A020714 -id:A020714 - cp's OEIS frontend

A146951 Numbers that are congruent to 0 or 6 mod 10.

0, 6, 10, 16, 20, 26, 30, 36, 40, 46, 50, 56, 60, 66, 70, 76, 80, 86, 90, 96, 100, 106, 110, 116, 120, 126, 130, 136, 140, 146, 150, 156, 160, 166, 170, 176, 180, 186, 190, 196, 200, 206, 210, 216, 220, 226, 230, 236, 240, 246, 250, 256, 260, 266, 270, 276, 280

Offset: 0

Paul Curtz, Nov 03 2008

Rank of terms of A061047 ending in with 0.

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

Cf. A001622, A020714, A030308, A061047.

I:=[0, 6, 10]; [n le 3 select I[n] else Self(n-1)+Self(n-2)-Self(n-3): n in [1..50]]; // Vincenzo Librandi, May 18 2012

CoefficientList[Series[x*(6+4*x)/((1-x)^2*(1+x)),{x,0,50}],x] (* Vincenzo Librandi, May 18 2012 *)

a(n) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=6 and b(k) = 5*2^k = A020714(k) for k > 0. - Philippe Deléham, Oct 18 2011
From Colin Barker, May 15 2012: (Start)
a(n) = 1/2 - (-1)^n/2 + 5*n.
a(n) = a(n-1) + a(n-2) - a(n-3).
G.f.: x*(6+4*x)/((1-x)^2*(1+x)). (End)
E.g.f.: 5*x*exp(x) + sinh(x). - Stefano Spezia, May 14 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = log(5)/8 - sqrt(1-2/sqrt(5))*Pi/20 - log(phi)/(4*sqrt(5)), where phi is the golden ratio (A001622). - Amiram Eldar, Sep 15 2022

Replaced definition by a comment from Philippe Deléham, Oct 18 2011. Afer the change this becomes a list, but it is better to keep the offset as 0. - N. J. A. Sloane, Sep 08 2022

A147675 Divide by 2, multiply by 4, repeat.

Original entry on oeis.org

10, 5, 20, 10, 40, 20, 80, 40, 160, 80, 320, 160, 640, 320, 1280, 640, 2560, 1280, 5120, 2560, 10240, 5120, 20480, 10240, 40960, 20480, 81920, 40960, 163840, 81920, 327680, 163840, 655360, 327680, 1310720, 655360, 2621440, 1310720, 5242880

Offset: 1

N. J. A. Sloane, Apr 21 2009

A147675-A147678 are from a quiz that someone asked me to help them with.

Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A020714, A135530.

LinearRecurrence[{0, 2}, {10, 5}, 50] (* Paolo Xausa, Jan 30 2024 *)

From R. J. Mathar, Apr 22 2009: (Start)
a(n) = 2*a(n-2) = 5*A135530(n-1).
G.f.: 5*x*(2+x)/(1-2*x^2). (End)
From Amiram Eldar, Feb 02 2024: (Start)
Sum_{n>=1} 1/a(n) = 3/5.
Sum_{n>=1} (-1)^n/a(n) = 1/5. (End)

More terms from R. J. Mathar, Apr 22 2009

A164682 a(n) = 2*a(n-2) for n > 2; a(1) = 5, a(2) = 8.

Original entry on oeis.org

5, 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 256, 320, 512, 640, 1024, 1280, 2048, 2560, 4096, 5120, 8192, 10240, 16384, 20480, 32768, 40960, 65536, 81920, 131072, 163840, 262144, 327680, 524288, 655360, 1048576, 1310720, 2097152, 2621440, 4194304

Offset: 1

Klaus Brockhaus, Aug 21 2009

Interleaving of A020714 and A000079 without initial terms 1, 2, 4.
First differences are in A162255.
Binomial transform is A135532 without initial terms -1, 3. Fourth binomial transform is A164537.

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

Equals A094958 (numbers of the form 2^n or 5*2^n) without initial terms 1, 2, 4.

Cf. A020714 (5*2^n), A000079 (powers of 2), A162255, A135532, A164537.

[ n le 2 select 2+3*n else 2*Self(n-2): n in [1..40] ];

LinearRecurrence[{0,2},{5,8},60] (* Harvey P. Dale, Jul 20 2022 *)

a(n) = (9-(-1)^n)*2^(1/4*(2*n-5+(-1)^n)).

G.f.: x*(5+8*x)/(1-2*x^2).

A303404 a(0) = 0, a(1) = 1; for n >= 1, a(2n) = a(2n-1) - 2a(n), a(2n+1) = n - 2*a(n).

Original entry on oeis.org

0, 1, -1, -1, 1, 4, 6, 5, 3, 2, -6, -3, -15, -6, -16, -3, -9, 2, -2, 5, 17, 22, 28, 17, 47, 42, 54, 25, 57, 46, 52, 21, 39, 34, 30, 13, 17, 22, 12, 9, -25, -14, -58, -23, -79, -34, -68, -11, -105, -70, -154, -59, -167, -82, -132, -23, -137, -86, -178, -63, -167, -74, -116, -11, -89, -46, -114, -35, -95, -26, -52, 9

Offset: 0

Altug Alkan, Aug 19 2018

Inspired by A002487.

A020714 is generally determinative for block structures of this sequence.

Altug Alkan, Table of n, a(n) for n = 0..10240
Altug Alkan, A scatterplot of a(n) for n <= 5*2^13
Altug Alkan, A scatterplot of first differences of a(n) for n <= 5*2^13

Cf. A002487, A305299.

A[0]:= 0: A[1]:= 1:
for n from 1 to 50 do
  A[2*n]:= A[2*n-1]-2*A[n];
  A[2*n+1]:= n - 2*A[n]
od:
seq(A[i],i=0..101); # Robert Israel, Aug 20 2018

a(n)=if(n<=1, n, if(n%2==0, a(n-1)-2*a(n/2), (n-1)/2-2*a((n-1)/2)));

G.f. g(x) satisfies g(x) + 2*g(x^2)*(1+x+x^2) = x + x^2 + x^3 + x^4 + 2*x^5. - Robert Israel, Aug 20 2018

A002066 a(n) = 10*4^n.

Original entry on oeis.org

10, 40, 160, 640, 2560, 10240, 40960, 163840, 655360, 2621440, 10485760, 41943040, 167772160, 671088640, 2684354560, 10737418240, 42949672960, 171798691840, 687194767360, 2748779069440, 10995116277760, 43980465111040, 175921860444160, 703687441776640, 2814749767106560

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (4).

Pairwise sums of A081294.

Cf. A000079, A003947, A004171, A020714.

[10*4^n: n in [0..30]]; // Vincenzo Librandi, May 19 2011

10*4^Range[0, 100] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)

a(n)=10*4^n \\ Charles R Greathouse IV, Apr 18 2012

From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 4*a(n-1), n > 0; a(0)=10.
G.f.: 10/(1-4*x). (End)
From Elmo R. Oliveira, Apr 02 2025: (Start)
E.g.f.: 10*exp(4*x).
a(n) = 5*A004171(n) = 2*A003947(n+1) = A020714(n)*A000079(n+1). (End)

A063757 G.f.: (1+3x+2x^2)/((1-x)(1-2x^2)).

Original entry on oeis.org

1, 4, 8, 14, 22, 34, 50, 74, 106, 154, 218, 314, 442, 634, 890, 1274, 1786, 2554, 3578, 5114, 7162, 10234, 14330, 20474, 28666, 40954, 57338, 81914, 114682, 163834, 229370, 327674, 458746, 655354, 917498, 1310714, 1835002, 2621434

Offset: 0

N. J. A. Sloane, Aug 14 2001

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 158.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

CoefficientList[Series[(1+3x+2x^2)/((1-x)(1-2x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{1,2,-2},{1,4,8},41] (* Harvey P. Dale, Jun 05 2012 *)

a(0)=1, a(1)=4, a(2)=8, a(n)=a(n-1)+2*a(n-2)-2*a(n-3) From Harvey P. Dale, Jun 05 2012
a(n)=2^((n-3)/2)*((5*Sqrt[2]-7)*(-1)^n+7+5*Sqrt[2])-6 From Harvey P. Dale, Jun 05 2012
a(2*n) = 7*2^n - 6 = A048489(n), a(2*n+1) = 10*2^n - 6 = A020714(n+1) - 6, a(n) = A070875(n+1) - 6. - Philippe Deléham, Apr 13 2013

A154407 a(n) = 52^(n-1) + 36^n/2.

Original entry on oeis.org

4, 14, 64, 344, 1984, 11744, 70144, 420224, 2520064, 15117824, 90701824, 544200704, 3265183744, 19591061504, 117546287104, 705277558784, 4231665025024, 25389989494784, 152339935657984, 914039611326464, 5484237662715904

Offset: 0

Paul Curtz, Jan 09 2009

One of the diagonals of the n-th differences of A154383.

			Sequence A154383 and its k-th iterated difference in the k-th row are
...1.....0.....4.....2.....16......8.....64.....32....256....128...1024.
..-1.....4....-2....14.....-8.....56....-32....224...-128....896...-512.
...5....-6....16...-22.....64....-88....256...-352...1024..-1408...4096.
.-11....22...-38....86...-152....344...-608...1376..-2432...5504..-9728.
..33...-60...124..-238....496...-952...1984..-3808...7936.-15232..31744.
.-93...184..-362...734..-1448...2936..-5792..11744.-23168..46976.-92672.
.277..-546..1096.-2182...4384..-8728..17536.-34912..70144.-139648.280576.
The sequence is the diagonal T(k,k+2) in this array.

Vincenzo Librandi, Table of n, a(n) for n = 0..150
Index entries for linear recurrences with constant coefficients, signature (8, -12).

[5*2^(n-1)+3*6^n/2: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

A154407:=n->5*2^(n-1)+3*6^n/2; seq(A154407(n), n=0..50); # Wesley Ivan Hurt, Nov 13 2013

Table[5*2^(n-1)+3*6^n/2, {n,0,50}] (* Wesley Ivan Hurt, Nov 13 2013 *)

a(n+1) = 6*a(n) - 10*2^n.
a(n) = 6*a(n) - 5*A020714(n+1).
G.f.: 2*(2 - 9*x)/((6*x-1)*(2*x-1)). - R. J. Mathar, May 21 2009
E.g.f.: (1/2)*( 5*exp(2*x) + 3*exp(6*x) ). - G. C. Greubel, Sep 16 2016

Edited by R. J. Mathar, May 21 2009

A253145 Triangular numbers (A000217) omitting the term 1.

Original entry on oeis.org

0, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275

Offset: 0

Paul Curtz, Mar 23 2015

The full triangle of the inverse Akiyama-Tanigawa transform applied to (-1)^n*A062510(n)=3*(-1)^n*A001045(n) yielding a(n) is
0,      3,    6,  10,   15,  21,  28, 36, ...
-3,    -6,  -12, -20,  -30, -42, -56, ...    essentially -A002378
3,     12,   24,  40,   60,  84, ...         essentially  A046092
-9,   -24,  -48, -80, -120, ...              essentially -A033996
15,    48,   96, 160, ...
-33,  -96, -192, ...
63,   192, ...
-129, ...
etc.
First column: (-1)^n*A062510(n).
The following columns are multiples of A122803(n)=(-2)^n. See A007283(n), A091629(n), A020714(n+1), A110286, A175805(n), 4*A005010(n).
An autosequence of the first kind is a sequence whose main diagonal is A000004 = 0's.
b(n) = 0, 0 followed by a(n) is an autosequence of the first kind.
The successive differences of b(n) are
0,   0,  0, 3, 6, 10, 15, 21, ...
0,   0,  3, 3, 4,  5,  6,  7, ...  see A194880(n)
0,   3,  0, 1, 1,  1,  1,  1, ...
3,  -3,  1, 0, 0,  0,  0,  0, ...
-6,  4, -1, 0, 0,  0,  0,  0, ...
10, -5,  1, 0, 0,  0,  0,  0, ...
-15, 6, -1, 0, 0,  0,  0,  0, ...
21, -7,  1, 0, 0,  0,  0,  0, ...
The inverse binomial transform (first column) is the signed sequence. This is general.
Also generalized hexagonal numbers without 1. - Omar E. Pol, Mar 23 2015

Muniru A Asiru, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A179865, A001045, A062510, A002378, A046092, A033996, A122803, A007283, A091629, A020714, A110286, A161680, A175805, A005010, A194880, A001840, A022003, A255935.

Concatenation([0],List([1..50],n->(n+1)*(n+2)/2)); # Muniru A Asiru, Oct 31 2018

Prepend[Table[(n + 1) (n + 2)/2, {n, 49}], 0] (* Michael De Vlieger, Mar 23 2015 *)

a(n)=if(n,(n+1)*(n+2)/2,0) \\ Charles R Greathouse IV, Mar 23 2015

Inverse Akiyama-Tanigawa transform of (-1)^n*A062510(n).
a(n) = (n+1)*(n+2)/2 for n > 0. - Charles R Greathouse IV, Mar 23 2015
a(n+1) = 3*A001840(n+1) + A022003(n).
a(n) = A161680(n+2) for n >= 1. - Georg Fischer, Oct 30 2018
From Stefano Spezia, May 28 2025: (Start)
G.f.: x*(3 - 3*x + x^2)/(1 - x)^3.
E.g.f.: exp(x)*(2 + 4*x + x^2)/2 - 1. (End)

A316528 a(n) = 3a(n-1) - a(n-2) - 2a(n-3) for n > 2, a(0)=1, a(1)=4, a(2)=10.

Original entry on oeis.org

1, 4, 10, 24, 54, 118, 252, 530, 1102, 2272, 4654, 9486, 19260, 38986, 78726, 158672, 319318, 641830, 1288828, 2586018, 5185566, 10393024, 20821470, 41700254, 83493244, 167136538, 334515862, 669424560, 1339484742, 2679997942, 5361659964, 10726012466, 21456381550

Offset: 0

Vincenzo Librandi, Jul 14 2018

Row sums of triangle A316939.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Cf. A000045, A020714, A116712, A118658, A316939.

a:=[1,4,10];; for n in [4..35] do a[n]:=3*a[n-1]-a[n-2]-2*a[n-3]; od; a; # Muniru A Asiru, Jul 14 2018

I:=[1,4,10]; [n le 3 select I[n] else 3*Self(n-1)-Self(n-2)-2*Self(n-3): n in [1..40]];

seq(coeff(series((1+x-x^2)/(1-3*x+x^2+2*x^3), x,n+1),x,n),n=0..35); # Muniru A Asiru, Jul 14 2018

RecurrenceTable[{a[n] == 3 a[n - 1] - a[n - 2] - 2 a[n - 3], a[0] == 1, a[1] == 4, a[2] == 10}, a, {n, 0, 40}]
Table[5 2^n - 2 Fibonacci[n + 3], {n, 0, 40}] (* Bruno Berselli, Jul 16 2018 *)
LinearRecurrence[{3,-1,-2},{1,4,10},40] (* Harvey P. Dale, Jul 18 2020 *)

Vec((1 + x - x^2)/((1 - 2*x)*(1 - x - x^2)) + O(x^40)) \\ Colin Barker, Jul 23 2018

G.f.: (1 + x - x^2)/((1 - 2*x)*(1 - x - x^2)).
a(n) = 2*A116712(n) for n > 0, a(0)=1.
a(n) = 5*2^n - 2*Fibonacci(n+3). - Bruno Berselli, Jul 16 2018
a(n) = (5*2^n - (2^(1-n)*((1-sqrt(5))^n*(-2+sqrt(5)) + (1+sqrt(5))^n*(2+sqrt(5))))/sqrt(5)). - Colin Barker, Jul 23 2018

A340717 Lexicographically earliest sequence of nonnegative integers with as many distinct values as possible such that for any n >= 0, a(rev(n)) = a(n) (where rev(n) = A030101(n) corresponds to the binary reversal of n).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 6, 4, 7, 1, 8, 5, 9, 3, 10, 6, 11, 2, 9, 6, 12, 4, 11, 7, 13, 1, 14, 8, 15, 5, 16, 9, 17, 3, 16, 10, 18, 6, 19, 11, 20, 2, 15, 9, 21, 6, 18, 12, 22, 4, 17, 11, 22, 7, 20, 13, 23, 1, 24, 14, 25, 8, 26, 15, 27, 5, 28, 16

Offset: 0

Rémy Sigrist, Jan 17 2021

The condition "with as many distinct values as possible" means here that for any distinct m and n, provided the orbits of m and n under the map x -> rev(x) do not merge, then a(m) <> a(n).

			The first terms, alongside rev(n), are:
  n   a(n)  rev(n)
  --  ----  ------
   0     0       0
   1     1       1
   2     1       1
   3     2       3
   4     1       1
   5     3       5
   6     2       3
   7     4       7
   8     1       1
   9     5       9
  10     3       5
  11     6      13
  12     2       3
  13     6      11
  14     4       7
  15     7      15

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Rémy Sigrist, Colored scatterplot of the first 2^16 terms (where the color is function of A007814(n), the 2-adic valuation of n)
Rémy Sigrist, PARI program for A340717

See A340716 for similar sequences.

Cf. A000079, A005009, A005010, A007283, A007814, A020714, A030101, A340718.

PARI
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See Links section.
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a(2*n) = a(n).
a(n) = 1 iff n is a power of 2.
a(n) = 2 iff n belongs to A007283.
a(n) = 3 iff n belongs to A020714.
a(n) = 4 iff n belongs to A005009.
a(n) = 5 iff n belongs to A005010.
a(A340718(n)) = n (and this is the first occurrence of n in the sequence).

cp's OEIS Frontend

A146951 Numbers that are congruent to 0 or 6 mod 10.

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A147675 Divide by 2, multiply by 4, repeat.

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A164682 a(n) = 2*a(n-2) for n > 2; a(1) = 5, a(2) = 8.

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A303404 a(0) = 0, a(1) = 1; for n >= 1, a(2*n) = a(2*n-1) - 2*a(n), a(2*n+1) = n - 2*a(n).

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A002066 a(n) = 10*4^n.

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A063757 G.f.: (1+3*x+2*x^2)/((1-x)*(1-2*x^2)).

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A154407 a(n) = 5*2^(n-1) + 3*6^n/2.

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A253145 Triangular numbers (A000217) omitting the term 1.

A303404 a(0) = 0, a(1) = 1; for n >= 1, a(2n) = a(2n-1) - 2a(n), a(2n+1) = n - 2*a(n).

A063757 G.f.: (1+3x+2x^2)/((1-x)(1-2x^2)).

A154407 a(n) = 52^(n-1) + 36^n/2.

A316528 a(n) = 3a(n-1) - a(n-2) - 2a(n-3) for n > 2, a(0)=1, a(1)=4, a(2)=10.