greubel, Aug 18 2023

A124469 Triangle, read by rows, where row n equals the inverse binomial transform of column n in the rectangular table A124460.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 8, 6, 1, 1, 22, 28, 11, 1, 1, 65, 120, 81, 20, 1, 1, 209, 500, 494, 219, 37, 1, 1, 730, 2088, 2733, 1812, 578, 70, 1, 1, 2743, 8884, 14411, 12904, 6299, 1518, 135, 1, 1, 10958, 38803, 74484, 84424, 56590, 21384, 4007, 264, 1, 1, 46057, 174366

Offset: 0

Paul D. Hanna, Nov 03 2006

In table A124460, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k * R_k(y)^n for n>=0.

			Triangle begins:
1;
1, 1;
1, 3, 1;
1, 8, 6, 1;
1, 22, 28, 11, 1;
1, 65, 120, 81, 20, 1;
1, 209, 500, 494, 219, 37, 1;
1, 730, 2088, 2733, 1812, 578, 70, 1;
1, 2743, 8884, 14411, 12904, 6299, 1518, 135, 1;
1, 10958, 38803, 74484, 84424, 56590, 21384, 4007, 264, 1;
1, 46057, 174366, 383391, 526121, 453082, 238853, 72076, 10693, 521, 1;

Cf. A124470 (row sums), A006127 (diagonal T(n+1, n)); A124460 (table).

{T(n,k)=local(R=vector(n+2,r,vector(n+2,c,binomial(r+c-2,c-1)))); for(i=0,n,for(r=0,n,R[r+1]=Vec(sum(c=0,n,x^c*Ser(R[c+1])^r+O(x^(n+1)))))); Vec(subst(Ser(vector(n+1,j,R[j][n+1])),x,x/(1+x))/(1+x))[k+1]}

Secondary diagonal T(n+1,n) = 2^n + n = A006127(n).

A135227 Triangle A000012 * A135225, read by rows.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 3, 1, 5, 4, 6, 4, 1, 6, 5, 10, 10, 5, 1, 7, 6, 15, 20, 15, 6, 1, 8, 7, 21, 35, 35, 21, 7, 1, 9, 8, 28, 56, 70, 56, 28, 8, 1, 10, 9, 36, 84, 126, 126, 84, 36, 9, 1, 11, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 12, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

Row sums = A006127: (1, 3, 6, 11, 20, 37, ...).

			First few rows of the triangle:
  1;
  2, 1;
  3, 2,  1;
  4, 3,  3,  1;
  5, 4,  6,  4,  1;
  6, 5, 10, 10,  5, 1;
  7, 6, 15, 20, 15, 6, 1;
...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A006127, A007318, A135225.

T:= function(n,k)
    if k=0 then return 1;
    else return Binomial(n,k);
    fi; end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 20 2019

[k eq 0 select n+1 else Binomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019

seq(seq( `if`(k=0, n+1, binomial(n,k)), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019

Table[If[k==0, n+1, Binomial[n, k]], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

T(n,k) = if(k==0, n+1, binomial(n,k)); \\ G. C. Greubel, Nov 20 2019

def T(n, k):
    if (k==0): return 1
    else: return binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019

A000012 * A135225 as infinite lower triangular matrices. Left border of 1's in Pascal's Triangle (A007318) is replaced with a column of (1,2,3,...).

T(n,k) = binomial(n,k), with T(n,0) = n+1. - G. C. Greubel, Nov 20 2019

More terms added by G. C. Greubel, Nov 20 2019

A143397 Triangle T(n,k)=number of forests of labeled rooted trees of height at most 1, with n labels and k nodes, where any root may contain >= 1 labels, n >= 0, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 6, 10, 0, 1, 11, 36, 41, 0, 1, 20, 105, 230, 196, 0, 1, 37, 285, 955, 1560, 1057, 0, 1, 70, 756, 3535, 8680, 11277, 6322, 0, 1, 135, 2002, 12453, 41720, 80682, 86800, 41393, 0, 1, 264, 5347, 43008, 186669, 485982, 773724, 708948, 293608

Offset: 0

Alois P. Heinz, Aug 12 2008

			T(3,2) = 6: {1,2}{3}, {1,3}{2}, {2,3}{1}, {1,2}<-3, {1,3}<-2, {2,3}<-1.
Triangle begins:
  1;
  0, 1;
  0, 1,  3;
  0, 1,  6,  10;
  0, 1, 11,  36,   41;
  0, 1, 20, 105,  230,  196;
  0, 1, 37, 285,  955, 1560,  1057;
  0, 1, 70, 756, 3535, 8680, 11277, 6322;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Index entries for sequences related to rooted trees

Columns k=0-2: A000007, A000012, A006127. Diagonal: A000248. See also A048993, A008277, A007318, A143405 for row sums.

T:= (n,k)-> add(binomial(n, k-t)*Stirling2(n-(k-t),t)*t^(k-t), t=0..k):
seq(seq(T(n, k), k=0..n), n=0..11);

T[n_, k_] := Sum[Binomial[n, k-t]*StirlingS2[n - (k-t), t]*t^(k-t), {t, 0, k}]; T[0, 0] = 1; T[_, 0] = 0;
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 31 2016, translated from Maple *)

T(n,k) = Sum_{t=0..k} C(n,k-t) * Stirling2(n-(k-t),t) * t^(k-t).

E.g.f.: exp(y*exp(x*y)*(exp(x)-1)). - Vladeta Jovovic, Dec 08 2008

A286228 Numbers n such that d(n) = 2^omega(n) + omega(n) where d = A000005 and omega = A001221.

Original entry on oeis.org

1, 4, 9, 12, 18, 20, 25, 28, 44, 45, 49, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 121, 124, 147, 148, 153, 164, 169, 171, 172, 175, 188, 207, 212, 236, 242, 244, 245, 261, 268, 275, 279, 284, 289, 292, 316, 325, 332, 333, 338, 356, 361, 363, 369, 387, 388, 404, 412, 423, 425, 428

Offset: 1

Juri-Stepan Gerasimov, May 04 2017

nonn

			1 is in this sequence because d(1) = 1 is equal to 2^omega(1) + omega(1) = 2^0 + 0 = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Supersequence of A001248 and of A054753.

Cf. A006127.

Select[Range@ 432, Function[f, DivisorSigma[0, #] == 2^f + f]@ PrimeNu@ # &] (* Michael De Vlieger, May 04 2017 *)

isok(n) = numdiv(n) == 2^omega(n) + omega(n); \\ Michel Marcus, May 07 2017

cp's OEIS Frontend

A188917 Where powers of 2 occur in the union of squares and powers of 2.

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A073113 a(n) = 2^(2^n + n).

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A134466 See A134457.

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A162574 Primes of the form 2^x+x+y+2^y, with x and y integers of any sign.

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Mathematica

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A198061 Array read by antidiagonals, m>=0, n>=0, A(m,n) = sum{k=0..n} sum{j=0..m} sum{i=0..m} (-1)^(j+i)*C(i,j)*n^j*k^(m-j).

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Maple

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A024038 a(n) = 4^n - n^2.

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Magma

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A124469 Triangle, read by rows, where row n equals the inverse binomial transform of column n in the rectangular table A124460.

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PARI

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A135227 Triangle A000012 * A135225, read by rows.

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A198061 Array read by antidiagonals, m>=0, n>=0, A(m,n) = sum{k=0..n} sum{j=0..m} sum{i=0..m} (-1)^(j+i)C(i,j)n^j*k^(m-j).