A005010 -id:A005010 - cp's OEIS frontend

A280345 a(0) = 3, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [1, -2].

3, 7, 12, 25, 48, 97, 192, 385, 768, 1537, 3072, 6145, 12288, 24577, 49152, 98305, 196608, 393217, 786432, 1572865, 3145728, 6291457, 12582912, 25165825, 50331648, 100663297, 201326592, 402653185, 805306368, 1610612737, 3221225472, 6442450945, 12884901888

Offset: 0

Paul Curtz, Jan 01 2017

a(n) mod 9 is a periodic sequence of length 2: repeat [3, 7].
From 7, the last digit is of period 4: repeat [7, 2, 5, 8].
(Main sequence for the signature (2,1,-2): 0, 0, 1, 2, 5, 10, 21, 42, ... = 0 followed by A000975(n) = b(n), which first differences are A001045(n) (Paul Barry, Oct 08 2005). Then, 0 followed by b(n) is an autosequence of the first kind. The corresponding autosequence of the second kind is 0, 0, 2, 3, 8, 15, 32, 63, ... . See A277078(n).)
Difference table of a(n):
3,   7, 12, 25, 48,  97, 192, ...
4,   5, 13, 23, 49,  95, 193, ...  = -(-1)^n* A140683(n)
1,   8, 10, 26, 46,  98, 190, ...  = A259713(n)
7,   2, 16, 20, 52,  92, 196, ...
-5, 14,  4, 32, 40, 104, 184, ...
... .

			a(0) = 3, a(1) = 2*3 + 1 = 7, a(2) = 2*7 - 2 = 12, a(3) = 2*12 + 1 = 25.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A005010, A051049, A140657, A140683, A164346, A199116, A259713.

a[0] = 3; a[n_] := a[n] = 2 a[n - 1] + 1 + (-3) Boole[EvenQ@ n]; Table[a@ n, {n, 0, 32}] (* or *)
CoefficientList[Series[(3 + x - 5 x^2)/((1 - x) (1 + x) (1 - 2 x)), {x, 0, 32}], x] (* Michael De Vlieger, Jan 01 2017 *)

Vec((3 + x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Jan 01 2017

a(2n) = 3*4^n, a(2n+1) = 6*4^n + 1.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), n>2.
a(n+2) = a(n) + 9*2^n.
a(n) = 2^(n+2) - A051049(n).
From Colin Barker, Jan 01 2017: (Start)
a(n) = 3*2^n for n even.
a(n) = 3*2^n + 1 for n odd.
G.f.: (3 + x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).
(End)
Binomial transform of 3, followed by (-1)^n* A140657(n).

More terms from Colin Barker, Jan 01 2017

A281166 a(n) = 3a(n-1) - 3a(n-2) + 2*a(n-3) for n>2, a(0)=a(1)=1, a(2)=3.

Original entry on oeis.org

1, 1, 3, 8, 17, 33, 64, 127, 255, 512, 1025, 2049, 4096, 8191, 16383, 32768, 65537, 131073, 262144, 524287, 1048575, 2097152, 4194305, 8388609, 16777216, 33554431, 67108863, 134217728, 268435457, 536870913, 1073741824, 2147483647, 4294967295, 8589934592

Offset: 0

Paul Curtz, Jan 16 2017

a(n) is the first sequence on three (with its first and second differences):
1, 1, 3, 8, 17, 33, 64, 127, ...;
0, 2, 5, 9, 16, 31, 63, 128, ..., that is 0 followed by A130752;
2, 3, 4, 7, 15, 32, 65, 129, ..., that is 2 followed by A130755;
1, 1, 3, 8, 17, 33, 64, 127, ..., this sequence.
The main diagonal is 2^n.
The sum of the first three lines is 3*2^n.
Alternated sum and subtraction of a(n) and its inverse binomial transform (period 3: repeat [1, 0, 2]) gives the autosequence of the first kind b(n):
   0,  1,  1,  9, 17, 35, 63, 127, ...
   1,  0,  8,  8, 18, 28, 64, 126, ...
  -1,  8,  0, 10, 10, 36, 62, 134, ...
   9, -8, 10,  0, 26, 26, 72, 118, ... .
The main diagonal is 0's. The first two upper diagonals are A259713.
The sum of the first three lines gives 9*A001045.
a(n) mod 9 gives a periodic sequence of length 6: repeat [1, 1, 3, 8, 8, 6].
a(n) = A130750(n-1) for n > 2. - Georg Fischer, Oct 23 2018

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

Cf. A000079, A001045, A005010, A007283, A014551 (a diagonal), A057079, A062510 (a diagonal), A128834, A130750, A130752, A130755, A153234, A153237, A259713.

I:=[1,1,3]; [n le 3 select I[n] else 3*Self(n-1) - 3*Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jan 15 2018

LinearRecurrence[{3, -3, 2}, {1, 1, 3}, 30] (* Jean-François Alcover, Jan 16 2017 *)

Vec((1 - 2*x + 3*x^2) / ((1 - 2*x)*(1 - x + x^2)) + O(x^40)) \\ Colin Barker, Jan 16 2017

Binomial transform of the sequence of length 3: repeat [1, 0, 2].
a(n+3) = -a(n) + 9*2^n.
a(n) = 2^n - periodic 6: repeat [0, 1, 1, 0, -1, -1, 0].
a(n+6) = a(n) + 63*2^n.
a(n+1) = 2*a(n) - period 6: repeat [1, -1, -2, -1, 1, 2].
a(n) = 2^n - 2*sin(Pi*n/3)/sqrt(3). - Jean-François Alcover and Colin Barker, Jan 16 2017
G.f.: (1 - 2*x + 3*x^2)/((1 - 2*x)*(1 - x + x^2)). - Colin Barker, Jan 16 2017

A336715 Numbers m that divide the product phi(m) * tau(m), where tau is the number of divisors function (A000005) and phi is the Euler totient function (A000010).

Original entry on oeis.org

1, 2, 8, 9, 12, 18, 32, 36, 72, 80, 96, 108, 128, 144, 243, 288, 324, 400, 448, 486, 512, 576, 625, 720, 768, 864, 972, 1152, 1200, 1250, 1344, 1620, 1944, 2000, 2025, 2048, 2304, 2500, 2560, 2592, 2916, 3136, 3600, 3888, 4032, 4050, 4608, 5000, 5103, 5625, 6144, 6561, 6912

Offset: 1

Bernard Schott, Aug 01 2020

nonn

Numbers of the form q = 2^(2k+1) with k>=0 (A004171) form a subsequence because tau(q) * phi(q) / q = k + 1.

Numbers of the form q = 9 * 2^k with k>=0 (A005010) form another subsequence because tau(q) * phi(q) / q = k+1 (also).

			For 80, phi(80) = 32, tau(80) = 10 and tau(80)*phi(80)/80 = 4, hence 80 is a term.

David A. Corneth, Table of n, a(n) for n = 1..12173 (terms <= 10^15)

Cf. A000010 (phi), A000005 (tau), A062355.

Subsequences: A004171, A005010.

with(numtheory):
filter:= m-> irem(phi(m)*tau(m), m)=0:
select(filter, [$1..7000])[];

Select[Range[7000], Divisible[DivisorSigma[0, #] * EulerPhi[#], #] &] (* Amiram Eldar, Aug 01 2020 *)

isok(m) = (eulerphi(m)*numdiv(m) % m) == 0; \\ Michel Marcus, Aug 02 2020

A155118 Array T(n,k) read by antidiagonals: the k-th term of the n-th iterated differences of A140429.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 3, 4, 6, 9, 5, 8, 12, 18, 27, 11, 16, 24, 36, 54, 81, 21, 32, 48, 72, 108, 162, 243, 43, 64, 96, 144, 216, 324, 486, 729, 85, 128, 192, 288, 432, 648, 972, 1458, 2187, 171, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 341, 512, 768, 1152, 1728, 2592, 3888, 5832, 8748, 13122, 19683

Offset: 0

Paul Curtz, Jan 20 2009

Deleting column k=0 and reading by antidiagonals yields A036561.

Deleting column k=0 and reading the antidiagonals downwards yields A175840.

			The array starts in row n=0 with columns k>=0 as:
   0   1    3    9    27    81    243    729    2187  ... A140429;
   1   2    6   18    54   162    486   1458    4374  ... A025192;
   1   4   12   36   108   324    972   2916    8748  ... A003946;
   3   8   24   72   216   648   1944   5832   17496  ... A080923;
   5  16   48  144   432  1296   3888  11664   34992  ... A257970;
  11  32   96  288   864  2592   7776  23328   69984  ...
  21  64  192  576  1728  5184  15552  46656  139968  ...
Antidiagonal triangle begins as:
   0;
   1,   1;
   1,   2,   3;
   3,   4,   6,   9;
   5,   8,  12,  18,  27;
  11,  16,  24,  36,  54,  81;
  21,  32,  48,  72, 108, 162, 243;
  43,  64,  96, 144, 216, 324, 486, 729;
  85, 128, 192, 288, 432, 648, 972, 1458, 2187; - _G. C. Greubel_, Mar 25 2021

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Cf. A001045, A004054, A046936.

t:= func< n,k | k eq 0 select (2^(n-k) -(-1)^(n-k))/3 else 2^(n-k)*3^(k-1) >;
[t(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 25 2021

T:=proc(n,k)if(k>0)then return 2^n*3^(k-1):else return (2^n - (-1)^n)/3:fi:end:
for d from 0 to 8 do for m from 0 to d do print(T(d-m,m)):od:od: # Nathaniel Johnston, Apr 13 2011

t[n_, k_]:= If[k==0, (2^(n-k) -(-1)^(n-k))/3, 2^(n-k)*3^(k-1)];
Table[t[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 25 2021 *)

def A155118(n,k): return (2^(n-k) -(-1)^(n-k))/3 if k==0 else 2^(n-k)*3^(k-1)
flatten([[A155118(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 25 2021

For the square array:
T(n,k) = 2^n*3^(k-1), k>0.
T(n,k) = T(n-1,k+1) - T(n-1,k), n>0.
Rows:
T(0,k) = A140429(k) = A000244(k-1).
T(1,k) = A025192(k).
T(2,k) = A003946(k).
T(3,k) = A080923(k+1).
T(4,k) = A257970(k+3).
Columns:
T(n,0) = A001045(n) (Jacobsthal numbers J_{n}).
T(n,1) = A000079(n).
T(n,2) = A007283(n).
T(n,3) = A005010(n).
T(n,4) = A175806(n).
T(0,k) - T(k+1,0) = 4*A094705(k-2).
From G. C. Greubel, Mar 25 2021: (Start)
For the antidiagonal triangle:
t(n, k) = T(n-k, k).
t(n, k) = (2^(n-k) - (-1)^(n-k))/3 (J_{n-k}) if k = 0 else 2^(n-k)*3^(k-1).
Sum_{k=0..n} t(n, k) = 3^n - J_{n+1}, where J_{n} = A001045(n).
Sum_{k=0..n} t(n, k) = A004054(n-1) for n >= 1. (End)

a(22) - a(57) from Nathaniel Johnston, Apr 13 2011

A159022 a(0)=29; a(n) = a(n-1) + floor(sqrt(a(n-1))), n > 0.

Original entry on oeis.org

29, 34, 39, 45, 51, 58, 65, 73, 81, 90, 99, 108, 118, 128, 139, 150, 162, 174, 187, 200, 214, 228, 243, 258, 274, 290, 307, 324, 342, 360, 378, 397, 416, 436, 456, 477, 498, 520, 542, 565, 588, 612, 636, 661, 686, 712, 738, 765, 792, 820, 848, 877, 906, 936, 966, 997, 1028

Offset: 0

Philippe Deléham, Apr 02 2009

nonn

Row 4 in square array A159016.
This sequence contains infinitely many squares. - Philippe Deléham, Apr 04 2009
There are 10 squares in the first 10000 terms. - Harvey P. Dale, Aug 26 2019

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A028392, A005010.

NestList[#+Floor[Sqrt[#]]&,29,60] (* Harvey P. Dale, Aug 26 2019 *)

More terms from Vincenzo Librandi, Apr 10 2009

A195332 Numbers such that the sum of the cube of the odd divisors is prime.

Original entry on oeis.org

9, 18, 36, 72, 121, 144, 242, 288, 484, 576, 968, 1152, 1936, 2304, 3872, 4608, 7744, 9216, 15488, 18432, 30976, 36481, 36864, 61952, 72361, 72962, 73728, 123904, 144722, 145924, 146689, 147456, 247808, 259081, 289444, 291848, 293378, 294912

Offset: 1

Michel Lagneau, Sep 15 2011

nonn

a(n) is of the form m^2 or 2*m^2.
See the comments in A195268 (numbers such that the sum of the odd divisors is prime).
It is interesting to observe that the intersection of this sequence with A195268 gives {9, 18, 36, 72, 144, 288, 576, 1152, 2304, 4608, 9216, 18432, 36864, 73728, 146689, 147456, 293378, 294912,...} and contains the sequence A005010(n) (numbers of the form 9*2^n), but is not equal to this sequence. For example, up to n = 400000, the numbers 146689 and 293378 are not divisible by 9.

			The divisors of 18 are  { 1, 2, 3, 6, 9, 18}, and the sum of the cube of the odd divisors 1^3 + 3^3 + 9^3 =757 is prime. Hence 18 is in the sequence.

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

Cf. A005010, A066100 (sqrt of odd numbers here), A195268.

with(numtheory):for n from 1 to 400000 do:x:=divisors(n):n1:=nops(x):s:=0:for m from 1 to n1 do:if irem(x[m],2)=1 then s:=s+x[m]^3:fi:od:if type(s,prime)=true  then printf(`%d, `,n): else fi:od:

  Module[{c=Range[800]^2,m},m=Sort[Join[c,2c]];Select[m,PrimeQ[Total[ Select[ Divisors[#],OddQ]^3]]&]](* Harvey P. Dale, Jul 31 2012 *)

A370882 Square array T(n,k) = 9*2^k - n read by ascending antidiagonals.

Original entry on oeis.org

9, 8, 18, 7, 17, 36, 6, 16, 35, 72, 5, 15, 34, 71, 144, 4, 14, 33, 70, 143, 288, 3, 13, 32, 69, 142, 287, 576, 2, 12, 31, 68, 141, 286, 575, 1152, 1, 11, 30, 67, 140, 285, 574, 1151, 2304, 0, 10, 29, 66, 139, 284, 573, 1150, 2303, 4608, -1, 9, 28, 65, 138, 283, 572, 1149, 2302, 4607, 9216

Offset: 0

Paul Curtz, Mar 05 2024

Just after A367559 and A368826.

			Table begins:
       k=0  1  2  3   4   5
  n=0:   9 18 36 72 144 288 ...
  n=1:   8 17 35 71 143 287 ...
  n=2:   7 16 34 70 142 286 ...
  n=3:   6 15 33 69 141 285 ...
  n=4:   5 14 32 68 140 284 ...
  n=5:   4 13 31 67 139 283 ...
Every line has the signature (3,-2). For n=1: 3*17 - 2*8 = 35.
Main diagonal's difference table:
  9   17   34   69   140   283   570  1145  ...  =  b(n)
  8   17   35   71   143   287   575  1151  ...  =  A052996(n+2)
  9   18   36   72   144   288   576  1152  ...  =  A005010(n)
  ...
b(n+1) - 2*b(n) = A023443(n).

Cf. A000225, A033484, A048491, A005010, A052996, A053209, A083329, A154251, A176449, A304383, A367559, A368826.

Cf. A023443.

T[n_, k_] := 9*2^k - n; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Mar 06 2024 *)

T(0,k) = 9*2^k     =   A005010(k);
T(1,k) = 9*2^k - 1 =   A052996(k+2);
T(2,k) = 9*2^k - 2 =   A176449(k);
T(3,k) = 9*2^k - 3 = 3*A083329(k);
T(4,k) = 9*2^k - 4 =   A053209(k);
T(5,k) = 9*2^k - 5 =   A304383(k+3);
T(6,k) = 9*2^k - 6 = 3*A033484(k);
T(7,k) = 9*2^k - 7 =   A154251(k+1);
T(8,k) = 9*2^k - 8 =   A048491(k);
T(9,k) = 9*2^k - 9 = 3*A000225(k).
G.f.: (9 - 9*y + x*(11*y - 10))/((1 - x)^2*(1 - y)*(1 - 2*y)). - Stefano Spezia, Mar 17 2024

cp's OEIS Frontend

A280345 a(0) = 3, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [1, -2].

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A281166 a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) for n>2, a(0)=a(1)=1, a(2)=3.

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A336715 Numbers m that divide the product phi(m) * tau(m), where tau is the number of divisors function (A000005) and phi is the Euler totient function (A000010).

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A155118 Array T(n,k) read by antidiagonals: the k-th term of the n-th iterated differences of A140429.

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A159022 a(0)=29; a(n) = a(n-1) + floor(sqrt(a(n-1))), n > 0.

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A195332 Numbers such that the sum of the cube of the odd divisors is prime.

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Maple

Mathematica

A370882 Square array T(n,k) = 9*2^k - n read by ascending antidiagonals.

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A281166 a(n) = 3a(n-1) - 3a(n-2) + 2*a(n-3) for n>2, a(0)=a(1)=1, a(2)=3.