A020714 -id:A020714 - cp's OEIS frontend

A129638 Number of meaningful differential operations of the k-th order on the space R^11.

11, 21, 40, 77, 148, 286, 552, 1069, 2068, 4010, 7768, 15074, 29225, 56736, 110055, 213705, 414676, 805314, 1562977, 3035514, 5892257, 11443768, 22215753, 43146726, 83766396, 162686691, 315860810, 613439352, 1191054193, 2313133481

Offset: 11

Branko Malesevic, May 31 2007

Also number of meaningful compositions of the k-th order of the differential operations and Gateaux directional derivative on the space R^10. - Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 20 2007

Also (starting 6,11,...) the number of zig-zag paths from top to bottom of a rectangle of width 12, whose color is that of the top right corner. [Joseph Myers, Dec 23 2008]

B. Malesevic, Some combinatorial aspects of differential operation composition on the space R^n, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33.
B. Malesevic and I. Jovovic, The Compositions of the Differential Operations and Gateaux Directional Derivative , arXiv:0706.0249 [math.CO], 2007.
Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
Index entries for linear recurrences with constant coefficients, signature (1,5,-4,-6,3,1).

Cf. A090989-A090995.

Cf. A000079, A007283, A020701, A020714.

NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n:=11; # <- DIMENSION Fun:=(i,j)->piecewise(((j=i+1) or (i+j=n+1)),1,0); Identity:=(i,j)->piecewise(i=j,1,0); v:=matrix(1,n,1); A:=piecewise(k>1,(matrix(n,n,Fun))^(k-1),k=1,matrix(n,n,Identity)); return(evalm(v&*A&*transpose(v))[1,1]); end:

LinearRecurrence[{1, 5, -4, -6, 3, 1}, {11, 21, 40, 77, 148, 286}, 30] (* Jean-François Alcover, Oct 10 2017 *)

a(k+6) = a(k+5) +5*a(k+4) -4*a(k+3) -6*a(k+2) +3*a(k+1) +a(k).

G.f.: -x^11*(6*x^5+21*x^4-24*x^3-36*x^2+10*x+11)/(x^6+3*x^5-6*x^4-4*x^3+5*x^2+x-1). [Colin Barker, Jul 08 2012]

More terms from Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 20 2007

More terms from Joseph Myers, Dec 23 2008

A164095 a(n) = 2*a(n-2) for n > 2; a(1) = 5, a(2) = 6.

Original entry on oeis.org

5, 6, 10, 12, 20, 24, 40, 48, 80, 96, 160, 192, 320, 384, 640, 768, 1280, 1536, 2560, 3072, 5120, 6144, 10240, 12288, 20480, 24576, 40960, 49152, 81920, 98304, 163840, 196608, 327680, 393216, 655360, 786432, 1310720, 1572864, 2621440, 3145728

Offset: 1

Klaus Brockhaus, Aug 10 2009

nonn

Interleaving of A020714 and A007283 without initial term 3.
Partial sums are in A164096.
Binomial transform is A048655 without initial 1, second binomial transform is A161941 without initial 2, third binomial transform is A164037, fourth binomial transform is A161731 without initial 1, fifth binomial transform is A164038, sixth binomial transform is A164110.

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

Cf. A020714 (5*2^n), A007283 (3*2^n), A164096, A048655, A161941, A164037, A161731, A164038, A164110, A070876.

[ n le 2 select n+4 else 2*Self(n-2): n in [1..40] ];

LinearRecurrence[{0,2},{5,6},50] (* or *) With[{nn=20},Riffle[NestList[ 2#&,5,nn],NestList[2#&,6,nn]]] (* Harvey P. Dale, Aug 15 2020 *)

a(n) = A070876(n)/3.
a(n) = (4-(-1)^n)*2^(1/4*(2*n-1+(-1)^n)).
G.f.: x*(5+6*x)/(1-2*x^2).

A110288 a(n) = 19*2^n.

Original entry on oeis.org

19, 38, 76, 152, 304, 608, 1216, 2432, 4864, 9728, 19456, 38912, 77824, 155648, 311296, 622592, 1245184, 2490368, 4980736, 9961472, 19922944, 39845888, 79691776, 159383552, 318767104, 637534208, 1275068416, 2550136832, 5100273664, 10200547328, 20401094656

Offset: 0

Alexandre Wajnberg, Sep 07 2005

The first differences are the sequence itself. Doubling the terms gives the same sequence (beginning one step further).

19 times powers of 2. - Omar E. Pol, Dec 17 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..235
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2).

Sequences of the form (2*m+1)*2^n: A000079 (m=0), A007283 (m=1), A020714 (m=2), A005009 (m=3), A005010 (m=4), A005015 (m=5), A005029 (m=6), A110286 (m=7), A110287 (m=8), this sequence (m=9), A175805 (m=10), A248646 (m=11), A164161 (m=12), A175806 (m=13), A257548 (m=15).

[19*2^n: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

19*2^Range[0, 60] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
NestList[2#&,19,30] (* Harvey P. Dale, May 11 2018 *)

[19*2^n for n in range(41)] # G. C. Greubel, Jan 04 2023

G.f.: 19/(1-2*x). - Philippe Deléham, Nov 23 2008
a(n) = A000079(n)*19. - Omar E. Pol, Dec 17 2008
E.g.f.: 19*exp(2*x). - G. C. Greubel, Jan 04 2023

Edited by Omar E. Pol, Dec 16 2008

A163888 a(n) = 2*a(n-2) for n > 2; a(1) = 5, a(2) = 4.

Original entry on oeis.org

5, 4, 10, 8, 20, 16, 40, 32, 80, 64, 160, 128, 320, 256, 640, 512, 1280, 1024, 2560, 2048, 5120, 4096, 10240, 8192, 20480, 16384, 40960, 32768, 81920, 65536, 163840, 131072, 327680, 262144, 655360, 524288, 1310720, 1048576, 2621440, 2097152, 5242880

Offset: 1

Klaus Brockhaus, Aug 06 2009

nonn

Interleaving of A020714 and A000079 without initial terms 1 and 2.

Binomial transform is A163607, second binomial transform is A163608, third binomial transform is A163609, fourth binomial transform is A163610, fifth binomial transform is A163611.

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

Cf. A020714 (5*2^n), A000079 (powers of 2), A163607, A163608, A163609, A163610, A163611.

[ n le 2 select 6-n else 2*Self(n-2): n in [1..41] ];

Transpose[NestList[{Last[#],2First[#]}&,{5,4},40]] [[1]]  (* Harvey P. Dale, Mar 14 2011 *)
LinearRecurrence[{0, 2},{5, 4},41] (* Ray Chandler, Aug 14 2015 *)

x='x+O('x^50); vec(x*(5+4*x)/(1-2*x^2)) \\ G. C. Greubel, Aug 07 2017

a(n) = (7 - 3*(-1)^n)*2^((2*n-5+(-1)^n)/4).

G.f.: x*(5+4*x)/(1-2*x^2).

A197652 Numbers that are congruent to 0 or 1 mod 10.

Original entry on oeis.org

0, 1, 10, 11, 20, 21, 30, 31, 40, 41, 50, 51, 60, 61, 70, 71, 80, 81, 90, 91, 100, 101, 110, 111, 120, 121, 130, 131, 140, 141, 150, 151, 160, 161, 170, 171, 180, 181, 190, 191, 200, 201, 210, 211, 220, 221, 230, 231, 240, 241, 250, 251, 260, 261, 270, 271

Offset: 1

Philippe Deléham, Oct 16 2011

From Wesley Ivan Hurt, Sep 26 2015: (Start)
Numbers with last digit 0 or 1.
Complement of (A260181 Union A262389). (End)
Numbers k such that floor(k/2) = 5*floor(k/10). - Bruno Berselli, Oct 05 2017

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

Cf. A020714, A030308, A260181, A262389.

[5*n-7-2*(-1)^n: n in [1..60]]; // Vincenzo Librandi, Jul 11 2012

A197652:=n->5*n-7-2*(-1)^n: seq(A197652(n), n=1..100); # Wesley Ivan Hurt, Sep 26 2015

CoefficientList[Series[x*(1+9*x)/((1+x)*(1-x)^2),{x,0,50}],x] (* Vincenzo Librandi, Jul 11 2012 *)

a(n)=n\2*10+n%2*9-9 \\ Charles R Greathouse IV, Oct 25 2011

def A197652(n): return 5*n-(5 if n&1 else 9) # Chai Wah Wu, Oct 29 2024

a(n) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=1 and b(k) = 5*2^k = A020714(k) for k>0.
From Zak Seidov, Oct 20 2011: (Start)
a(n) = a(n-2) + 10.
a(n) = 5*n - 7 - 2*(-1)^n. (End)
From Vincenzo Librandi, Jul 11 2012: (Start)
G.f.: x^2*(1+9*x)/((1+x)*(1-x)^2).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3. (End)
E.g.f.: 9 + (5*x - 7)*exp(x) - 2*exp(-x). - David Lovler, Sep 03 2022

A290002 Numbers k such that psi(phi(k)) = phi(psi(k)).

Original entry on oeis.org

1, 10, 18, 20, 36, 40, 54, 70, 72, 78, 80, 108, 110, 140, 144, 156, 160, 162, 174, 198, 216, 220, 222, 230, 234, 246, 280, 288, 294, 312, 320, 324, 348, 396, 414, 426, 432, 438, 440, 444, 450, 460, 468, 470, 486, 492, 534, 560, 576, 588, 594, 624, 640, 648, 666, 696, 702, 770, 792, 828, 846, 852

Offset: 1

Altug Alkan, Sep 03 2017

Squarefree terms are 1, 10, 70, 78, 110, 174, 222, 230, 246, 426, 438, ...
Common terms of this sequence and A033632 are 1, 14406, 544500, 141118050, ...
From Robert Israel, Sep 03 2017: (Start)
Includes 2^i*3^j if i >= 1 and j >= 2, i.e., 3*A033845, and A020714(n) for n >= 1.
If an even number m is in the sequence, then so is 2*m.
Are there any odd terms other than 1? (End)
a(1) = 1 is the only odd term. LHS of equation allows for 1 and 3 but only for k <= 6. RHS allows for 1 and only for k = 1. - Torlach Rush, Jul 28 2023

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A001615, A020714, A033632, A033845.

psi:= proc(n)  n*mul((1+1/i[1]), i=ifactors(n)[2]) end:
select(psi @ numtheory:-phi = numtheory:-phi @ psi, [$1..1000]); # Robert Israel, Sep 03 2017

f[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors@ n}]; Select[Range[10^3], f[EulerPhi@ #] == EulerPhi[f@ #] &] (* Michael De Vlieger, Sep 03 2017 *)

a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
isok(n) = eulerphi(a001615(n))==a001615(eulerphi(n)); \\ after Charles R Greathouse IV at A001615

A327539 Starting from n: as long as the decimal representation starts with a positive even number, divide the largest such prefix by 2; a(n) corresponds to the final value.

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 1, 17, 9, 19, 5, 11, 11, 13, 3, 15, 13, 17, 7, 19, 15, 31, 1, 33, 17, 35, 9, 37, 19, 39, 5, 11, 11, 13, 11, 15, 13, 17, 3, 19, 15, 51, 13, 53, 17, 55, 7, 57, 19, 59, 15, 31, 31, 33, 1, 35, 33, 37, 17, 39, 35, 71

Offset: 0

Rémy Sigrist, Nov 29 2019

For n > 0, as long as we have a number whose decimal representation is the concatenation of a positive even number, say u, and a possibly empty string of odd digits, say v, we replace this number with the concatenation of u/2 and v; eventually only odd digits remain.

			For n = 10000:
- 10000 gives 10000/2 = 5000,
- 5000 gives 5000/2 = 2500,
- 2500 gives 2500/2 = 1250,
- 1250 gives 125/2 = 625,
- 625 gives 62/2 followed by 5 = 315,
- 315 has only odd digits, so a(10000) = 315.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Logarithmic scatterplot of the first 1000000 terms
Rémy Sigrist, Scatterplot of the ordinal transform of the first 1000000 terms

See A329249, A329424 and A329428 for similar sequences.

Cf. A007283, A014261, A020714, A005009, A005010, A215145.

Array[FixedPoint[If[AllTrue[#, OddQ], FromDigits@ #, FromDigits@ Flatten@ Join[IntegerDigitsFromDigits[First[#]]/2, Last[#]] &@ TakeDrop[#, Position[#, ?EvenQ][[-1, -1]] ] ] &@ IntegerDigits[#] &, #] &, 71] (* _Michael De Vlieger, Dec 01 2019 *)

a(n) = if (n==0, 0, n%2==0, a(n/2), 10*a(n\10)+(n%10))

a(n) <= n with equality iff n = 0 or n belongs to A014261.
a(2*n) = a(n).
a(10*k + v) = 10*a(k) + v for any k >= 0 and v in {1, 3, 5, 7, 9}.
a(n) = 1 iff n is a power of 2.
a(n) = 3 iff n belongs to A007283.
a(n) = 5 iff n belongs to A020714.
a(n) = 7 iff n belongs to A005009.
a(n) = 9 iff n belongs to A005010.
a(n) = a(n+1) iff n belongs to A215145.

A376192 Indices n where a run of primes begins in A375564.

Original entry on oeis.org

2, 8, 18, 40, 84, 162, 321, 649, 1286, 2550, 5096, 10188, 20406, 40883, 81932, 164190, 328490, 657509, 1316258, 2635513, 5276876, 10565366, 21155215, 42355195, 84797387, 169759097

Offset: 1

N. J. A. Sloane, Sep 26 2024

			A375564 begins 1, 2, 3, 4, 6, 8, 10, 5, 7, 9, 12, ..., so the present sequence begins 2, 8, ...

Cf. A020714, A375564, A376191, A376193.

a(n) is roughly equal to 5*2^(n-1) (or A020714[n-1]).

a(16)-a(19) from Scott R. Shannon, Sep 27 2024

a(20)-a(26) from Scott R. Shannon, Oct 02 2024

A288732 a(n) = a(n-1) + 2a(n-4) - 2a(n-5) for n >= 5, where a(0) = 2, a(1) = 4, a(2) = 6, a(3) = 8, a(4) = 10.

Original entry on oeis.org

2, 4, 6, 8, 10, 14, 18, 22, 26, 34, 42, 50, 58, 74, 90, 106, 122, 154, 186, 218, 250, 314, 378, 442, 506, 634, 762, 890, 1018, 1274, 1530, 1786, 2042, 2554, 3066, 3578, 4090, 5114, 6138, 7162, 8186, 10234, 12282, 14330, 16378, 20474, 24570, 28666, 32762

Offset: 0

Clark Kimberling, Jun 16 2017

Conjecture:  a(n) is the number of letters (0's and 1's) in the n-th iterate of the mapping 00->1000, 10->01, starting with 00; see A288729.
From Michel Dekking, Mar 25 2018: (Start)
Note that a(n) - a(n-1) = 2*[a(n-4) - a(n-5)] for n>4.
It follows that this sequence is a union of four simple sequences:
a(4k-4) = 4*2^k - 6 = A131130(k) for k = 1,2,3,...
a(4k-3) = 5*2^k - 6 = A020714(k) - 6 for k = 1,2,3...
a(4k-2) = 6*2^k - 6 = A007283(k+1) - 6 for k = 1,2,3, ...
a(4k-1) = 7*2^k - 6 = A048489(k) for k = 1,2,3...
(End)

Clark Kimberling, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 2, -2).

Cf. A288729.

a:=[2,4,6,8,10];; for n in [6..45] do a[n]:=a[n-1]+2*a[n-4]-2*a[n-5]; od; a; # Muniru A Asiru, Mar 22 2018

f:= gfun:-rectoproc({a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5),
a(0) = 2, a(1) = 4, a(2) = 6, a(3) = 8, a(4) = 10},a(n),remember):
map(f, [$0..50]); # Robert Israel, Mar 25 2018

LinearRecurrence[{1, 0, 0, 2, -2}, {2, 4, 8, 8, 10}, 40]

x='x+O('x^99); Vec(2*(1+x+x^2+x^3-x^4)/(1-x-2*x^4+2*x^5)) \\ Altug Alkan, Mar 22 2018

a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) for n >= 5, where a(0) = 2, a(1) = 4, a(2) = 6, a(3) = 8, a(4) = 10.

G.f.: -((2*(-1 - x - x^2 - x^3 + x^4))/(1 - x - 2*x^4 + 2*x^5)).

a(41)-a(49) from Muniru A Asiru, Mar 22 2018

A084640 Generalized Jacobsthal numbers.

Original entry on oeis.org

0, 1, 5, 11, 25, 51, 105, 211, 425, 851, 1705, 3411, 6825, 13651, 27305, 54611, 109225, 218451, 436905, 873811, 1747625, 3495251, 6990505, 13981011, 27962025, 55924051, 111848105, 223696211, 447392425, 894784851, 1789569705, 3579139411

Offset: 0

Paul Barry, Jun 06 2003

This is the sequence A(0,1;1,2;4) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. [From _Wolfdieter Lang_, Oct 18 2010]
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A000975, A000225, A084639.

a084640 n = a084640_list !! n
a084640_list = 0 : 1 : (map (+ 4) $
   zipWith (+) (map (* 2) a084640_list) (tail a084640_list))
-- Reinhard Zumkeller, May 23 2013

[5*2^n/3+(-1)^n/3-2: n in [0..35]]; // Vincenzo Librandi, Jun 15 2011

LinearRecurrence[{2,1,-2},{0,1,5},40] (* Harvey P. Dale, Oct 27 2015 *)

x='x+O('x^50); Vec(x*(1+3*x)/((1-x^2)*(1-2*x))) \\ G. C. Greubel, Sep 26 2017

G.f.: x*(1+3*x)/((1-x^2)*(1-2*x)).
a(n) = a(n-1) + 2a(n-2) + 4, a(0)=0, a(1)=1.
a(n) = (5*2^n + (-1)^n - 6)/3.
a(n) = A001045(n+2) + 4*A000975(n-3).
a(n+1) - 2*a(n) = period 2: repeat 1, 3. - Paul Curtz, Apr 03 2008
Contribution from Paul Curtz, Dec 10 2009: (Start)
a(n+2) - a(n) = A020714(n).
Le the array D(n,k) of the first differences be defined via D(0,k) = a(k); D(n+1,k) = D(n,k+1)-D(n,k).
Then D(n,n) = 4*A131577(n); D(1,k) = A084214(k+1); D(2,k) = A115102(k-1) for k>0; D(3,k) = (-1)^(k+1)*A083581(k). (End)
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), a(0)=0, a(1)=1, a(2)=5. Observed by G. Detlefs. See the W. Lang link. - Wolfdieter Lang, Oct 18 2010

cp's OEIS Frontend

A129638 Number of meaningful differential operations of the k-th order on the space R^11.

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A164095 a(n) = 2*a(n-2) for n > 2; a(1) = 5, a(2) = 6.

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A110288 a(n) = 19*2^n.

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A163888 a(n) = 2*a(n-2) for n > 2; a(1) = 5, a(2) = 4.

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A197652 Numbers that are congruent to 0 or 1 mod 10.

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A290002 Numbers k such that psi(phi(k)) = phi(psi(k)).

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A327539 Starting from n: as long as the decimal representation starts with a positive even number, divide the largest such prefix by 2; a(n) corresponds to the final value.

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A288732 a(n) = a(n-1) + 2a(n-4) - 2a(n-5) for n >= 5, where a(0) = 2, a(1) = 4, a(2) = 6, a(3) = 8, a(4) = 10.