A000918 -id:A000918 - cp's OEIS frontend

A075427 a(0) = 1; a(n) = a(n-1)+1 if n is even, otherwise a(n) = 2*a(n-1).

1, 2, 3, 6, 7, 14, 15, 30, 31, 62, 63, 126, 127, 254, 255, 510, 511, 1022, 1023, 2046, 2047, 4094, 4095, 8190, 8191, 16382, 16383, 32766, 32767, 65534, 65535, 131070, 131071, 262142, 262143, 524286, 524287, 1048574, 1048575, 2097150, 2097151, 4194302, 4194303, 8388606

Offset: 0

Reinhard Zumkeller, Sep 15 2002

Fixed points for permutations A180200, A180201, A180198, and A180199. - Reinhard Zumkeller, Aug 15 2010

The Kn22 sums, see A180662, of triangle A194005 equal the terms of this sequence. - Johannes W. Meijer, Aug 16 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
R. Hinze, Concrete stream calculus: An extended study, J. Funct. Progr. 20 (5-6) (2010) 463-535, doi.
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-2).

Cf. A075426, A066880, A083416, A000225 (bisection), A000918 (bisection).

Cf. A180198, A180199, A180200, A180201, A180662, A194005.

a075427 n = a075427_list !! n
a075427_list = 1 : f 1 1 where
   f x y = z : f (x + 1) z where z = (1 + x `mod` 2) * y + 1 - x `mod` 2
-- Reinhard Zumkeller, Feb 27 2012

[2^Floor((n+3)/2)-3/2+(-1)^n/2: n in [0..30]]; // Vincenzo Librandi, Aug 17 2011

A075427 := proc(n) if type(n,'even') then 2^(n/2+1)-1 ; else 2^(1+(n+1)/2)-2 ; end if; end proc: seq(A075427(n), n=0..40); # R. J. Mathar, Feb 18 2011
isA := proc(n) convert(n, base, 2): 1 - %[1] = nops(%) - add(%) end:
select(isA, [$1..4095]); # Peter Luschny, Oct 27 2022

a[0]=1; a[n_]:=a[n]=If[EvenQ[n],a[n-1]+1,2*a[n-1]]; Table[a[n],{n,0,40}] (* Jean-François Alcover, Mar 20 2011 *)
nxt[{n_,a_}]:={n+1,If[OddQ[n],a+1,2a]}; Transpose[NestList[nxt,{0,1},40]][[2]] (* or *) LinearRecurrence[{0,3,0,-2},{1,2,3,6},50] (* Harvey P. Dale, Mar 12 2016 *)

a(n)=2^((n+3)\2)-3/2+(-1)^n/2 \\ Charles R Greathouse IV, Feb 06 2017

def A075427(n): return (1<<(n>>1)+2)-2 if n&1 else (1<<(n>>1)+1)-1 # Chai Wah Wu, Apr 23 2023

a(0) = 1; for n >= 1, a(2*n) = 2^(n+1)-1, a(2*n-1) = 2^(n+1)-2; a(n) = 2^floor((n+3)/2) - 3/2 + (-1)^n/2. - Benoit Cloitre, Sep 17 2002 [corrected by Robert FERREOL, Jan 26 2011]
a(n) = (-1)^n/2 - 3/2 + 2^(n/2)*(1 + sqrt(2) + (1-sqrt(2))*(-1)^n). - Paul Barry, Apr 22 2004
From Paul Barry, Jul 30 2004: (Start)
Interleaved Mersenne numbers: interleaves 2*2^n-1 and 2(2*2^n-1) (A000225(n+1) and 2*A000225(n+1)).
G.f.: (1+2*x)/((1-x^2)*(1-2*x^2));
a(n) = 3*a(n-2) - 2*a(n-4);
a(n) = Sum_{k=0..n} binomial(floor((n+1)/2), floor((k+1)/2)). (End)
For n > 0: a(n) = (1 + n mod 2) * a(n-1) + 1 - (n mod 2). - Reinhard Zumkeller, Feb 27 2012
E.g.f.: 2*(cosh(sqrt(2)*x) - sinh(x) + sqrt(2)*sinh(sqrt(2)*x)) - cosh(x). - Stefano Spezia, Jul 11 2023
From Alois P. Heinz, Dec 27 2023: (Start)
a(n) = 2^floor((n+3)/2)-1-(n mod 2).
a(n) = A066880(n) for n>=1. (End)

Formulae corrected and minor edits by Johannes W. Meijer, Aug 16 2011

A014410 Elements in Pascal's triangle (by row) that are not 1.

Original entry on oeis.org

2, 3, 3, 4, 6, 4, 5, 10, 10, 5, 6, 15, 20, 15, 6, 7, 21, 35, 35, 21, 7, 8, 28, 56, 70, 56, 28, 8, 9, 36, 84, 126, 126, 84, 36, 9, 10, 45, 120, 210, 252, 210, 120, 45, 10, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 13, 78

Offset: 2

Mohammad K. Azarian

Also, rows of triangle formed using Pascal's rule except begin and end n-th row with n+2. - Asher Auel.
Row sums are A000918. - Roger L. Bagula and Gary W. Adamson, Jan 15 2009
Given the triangle signed by rows (+ - + ...) = M, with V = a variant of the Bernoulli numbers starting [1/2, 1/6, 0, -1/30, 0, 1/42, ...]; M*V = [1, 1, 1, ...]. - Gary W. Adamson, Mar 05 2012
Also A014410 * [1/2, 1/6, 0, -1/30, 0, 1/42, 0, ...] = [1, 2, 3, 4, ...]. For an alternative way to derive the Bernoulli numbers from a modified version of Pascal's triangle see A135225. - Peter Bala, Dec 18 2014
T(n,k) mod n = A053201(n,k), k=1..n-1. - Reinhard Zumkeller, Aug 17 2013
From Wolfdieter Lang, May 22 2015: (Start)
This is Johannes Scheubel's (1494-1570) (also Scheybl, Schöblin) version of the arithmetical triangle from his 1545 book "De numeris et diversis rationibus". See the Kac reference, p. 396 and the Table 12.1 on p. 395.
The row sums give 2*A000225(n-1) = A000918(n) = 2*(2^n - 1), n >= 2. (See the second comment above).
The alternating row sums give repeat(2,0) = 2*A059841(n), n >= 2. (End)
T(n+1,k) is the number of k-facets of the n-simplex. - Jianing Song, Oct 22 2023

			The triangle T(n,k) begins:
n\k  1  2   3   4    5    6    7    8   9  10 11
2:   2
3:   3  3
4:   4  6   4
5:   5 10  10   5
6:   6 15  20  15    6
7:   7 21  35  35   21    7
8:   8 28  56  70   56   28    8
9:   9 36  84 126  126   84   36    9
10: 10 45 120 210  252  210  120   45  10
11: 11 55 165 330  462  462  330  165  55  11
12: 12 66 220 495  792  924  792  495 220  66 12
... reformatted. - _Wolfdieter Lang_, May 22 2015

Victor J. Kac, A History of Mathematics, third edition, Addison-Wesley, 2009, pp. 395, 396.

Reinhard Zumkeller, Rows n=2..150 of triangle, flattened
Carl McTague, On the Greatest Common Divisor of binomial(qn, q), binomial(qn,2q), ..., binomial(qn, qn-q), arXiv:1510.06696 [math.CO], 2015.
Wikipedia, Johannes Scheubel (in German).
Wikipedia, Simplex

Cf. A007318, A000918, A027641.
A180986 is the same sequence but regarded as a square array.
Cf. A000225,A059841, A257241 (Stifel's version).

a014410 n k = a014410_tabl !! (n-2) !! (k-1)
a014410_row n = a014410_tabl !! (n-2)
a014410_tabl = map (init . tail) $ drop 2 a007318_tabl
-- Reinhard Zumkeller, Mar 12 2012

for i from 0 to 12 do seq(binomial(i, j)*1^(i-j), j = 1 .. i-1) od; # Zerinvary Lajos, Dec 02 2007

Select[ Flatten[ Table[ Binomial[ n, i ], {n, 0, 13}, {i, 0, n} ] ], #>1& ]

T(n,k) = binomial(n,k) = A007318(n,k), n >= 2, k = 1, 2, ..., n-1.
a(n) = C(A003057(n),A002260(n)) = C(A003057(n),A004736(n)). - Lekraj Beedassy, Jul 29 2006
T(n,k) = A028263(n,k) - A007318(n,k). - Reinhard Zumkeller, Mar 12 2012
gcd_{k=1..n-1} T(n, k) = A014963(n), see Theorem 1 of McTague link. - Michel Marcus, Oct 23 2015

More terms from Erich Friedman

A122746 G.f.: 1/((1-2x)(1-2*x^2)).

Original entry on oeis.org

1, 2, 6, 12, 28, 56, 120, 240, 496, 992, 2016, 4032, 8128, 16256, 32640, 65280, 130816, 261632, 523776, 1047552, 2096128, 4192256, 8386560, 16773120, 33550336, 67100672, 134209536, 268419072, 536854528, 1073709056, 2147450880, 4294901760, 8589869056

Offset: 0

N. J. A. Sloane, Sep 24 2006

Equals row sums of triangle A156665. - Gary W. Adamson, Feb 12 2009
a(n) is the number of subsets of {1,2,...,n+1} that contain at least one odd integer. - Geoffrey Critzer, Mar 03 2009
a(n-3) is the number of chiral pairs of color patterns of length n using two colors. Two color patterns are equivalent if the colors are permuted. For example, a string of five colors using exactly two different colors has six chiral pairs: AAAAB-ABBBB, AAABA-ABAAA, AAABB-AABBB, AABAB-ABABB, AABBA-ABBAA, and ABAAB-ABBAB. The number of color patterns of length n using exactly k colors when chiral pairs are counted twice is the Stirling subset number S2(n,k). The number of achiral color patterns of length n using exactly 2 colors is S2(floor(n/2)+1,2). The value of a(n-3) is half the difference of these two. - Robert A. Russell, Feb 01 2018
a(n-2) is the number of chiral pairs for a row of n colors with exactly 2 different colors. If the reverse of a sequence is different, the combination of the two is a chiral pair. For a row of 4 colors using exactly 2 different colors, the chiral pairs are AAAB-BAAA, AABA-ABAA, AABB-BBAA, ABAB-BABA, ABBB-BBBA, and BABB-BBAB. Thus a(4-2) = a(2) = 6. - Robert A. Russell, Jun 10 2018

			G.f. = 1 + 2*x + 6*x^2 + 12*x^3 + 28*x^4 + 56*x^5 + 120*x^6 + 240*x^7 + 496*x^8 + ... - _Michael Somos_, Jul 01 2018

Harvey P. Dale, Table of n, a(n) for n = 0..1000
S. J. Cyvin et al., Theory of polypentagons, J. Chem. Inf. Comput. Sci., 33 (1993), 466-474.
E. Filiol and C. Fontaine, Highly nonlinear balanced Boolean functions with a good correlation-immunity, Lect. Not. Comp. Sci 1403 (1998), 475-488, NL(F_n).
Juan B. Gil and Jessica A. Tomasko, Restricted Grassmannian permutations, arXiv:2112.03338 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Essentially the same as A032085.
Cf. A000079, A060546.
Cf. A006516, A020522, A156665.
Cf. A000918, A056309, A056453.

List([0..35],n->2^(n+1)-2^(QuoInt(n+1,2))); # Muniru A Asiru, Sep 27 2018

seq(coeff(series(((1-2*x)*(1-2*x^2))^(-1),x,n+1), x, n), n = 0..35); # Muniru A Asiru, Sep 27 2018

RecurrenceTable[{a[n] == 2 (BitOr[a[n - 1], a[n - 2]]), a[0] == 1, a[1] == 2}, a, {n, 0, 32}] (* Geoffrey Critzer, Jan 09 2011 *)
CoefficientList[Series[1/((1-2x)(1-2x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{2,2,-4},{1,2,6},40] (* Harvey P. Dale, Jun 25 2013 *)
Table[(StirlingS2[n,2] - StirlingS2[Floor[n/2]+1,2])/2, {n,3,30}] (* Robert A. Russell, Jan 29 2018 *)
a[ n_] := 2^(n + 1) - 2^Quotient[n + 1, 2]; (* Michael Somos, Jul 01 2018 *)

{a(n) = 2^(n+1) - 2^((n+1)\2)}; /* Michael Somos, Jul 01 2018 */

From Alexander Adamchuk, Sep 25 2006: (Start)
a(2k) = A006516(k+1) = 2^k*(2^(k+1) - 1) = A020522(k+1) /2.
a(2k+1) = 2*A006516(k+1) = 2^(k+1)*(2^(k+1) - 1) = A020522(k+1). (End)
a(n) = 2^(n+1) - 2^(floor((n+1)/2)). - Geoffrey Critzer, Mar 03 2009
a(n) = 2*(a(n-1) bitwiseOR a(n-2)), a(0)=1, a(1)=2. - Pierre Charland, Dec 12 2010
G.f.: (1+x*Q(0))/(1-x)^2, where Q(k)= 1 - 1/(2^k - 2*x*2^(2*k)/(2*x*2^k - 1/(1 + 1/(2*2^k - 8*x*2^(2*k)/(4*x*2^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 23 2013
a(0)=1, a(1)=2, a(2)=6, a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3). - Harvey P. Dale, Jun 25 2013
a(n) = (A000079(n+2) - A060546(n+2))/ 2. - Robert A. Russell, Jun 19 2018
a(n) = -a(-3-n) * 2^(n+2 + floor((n+1)/2)) for all n in Z. - Michael Somos, Jul 01 2018
a(n) = (A000918(n+2) - A056453(n+2)) / 2 = A000918(n+2) - A056309(n+2) = A056309(n+2) - A056453(n+2). - Robert A. Russell, Sep 26 2018

A228275 A(n,k) = Sum_{i=1..k} n^i; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 6, 3, 0, 0, 4, 14, 12, 4, 0, 0, 5, 30, 39, 20, 5, 0, 0, 6, 62, 120, 84, 30, 6, 0, 0, 7, 126, 363, 340, 155, 42, 7, 0, 0, 8, 254, 1092, 1364, 780, 258, 56, 8, 0, 0, 9, 510, 3279, 5460, 3905, 1554, 399, 72, 9, 0

Offset: 0

Alois P. Heinz, Aug 19 2013

A(n,k) is the total sum of lengths of longest ending contiguous subsequences with the same value over all s in {1,...,n}^k:
A(4,1) = 4 = 1+1+1+1: [1], [2], [3], [4].
A(1,4) = 4: [1,1,1,1].
A(3,2) = 12 = 2+1+1+1+2+1+1+1+2: [1,1], [1,2], [1,3], [2,1], [2,2], [2,3], [3,1], [3,2], [3,3].
A(2,3) = 14 = 3+1+1+2+2+1+1+3: [1,1,1], [1,1,2], [1,2,1], [1,2,2], [2,1,1], [2,1,2], [2,2,1], [2,2,2].

			Square array A(n,k) begins:
  0, 0,  0,   0,    0,     0,      0,      0, ...
  0, 1,  2,   3,    4,     5,      6,      7, ...
  0, 2,  6,  14,   30,    62,    126,    254, ...
  0, 3, 12,  39,  120,   363,   1092,   3279, ...
  0, 4, 20,  84,  340,  1364,   5460,  21844, ...
  0, 5, 30, 155,  780,  3905,  19530,  97655, ...
  0, 6, 42, 258, 1554,  9330,  55986, 335922, ...
  0, 7, 56, 399, 2800, 19607, 137256, 960799, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000004, A001477, A002378, A027444, A027445, A152031, A228290, A228291, A228292, A228293, A228294.
Rows n=0-11 give: A000004, A001477, A000918(k+1), A029858(k+1), A080674, A104891, A105281, A104896, A052379(k-1), A052386, A105279, A105280.
Main diagonal gives A031972.
Lower diagonal gives A226238.
Cf. A228250.

A:= (n, k)-> `if`(n=1, k, (n/(n-1))*(n^k-1)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

a[0, 0] = 0; a[1, k_] := k; a[n_, k_] := n*(n^k-1)/(n-1); Table[a[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 16 2013 *)

A(1,k) = k, else A(n,k) = n/(n-1)*(n^k-1).
A(n,k) = Sum_{i=1..k} n^i.
A(n,k) = Sum_{i=1..k+1} binomial(k+1,i)*A(n-i,k)*(-1)^(i+1) for n>k, given values A(0,k), A(1,k),..., A(k,k). - Yosu Yurramendi, Sep 03 2013

A069532 Smallest even number with digit sum n.

Original entry on oeis.org

10, 2, 12, 4, 14, 6, 16, 8, 18, 28, 38, 48, 58, 68, 78, 88, 98, 198, 298, 398, 498, 598, 698, 798, 898, 998, 1998, 2998, 3998, 4998, 5998, 6998, 7998, 8998, 9998, 19998, 29998, 39998, 49998, 59998, 69998, 79998, 89998, 99998, 199998, 299998, 399998, 499998

Offset: 1

Amarnath Murthy, Apr 01 2002

Chai Wah Wu, Table of n, a(n) for n = 1..8999
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,10,-10).

Cf. A069521 to A069530.
Cf. A000918 (smallest even number with bit sum n), A051885 (smallest number with digit sum n).
Cf. A077491.

t={}; Do[i=2; While[Total[IntegerDigits[i]]!=n,i=i+2]; AppendTo[t,i],{n,48}]; t (* Jayanta Basu, May 18 2013 *)

a(n) = {my(k = 2); while(sumdigits(k) != n, k+=2); k;} \\ Michel Marcus, Mar 18 2016

From Chai Wah Wu, Sep 15 2020: (Start)
a(n) = a(n-1) + 10*a(n-9) - 10*a(n-10) for n > 17.
G.f.: 2*x*(45*x^16 - 45*x^15 + 45*x^14 - 45*x^13 + 45*x^12 - 45*x^11 + 45*x^10 - 45*x^9 + 5*x^8 - 4*x^7 + 5*x^6 - 4*x^5 + 5*x^4 - 4*x^3 + 5*x^2 - 4*x + 5)/((x - 1)*(10*x^9 - 1)). (End)
a(n) = 2 * A077491(n). - Alois P. Heinz, Sep 15 2020

More terms from Ray Chandler, Jul 28 2003

A173787 Triangle read by rows: T(n,k) = 2^n - 2^k, 0 <= k <= n.

Original entry on oeis.org

0, 1, 0, 3, 2, 0, 7, 6, 4, 0, 15, 14, 12, 8, 0, 31, 30, 28, 24, 16, 0, 63, 62, 60, 56, 48, 32, 0, 127, 126, 124, 120, 112, 96, 64, 0, 255, 254, 252, 248, 240, 224, 192, 128, 0, 511, 510, 508, 504, 496, 480, 448, 384, 256, 0, 1023, 1022, 1020, 1016, 1008, 992, 960, 896, 768, 512, 0

Offset: 0

Reinhard Zumkeller, Feb 28 2010

			Triangle begins as:
   0;
   1,  0;
   3,  2,  0;
   7,  6,  4,  0;
  15, 14, 12,  8,  0;
  31, 30, 28, 24, 16, 0;

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

[2^n -2^k: k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 13 2021

Table[2^n -2^k, {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 13 2021 *)

flatten([[2^n -2^k for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 13 2021

A000120(T(n,k)) = A025581(n,k).
Row sums give A000337.
Central terms give A020522.
T(2*n+1, n) = A006516(n+1).
T(2*n+3, n+2) = A059153(n).
T(n, k) = A140513(n,k) - A173786(n,k), 0 <= k <= n.
T(n, k) = A173786(n,k) - A059268(n+1,k+1), 0 < k <= n.
T(2*n, 2*k) = T(n,k) * A173786(n,k), 0 <= k <= n.
T(n, 0) = A000225(n).
T(n, 1) = A000918(n) for n>0.
T(n, 2) = A028399(n) for n>1.
T(n, 3) = A159741(n-3) for n>3.
T(n, 4) = A175164(n-4) for n>4.
T(n, 5) = A175165(n-5) for n>5.
T(n, 6) = A175166(n-6) for n>6.
T(n, n-4) = A110286(n-4) for n>3.
T(n, n-3) = A005009(n-3) for n>2.
T(n, n-2) = A007283(n-2) for n>1.
T(n, n-1) = A000079(n-1) for n>0.
T(n, n) = A000004(n).

A285332 a(0) = 1, a(1) = 2, a(2n) = A019565(a(n)), a(2n+1) = A065642(a(n)).

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 5, 8, 15, 12, 14, 27, 10, 25, 7, 16, 210, 45, 35, 18, 105, 28, 462, 81, 21, 20, 154, 125, 30, 49, 11, 32, 10659, 420, 910, 75, 78, 175, 33, 24, 3094, 315, 385, 56, 780045, 924, 374, 243, 110, 63, 55, 40, 4389, 308, 170170, 625, 1155, 60, 286, 343, 42, 121, 13, 64, 54230826, 31977, 28405, 630, 1330665, 1820, 714

Offset: 0

Antti Karttunen, Apr 17 2017

Note the indexing: the domain starts from 0, while the range excludes zero.
This sequence can be represented as a binary tree. Each left hand child is produced as A019565(n), and each right hand child as A065642(n), when the parent node contains n >= 2:
                                     1
                                     |
                  ...................2...................
                 3                                       4
       6......../ \........9                   5......../ \........8
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  15       12         14       27         10       25          7       16
210 45   35  18    105  28  462  81     21  20  154  125     30 49   11  32
etc.
Where will 38 appear in this tree? It is a reasonable assumption that by iterating A087207 starting from 38, as A087207(38) = 129, A087207(129) = 8194, A087207(8194) = 1501199875790187, ..., we will eventually hit a prime A000040(k), most likely with a largish index k. This prime occurs at the penultimate edge at right, as a(A000918(k)) = a((2^k)-2), and thus 38 occurs somewhere below it as a(m) = 38, m > k. All the numbers that share prime factors with 38, namely 76, 152, 304, 608, 722, ..., occur similarly late in this tree, as they form the rightward branch starting from 38. Alternatively, by iterating A285330 (each iteration moves one step towards the root) starting from 38, we might instead first hit some power of 3, or say, one of the terms of A033845 (the rightward branch starting from 6), in which case the first prime encountered would be a(2)=3 and 38 would appear on the left-hand side instead of the right-hand side subtree.
As long as it remains conjecture that A019565 has no cycles, it is certainly also an open question whether this is a permutation of the natural numbers: If A019565 has any cycles, then neither any of the terms in those cycles nor any A065642-trajectories starting from those terms (that is, numbers sharing same prime factors) may occur in this tree.
Sequence exhibits some outrageous swings, for example, a(703) = 224, but a(704) is 1427 decimal digits (4739 binary digits) long, thus it no longer fits into a b-file.
However, the scatter plot of A286543 gives some flavor of the behavior of this sequence even after that point. - Antti Karttunen, Dec 25 2017

Antti Karttunen, Table of n, a(n) for n = 0..703
Michael De Vlieger, Diagram of the binary tree of a(n) showing 1 <= n <= 2^8.
Antti Karttunen and David J. Seal, Discussion on SeqFan mailing list
Index entries for sequences that are permutations of the natural numbers

Inverse: A285331.
Cf. A007947, A019565, A033845, A048675, A065642, A087207, A109162, A285319, A285320, A285328, A285330, A285333, A286542, A286543.
Compare also to permutation A285112 and array A285321.

Block[{a = {1, 2}}, Do[AppendTo[a, If[EvenQ[i], Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[a[[i/2 + 1]], 2], If[# == 1, 1, Function[{n, c}, SelectFirst[Range[n + 1, n^2], Times @@ FactorInteger[#][[All, 1]] == c &]] @@ {#, Times @@ FactorInteger[#][[All, 1]]}] &[a[[(i - 1)/2 + 1]] ] ]], {i, 2, 70}]; a] (* Michael De Vlieger, Mar 12 2021 *)

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A007947(n) = factorback(factorint(n)[, 1]); \\ From Andrew Lelechenko, May 09 2014
A065642(n) = { my(r=A007947(n)); if(1==n,n,n = n+r; while(A007947(n) <> r, n = n+r); n); };
A285332(n) = { if(n<=1,n+1,if(!(n%2),A019565(A285332(n/2)),A065642(A285332((n-1)/2)))); };
for(n=0, 4095, write("b285332.txt", n, " ", A285332(n)));

from operator import mul
from sympy import prime, primefactors
def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1 # This function from Chai Wah Wu
def a065642(n):
    if n==1: return 1
    r=a007947(n)
    n = n + r
    while a007947(n)!=r:
        n+=r
    return n
def a(n):
    if n<2: return n + 1
    if n%2==0: return a019565(a(n//2))
    else: return a065642(a((n - 1)//2))
print([a(n) for n in range(51)]) # Indranil Ghosh, Apr 18 2017

;; With memoization-macro definec.
(definec (A285332 n) (cond ((<= n 1) (+ n 1)) ((even? n) (A019565 (A285332 (/ n 2)))) (else (A065642 (A285332 (/ (- n 1) 2))))))

a(0) = 1, a(1) = 2, a(2n) = A019565(a(n)), a(2n+1) = A065642(a(n)).
For n >= 0, a(2^n) = A109162(2+n). [The left edge of the tree.]
For n >= 0, a(A000225(n)) = A000079(n). [Powers of 2 occur at the right edge of the tree.]
For n >= 2, a(A000918(n)) = A000040(n). [And the next vertices inwards contain primes.]
For n >= 2, a(A036563(1+n)) = A001248(n). [Whose right children are their squares.]
For n >= 0, a(A055010(n)) = A000244(n). [Powers of 3 are at the rightmost edge of the left subtree.]
For n >= 2, a(A129868(n-1)) = A062457(n).
A048675(a(n)) = A285333(n).
A046523(a(n)) = A286542(n).

A000919 a(n) = 4^n - C(4,3)3^n + C(4,2)2^n - C(4,1).

Original entry on oeis.org

0, 0, 0, 24, 240, 1560, 8400, 40824, 186480, 818520, 3498000, 14676024, 60780720, 249401880, 1016542800, 4123173624, 16664094960, 67171367640, 270232006800, 1085570781624, 4356217681200, 17466686971800, 69992221794000, 280345359228024, 1122510953731440

Offset: 1

N. J. A. Sloane

Differences of 0: 4!*S(n,4).
Number of surjections from an n-element set onto a four-element set. - David Wasserman, Jun 06 2007
Number of rows of n colors using exactly four colors.  For n=4, the 24 rows are the 24 permutations of ABCD. - Robert A. Russell, Sep 25 2018

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 212.
K. S. Immink, Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols, http://www.exp-math.uni-essen.de/~immink/pdf/jsac13.pdf, 2013. [This link no longer works, but please do not delete this reference, for historical reasons. Michel Marcus has suggested that the Immink link below points to the published version of the original reference, and I agree. - N. J. A. Sloane, May 29 2023]
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54.

T. D. Noe, Table of n, a(n) for n = 1..201
K. S. Immink, Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols, Electronics Letters 50(1):20-22, January 2014.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Ethan Soloway, Megan Triplett, and Wenshi Zhao, On the Existence of Partition of the Hypercube Graph into 3 Initial Segments, arXiv:2501.05827 [math.CO], 2025. See p. 23.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only]
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A001117, A001118.

Column 4 of A019538.

with (combstruct):ZL:=[S,{S=Sequence(U,card=r),U=Set(Z,card>=1)}, labeled]: seq(count(subs(r=4,ZL),size=m),m=1..25); # Zerinvary Lajos, Mar 09 2007
A000919:=24/(z-1)/(3*z-1)/(2*z-1)/(4*z-1); # Simon Plouffe in his 1992 dissertation

nn = 25; CoefficientList[Series[24 x^3/((1 - x) (1 - 2 x) (1 - 3 x) (1 - 4 x)), {x, 0, nn}], x] (* T. D. Noe, Jun 20 2012 *)
k=4; Table[k!StirlingS2[n,k],{n,1,30}] (* Robert A. Russell, Sep 25 2018 *)

a(n) = 4!*stirling(n, 4, 2); \\ Altug Alkan, Sep 25 2018

G.f.: 24*x^3/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)).
a(n) = 4^n - binomial(4,3)*3^n + binomial(4,2)*2^n - binomial(4,1) = 24*A000453(n). - David Wasserman, Jun 06 2007
E.g.f.: (exp(x)-1)^4. - Geoffrey Critzer, Feb 11 2009
For n >= 4: a(n+1) = 4*a(n) + 4*(3^n - 3*2^n + 3) = 4*a(n) + 4*A001117(n). - Geoffrey Critzer, Feb 27 2009
a(n) = k!*S2(n,k), where k=4 is the number of colors and S2 is the Stirling subset number. - Robert A. Russell, Sep 25 2018

A001118 Number of labeled ordered set partitions into 5 parts for n>=1, a(0)=1.

Original entry on oeis.org

1, 0, 0, 0, 0, 120, 1800, 16800, 126000, 834120, 5103000, 29607600, 165528000, 901020120, 4809004200, 25292030400, 131542866000, 678330198120, 3474971465400, 17710714165200, 89904730860000, 454951508208120, 2296538629446600, 11570026582092000, 58200094019430000

Offset: 0

N. J. A. Sloane

Previous name: Differences of 0; labeled ordered partitions into 5 parts.
Number of surjections from an n-element set onto a five-element set, with n >= 5. - Mohamed Bouhamida, Dec 15 2007
For n > 0, the number of rows of n colors using exactly five colors. For n=5, the 120 rows are the 120 permutations of ABCDE. - Robert A. Russell, Sep 25 2018

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 212.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
K. S. Immink, Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols, 2013.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Leipzig, 1911.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only]
Index entries for linear recurrences with constant coefficients, signature (15, -85, 225, -274, 120).

Cf. A001117, A000919, A000920.

Column 5 of A019538, n > 0.

A001118:=-120/(z-1)/(4*z-1)/(3*z-1)/(2*z-1)/(5*z-1); # Conjectured (correctly) by Simon Plouffe in his 1992 dissertation. Gives sequence except for 5 leading terms.

CoefficientList[Series[(-1-274*x^4+225*x^3-85*x^2+15*x)/((x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1)),{x,0,30}],x] (* Vincenzo Librandi, Apr 11 2012 *)
k=5; Prepend[Table[k!StirlingS2[n,k],{n,1,30}],1] (* Robert A. Russell, Sep 25 2018 *)

a(n) = sum(i=0, 4, (-1)^i*binomial(5, i)*(5-i)^n); \\ Altug Alkan, Dec 04 2015

Vec((-274*x^4 + 225*x^3 - 85*x^2 + 15*x - 1)/((x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1))+O(x^30)) \\ Stefano Spezia, Oct 16 2018

a(n) = Sum_{i=0..4} (-1)^i*binomial(5, i)*(5-i)^n.
a(n) = [n=0] + 5!*S(n, 5).
E.g.f.: 1 + (e^x-1)^5.
a(n) = 5^n - C(5,4)*4^n + C(5,3)*3^n - C(5,2)*2^n + C(5,1). - Mohamed Bouhamida, Dec 15 2007
G.f.: (-274*x^4 + 225*x^3 - 85*x^2 + 15*x - 1)/((x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009

Extended with formula and alternate description by Christian G. Bower, Aug 15 1998

Name edited by Harry Richman, Mar 31 2023

A003063 a(n) = 3^(n-1) - 2^n.

Original entry on oeis.org

-1, -1, 1, 11, 49, 179, 601, 1931, 6049, 18659, 57001, 173051, 523249, 1577939, 4750201, 14283371, 42915649, 128878019, 386896201, 1161212891, 3484687249, 10456158899, 31372671001, 94126401611, 282395982049, 847221500579, 2541731610601, 7625329049531, 22876255584049

Offset: 1

Henrik Johansson (Henrik.Johansson(AT)Nexus.SE)

Binomial transform of A000918: (-1, 0, 2, 6, 14, 30, ...). - Gary W. Adamson, Mar 23 2012

This sequence demonstrates 2^n as a loose lower bound for g(n) in Waring's problem. Since 3^n > 2(2^n) for all n > 2, the number 2^(n + 1) - 1 requires 2^n n-th powers for its representation since 3^n is not available for use in the sum: the gulf between the relevant powers of 2 and 3 widens considerably as n gets progressively larger. - Alonso del Arte, Feb 01 2013

			a(3) = 1 because 3^2 - 2^3 = 9 - 8 = 1.
a(4) = 11 because 3^3 - 2^4 = 27 - 16 = 11.
a(5) = 49 because 3^4 - 2^5 = 81 - 32 = 49.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
D. Knuth, Letter to N. J. A. Sloane, date unknown
Index entries for linear recurrences with constant coefficients, signature (5,-6).

Cf. A000918, A056182 (first differences), A064686, A083313, A214091, A369490.
Cf. A363024 (prime terms).
From the third term onward the first differences of A005173.
Difference between two leftmost columns of A090888.
A diagonal in A254027.
Right edge of irregular triangle A252750.

[3^(n-1) -2^n: n in [1..30]]; // G. C. Greubel, Nov 03 2022

Table[3^(n-1) - 2^n, {n, 25}] (* Alonso del Arte, Feb 01 2013 *)
LinearRecurrence[{5,-6},{-1,-1},30] (* Harvey P. Dale, Feb 02 2015 *)

a(n)=3^(n-1)-2^n \\ Charles R Greathouse IV, Oct 07 2015

[3^(n-1) -2^n for n in range(1,31)] # G. C. Greubel, Nov 03 2022

Let b(n) = 2*(3/2)^n - 1. Then a(n) = -b(1-n)*3^(n-1) for n > 0. A083313(n) = A064686(n) = b(n)*2^(n-1) for n > 0. - Michael Somos, Aug 06 2006
From Colin Barker, May 27 2013: (Start)
a(n) = 5*a(n-1) - 6*a(n-2).
G.f.: -x*(1-4*x) / ((1-2*x)*(1-3*x)). (End)
E.g.f.: (1/3)*(2 - 3*exp(2*x) + exp(3*x)). - G. C. Greubel, Nov 03 2022

A few more terms from Alonso del Arte, Feb 01 2013

cp's OEIS Frontend

A075427 a(0) = 1; a(n) = a(n-1)+1 if n is even, otherwise a(n) = 2*a(n-1).

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A014410 Elements in Pascal's triangle (by row) that are not 1.

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A122746 G.f.: 1/((1-2*x)*(1-2*x^2)).

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A228275 A(n,k) = Sum_{i=1..k} n^i; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A069532 Smallest even number with digit sum n.

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A173787 Triangle read by rows: T(n,k) = 2^n - 2^k, 0 <= k <= n.

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A122746 G.f.: 1/((1-2x)(1-2*x^2)).

A000919 a(n) = 4^n - C(4,3)3^n + C(4,2)2^n - C(4,1).