A084215 -id:A084215 - cp's OEIS frontend

A047215 Numbers that are congruent to {0, 2} mod 5.

0, 2, 5, 7, 10, 12, 15, 17, 20, 22, 25, 27, 30, 32, 35, 37, 40, 42, 45, 47, 50, 52, 55, 57, 60, 62, 65, 67, 70, 72, 75, 77, 80, 82, 85, 87, 90, 92, 95, 97, 100, 102, 105, 107, 110, 112, 115, 117, 120, 122, 125, 127, 130, 132, 135, 137, 140, 142, 145, 147, 150, 152, 155, 157

Offset: 0

N. J. A. Sloane

Number of partitions of 5n into exactly 2 parts. - Colin Barker, Mar 23 2015

Numbers k such that k^2/5 + k*(k + 1)/5 = k*(2*k + 1)/5 is a nonnegative integer. - Bruno Berselli, Feb 14 2017

David Lovler, Table of n, a(n) for n = 0..1000
Robbert Fokkink and Gandhar Joshi, Anti-recurrence sequences, arXiv:2506.13337 [math.NT], 2025. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Different from A038126.
Cf. A001622, A010693, A030308, A084215
Cf. A249547 (partial sums), A010693 (1st differences).

[(5*n - (n mod 2))/2: n in [1..100]]; // G. C. Greubel, Jun 23 2024

Table[Floor[5 n/2], {n, 0, 100}] (* or *) LinearRecurrence[{1, 1, -1}, {0, 2, 5},101] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)

a(n)=5*n\2 \\ Charles R Greathouse IV, Oct 17 2011

[int(5*n//2) for n in range(1,101)] # G. C. Greubel, Jun 23 2024

a(n) = floor(5*n/2).
From R. J. Mathar, Sep 23 2008: (Start)
G.f.: x*(2 + 3*x)/((1 + x)*(1 - x)^2).
a(n) = 5*n/2 + ((-1)^n-1)/4.
a(n+1) - a(n) = A010693(n+1). (End)
a(n) = 5*n - a(n-1) - 8 with a(1)=0. - Vincenzo Librandi, Aug 05 2010
a(n+1) = Sum_{k>=0} A030308(n,k)*A084215(k+1). - Philippe Deléham, Oct 17 2011
a(n) = 2*n + floor(n/2). - Arkadiusz Wesolowski, Sep 19 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = log(5)/4 - sqrt(5)*log(phi)/10 + sqrt(1-2/sqrt(5))*Pi/10, where phi is the golden ratio (A001622). - Amiram Eldar, Dec 07 2021
E.g.f.: (5*x*exp(x) - sinh(x))/2. - David Lovler, Aug 22 2022

A011558 Expansion of (x + x^3)/(1 + x + ... + x^4) mod 2.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0

Offset: 0

N. J. A. Sloane

Multiplicative with a(5^e) = 0, a(p^e) = 1 otherwise. - David W. Wilson, Jun 12 2005
Characteristic function of numbers coprime to 5. - Reinhard Zumkeller, Nov 30 2009
From R. J. Mathar, Jul 15 2010: (Start)
The sequence is the principal Dirichlet character mod 5. (The other real character mod 5 is A080891.)
Associated Dirichlet L-functions are for example L(2,chi) = Sum_{n>=1} a(n)/n^2 = 1.5791367... = (psi'(1/5) + psi'(2/5) + psi'(3/5) + psi'(4/5))/25 or L(3,chi) = Sum_{n>=1} a(n)/n^3 = 1.192440... = -(psi''(1/5) + psi''(2/5) + psi''(3/5) + psi''(4/5))/250, where psi' and psi'' are the trigamma and tetragamma functions. (End)
a(n) is for n >= 1 also the characteristic function for rational g-adic integers (+n/5)A047201).%20See%20the%20definition%20in%20the%20Mahler%20reference,%20p.%207%20and%20also%20p.%2010.%20-%20_Wolfdieter%20Lang">g and also (-n/5)_g for all integers g >= 2 without a factor of 5 (A047201). See the definition in the Mahler reference, p. 7 and also p. 10. - _Wolfdieter Lang, Jul 11 2014
Conjecture: a(n+1) is the number of ways of partitioning n into distinct parts of A084215. - R. J. Mathar, Mar 01 2023

			G.f. = x + x^2 + x^3 + x^4 + x^6 + x^7 + x^8 + x^9 + x^11 + x^12 + ...

Arthur Gill, Linear Sequential Circuits, McGraw-Hill, 1966, Eq. (17-10).
K. Mahler, p-adic numbers and their functions, 2nd ed., Cambridge University press, 1981.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Michael Gilleland, Some Self-Similar Integer Sequences
R. Gold, Characteristic linear sequences and their coset functions, J. SIAM Applied. Math., 14 (1966), 980-985.
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A000035, A011655, A109720 coprimality with 2, 3, 7, respectively.
Cf. A168185, A145568, A168184, A168182, A168181, A097325, A166486.
Cf. A080891, A100047.

seq(n&^4 mod 5, n=0..50); # Gary Detlefs, Mar 20 2010

Mod[#,2]&/@CoefficientList[Series[(x+x^3)/(1+x+x^2+x^3+x^4) ,{x,0,100}], x] (* or *) Flatten[Table[{0,1,1,1,1},{30}]] (* Harvey P. Dale, May 15 2011 *)
a[ n_] := Sign@Mod[ n, 5]; (* Michael Somos, May 24 2015 *)

a(n)=!!(n%5) \\ Charles R Greathouse IV, Sep 23 2012

{a(n) = n%5>0}; /* Michael Somos, May 24 2015 */

(define (A011558 n) (if (zero? (modulo n 5)) 0 1)) ;; Antti Karttunen, Dec 21 2017

O.g.f.: x*(1+x+x^2+x^3)/(1-x^5). - Wolfdieter Lang, Feb 05 2009
From Reinhard Zumkeller, Nov 30 2009: (Start)
a(n) = 1 - A079998(n).
a(A047201(n))=1, a(A008587(n))=0.
A033437(n) = Sum_{k=0..n} a(k)*(n-k). (End)
a(n) = n^4 mod 5. - Gary Detlefs, Mar 20 2010
Sum_{n>=1} a(n)/n^s = L(s,chi) = (1-1/5^s)*Riemann_zeta(s), s > 1. - R. J. Mathar, Jul 31 2010
For the general case. The characteristic function of numbers that are not multiples of m is a(n) = floor((n-1)/m) - floor(n/m) + 1, m,n > 0. - Boris Putievskiy, May 08 2013
a(n) = sgn(n mod 5). - Wesley Ivan Hurt, Jun 30 2013
Euler transform of length 5 sequence [ 1, 0, 0, -1, 1]. - Michael Somos, May 24 2015
Moebius transform is length 5 sequence [ 1, 0, 0, 0, -1]. - Michael Somos, May 24 2015
G.f.: f(x) - f(x^5) where f(x) := x / (1 - x). - Michael Somos, May 24 2015
|a(n)| = |A080891(n)| = |A100047(n)|. - Michael Somos, May 24 2015

More terms from Antti Karttunen, Dec 21 2017

A257113 a(1) = 2, a(2) = 3; thereafter a(n) is the sum of all the previous terms.

Original entry on oeis.org

2, 3, 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, 10240, 20480, 40960, 81920, 163840, 327680, 655360, 1310720, 2621440, 5242880, 10485760, 20971520, 41943040, 83886080, 167772160, 335544320, 671088640, 1342177280, 2684354560, 5368709120, 10737418240

Offset: 1

Giovanni Teofilatto, Apr 24 2015

Except for first three terms, a(n) is 10 times 2^(n-4).

These values comprise the tile values used in the "fives" variant of the game 2048, including 1 as the zeroth term. - Michael De Vlieger, Jul 18 2018

Colin Barker, Table of n, a(n) for n = 1..1000
E. R. Berlekamp, A contribution to mathematical psychometrics, Unpublished Bell Labs Memorandum, Feb 08 1968 [Annotated scanned copy]
David Eppstein, Making Change in 2048, arXiv:1804.07396 [cs.DM], 2018.
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A000079, A020714. Essenitally the same as A084215.

[2,3] cat [5*2^n: n in [0..35]]; // Vincenzo Librandi, May 15 2015

t = {2, 3}; For[k = 3, k <= 27, k++, AppendTo[t, Total@ t]]; t (* Michael De Vlieger, May 14 2015 *)
Join[{2, 3}, Table[5 2^n, {n, 0, 40}]] (* Vincenzo Librandi, May 15 2015 *)
Join[{2,3},NestList[2#&,5,40]] (* Harvey P. Dale, Apr 06 2018 *)

a(n) = if(n<3, n+1, 5*2^(n-3)); \\ Altug Alkan, Jul 18 2018

Vec(x*(1 - x)*(2 + x) / (1 - 2*x) + O(x^40)) \\ Colin Barker, Nov 17 2018

a(n) = ceil(5*2^(n-3)) \\ Alan Michael Gómez Calderón, Mar 30 2022

a(n) = A020714(n-3) for n>2.
a(n) = A146523(n-2) for n>2. - R. J. Mathar, May 14 2015
G.f.: x*(1 - x)*(2 + x) / (1 - 2*x). - Colin Barker, Nov 17 2018

A216228 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=1 or if k-n>=3, T(0,0) = T(0,1) = T(0,2) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 32

Offset: 0

Philippe Deléham, Mar 13 2013

An arithmetic hexagon of E. Lucas.

			Square array begins:
1, 1, 1, 0, 0,  0,  0,  0, ... row n=0
0, 1, 2, 2, 0,  0,  0,  0, ... row n=1
0, 0, 2, 4, 4,  0,  0,  0, ... row n=2
0, 0, 0, 4, 8,  8,  0,  0, ... row n=3
0, 0, 0, 0, 8, 16, 16,  0, ... row n=4
0, 0, 0, 0, 0, 16, 32, 32, ... row n=5
...

E. Lucas, Théorie des nombres, Albert Blanchard, Paris 1958, Tome 1, p.89

E. Lucas, Théorie des nombres, Tome 1, Jacques Gabay, Paris, 1991, p.89

Cf. A000079, A011782, A016116, A068914, A084215.

T(n,n) = A011782(n).
T(n,n+1) = T(n,n+2) = 2^n = A000079(n).
Sum_{k, 0<=k<=n} T(n-k,k) = A016116(n).
Sum_{n, n>=0} T(n,k) = A084215(k).
Sum_{k, k>=0} T(n,k) = A084215(n+1), n>=1.

A146523 Binomial transform of A010685.

Original entry on oeis.org

1, 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, 10240, 20480, 40960, 81920, 163840, 327680, 655360, 1310720, 2621440, 5242880, 10485760, 20971520, 41943040, 83886080, 167772160, 335544320, 671088640, 1342177280, 2684354560, 5368709120, 10737418240

Offset: 0

Philippe Deléham, Oct 30 2008

Linked to A029609 by a Catalan transform.

Hankel transform is (1, -15, 0, 0, 0, 0, 0, 0, 0, ...).

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

Cf. A010685, A020714, A029609, A084215, A109466.

CoefficientList[Series[(1+3x)/(1-2x), {x,0,50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 21 2011 *)
Join[{1}, 5*2^(Range[40] -1)] (* G. C. Greubel, Nov 23 2021 *)

a(n)=if(n,5<<(n-1),1) \\ Charles R Greathouse IV, Jan 17 2012

[1]+[5*2^(n-1) for n in (1..50)] # G. C. Greubel, Nov 23 2021

a(n) = 5*2^(n-1) for n >= 1, a(0) = 1.
a(n) = Sum_{k=0..n} A109466(n,k)*A029609(k).
a(n) = A084215(n+1) = A020714(n-1), n > 0. - R. J. Mathar, Nov 02 2008
G.f.: (1 + 3*x)/(1 - 2*x). - Vladimir Joseph Stephan Orlovsky, Jun 21 2011
G.f.: G(0), where G(k)= 1 + 3*x/(1 -  2*x/(2*x + 3*x/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 03 2013
E.g.f.: (5*exp(2*x) - 3)/2. - Stefano Spezia, Feb 20 2023

A141495 a(n) = 3*a(n-1) for n>2; a(0)=1, a(1)=3, a(2)=7.

Original entry on oeis.org

1, 3, 7, 21, 63, 189, 567, 1701, 5103, 15309, 45927, 137781, 413343, 1240029, 3720087, 11160261, 33480783, 100442349, 301327047, 903981141, 2711943423, 8135830269, 24407490807, 73222472421, 219667417263, 659002251789

Offset: 0

Roger L. Bagula, Aug 10 2008

A sequence of the form: a(0)=1, a(1)=prime(m), a(2)=prime(m+2), a(n)=a(1)*a(n-1).

a(n) is divisible by 7 for n>1. - Colin Barker, Jan 09 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3).

Essentially the same as A005032 and A084215. For other examples see A141496, etc.

a[0] = 1; a[1] = 3; a[2] = 7; a[n_] := a[n] = a[1]*a[n - 1]; Table[a[n], {n, 0, 30}]
Join[{1,3},NestList[3#&,7,30]] (* Harvey P. Dale, Aug 05 2024 *)

a(n) = A082541(n-1), n>1. - R. J. Mathar, Aug 27 2008

a(n) = 7*3^(n-2) for n>1. a(n)=3*a(n-1) for n>2. G.f.: (2*x^2-1) / (3*x-1). - Colin Barker, Jan 09 2014

Edited by N. J. A. Sloane, Aug 16 2008

A129687 A129686 * A007318.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 4, 3, 1, 2, 6, 7, 4, 1, 2, 8, 13, 11, 5, 1, 2, 10, 21, 24, 16, 6, 1, 2, 12, 31, 45, 40, 22, 7, 1, 2, 14, 43, 76, 85, 62, 29, 8, 1, 2, 16, 57, 119, 161, 147, 91, 37, 9, 1, 2, 18, 73, 176, 280, 308, 238, 128, 46, 10, 1, 2, 20, 91, 249, 456

Offset: 0

Gary W. Adamson, Apr 28 2007

Row sums = A084215: (1, 2, 5, 10, 20, 40, 80, ...). A007318 * A129686 = A124725.
From Philippe Deléham, Feb 12 2014: (Start)
Riordan array ((1+x^2)/(1-x), x/(1-x)).
Diagonal sums are A000032(n) - 0^n (cf. A000204).
T(n,0) = A046698(n+1).
T(n+1,1) = A004277(n).
T(n+2,2) = A002061(n+1).
T(n+3,3) = A006527(n+1) = A167875(n).
T(n+4,4) = A006007(n+1).
T(n+5,5) = A081282(n+1). (End)

			First few rows of the triangle:
  1;
  1,   1;
  2,   2,   1;
  2,   4,   3,   1;
  2,   6,   7,   4,   1;
  2,   8,  13,  11,   5,   1;
  2,  10,  21,  24,  16,   6,   1;
  2,  12,  31,  45,  40,  22,   7,   1;
  2,  14,  43,  76,  85,  62,  29,   8,   1;
  2,  16,  57, 119, 161, 147,  91,  37,   9,   1;
  ...

Cf. A129686, A007318, A124725.

Cf. Columns: A046698, A004277, A002061, A006527, A006007, A081282

A129686 * A007318 (Pascal's Triangle), as infinite lower triangular matrices.

T(n,k) = T(n-1,k) + T(n-1,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = T(2,1) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Feb 12 2014

More terms from Philippe Deléham, Feb 12 2014

A287798 Least k such that A006667(k)/A006577(k) = 1/n.

Original entry on oeis.org

159, 6, 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, 10240, 20480, 40960, 81920, 163840, 327680, 655360, 1310720, 2621440, 5242880, 10485760, 20971520, 41943040, 83886080, 167772160, 335544320, 671088640, 1342177280, 2684354560, 5368709120, 10737418240

Offset: 3

Michel Lagneau, Jun 01 2017

A006667: number of tripling steps to reach 1 in '3x+1' problem.
A006577: number of halving and tripling steps to reach 1 in '3x+1' problem.
a(n) = {159, 6} union {A020714}.

			a(3) = 159 because A006667(159)/A006577(159) = 18/54 = 1/3.

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

Cf. A006577, A006666, A006667. Essentially the same as A020714, A084215, A146523 and A257113.

nn:=10^12:
for n from 3 to 35 do:
ii:=0:
for k from 2 to 10^6 while(ii=0) do:
  m:=k:s1:=0:s2:=0:
   for i from 1 to nn while(m<>1) do:
    if irem(m,2)=0
     then
     s2:=s2+1:m:=m/2:
     else
     s1:=s1+1:m:=3*m+1:
    fi:
   od:
    if n*s1=s1+s2
     then
     ii:=1: printf(`%d, `,k):
     else
    fi:
od:od:

f[u_]:=Module[{a=u,k=0},While[a!=1,k++;If[EvenQ[a],a=a/2,a=a*3+1]];k];Table[f[u],{u,10^7}];g[v_]:=Count[Differences[NestWhileList[If[EvenQ[#],#/2,3#+1]&,v,#>1&]],_?Positive];Table[g[v],{v,10^7}];Do[k=3;While[g[k]/f[k]!=1/n,k++];Print[n," ",k],{n,3,35}]

a(n) = if(n < 5, [0,0,159,6][n], 5<<(n-5)) \\ David A. Corneth, Jun 01 2017

Vec(x^3*(159 - 312*x - 7*x^2) / (1 - 2*x) + O(x^50)) \\ Colin Barker, Jun 01 2017

For n >= 5, a(n) = 5*2^n/32. - David A. Corneth, Jun 01 2017
From Colin Barker, Jun 01 2017: (Start)
G.f.: x^3*(159 - 312*x - 7*x^2) / (1 - 2*x).
a(n) = 2*a(n-1) for n>5.
(End)

A124459 Square array resulting from the bisection of array A124458. (The other array is A093560.)

Original entry on oeis.org

2, 3, 2, 3, 5, 2, 3, 8, 7, 2, 3, 11, 15, 9, 2, 3, 14, 26, 24, 11, 2, 3, 17, 40, 50, 35, 13, 2, 3, 20, 57, 90, 85, 48, 15, 2, 3, 23, 77, 147, 175, 133, 63, 17, 2, 3, 26, 100, 224, 322, 308, 196, 80, 19, 2, 3, 29, 126, 324, 546, 630, 504, 276, 99, 21, 2, 3, 32, 155, 450, 870, 1176

Offset: 1

Alford Arnold, Nov 09 2006

Apparently the same as A029618 if the first term is ignored. - R. J. Mathar, Jun 18 2008

			Given the square array
1 2 3 3 3 3 3 3 3 3
1 2 4 5 7 8 10 11 13
1 2 5 7 12 15 22 26
1 2 6 9 18 24 40
1 2 7 11 25 35
1 2 8 13 33 (Table A124458)
1 2 9 15
1 2 10
1 2
1
Omit these odd columns:
1 3 3 3 3 3 3 3 3 3 3
1 4 7 10 13 16 19 22 25 28
1 5 12 22 35 51 70 92 117
1 6 18 40 75 126 196 288
1 7 25 65 140 266 462
1 8 33 98 238 504
1 9 42 140 378
1 10 52 192 (Table A093560)
1 11 63
1 12
1
which yields the square array A124459

Cf. A084215 (antidiagonal sums).

Reppasc := proc(n,k) binomial(n+floor(k/2),n) ; end: A124458 := proc(n,k) add(Reppasc(n,i), i=max(0,k-3)..k-1) ; end: A124459 := proc(n,k) A124458(n,2*k) ; end: for d from 1 to 19 do for k from d to 1 by -1 do n := d-k ; printf("%d,",A124459(n,k)) ; od: od: # R. J. Mathar, Jun 18 2008

More terms from R. J. Mathar, Jun 18 2008

A141496 a(0)=1; a(1)=5; a(2)=11; a(n)=a(1)*a(n-1).

Original entry on oeis.org

1, 5, 11, 55, 275, 1375, 6875, 34375, 171875, 859375, 4296875, 21484375, 107421875, 537109375, 2685546875, 13427734375, 67138671875, 335693359375, 1678466796875, 8392333984375, 41961669921875, 209808349609375

Offset: 0

Roger L. Bagula, Aug 10 2008

Index entries for linear recurrences with constant coefficients, signature (5).

Cf. A084215, A141495.

Clear[a] a[0] = 1; a[1] = 5; a[2] = 11; a[n_] := a[n] = a[1]*a[n - 1]; Table[a[n], {n, 0, 30}]

a(n) = 11*5^(n-2) for n>1. a(n) = 5*a(n-1) for n>2. G.f.: (1-14*x^2)/(1-5*x). [Colin Barker, Oct 13 2012]

Edited by N. J. A. Sloane, Aug 16 2008

cp's OEIS Frontend

A047215 Numbers that are congruent to {0, 2} mod 5.

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A011558 Expansion of (x + x^3)/(1 + x + ... + x^4) mod 2.

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A257113 a(1) = 2, a(2) = 3; thereafter a(n) is the sum of all the previous terms.

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A216228 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=1 or if k-n>=3, T(0,0) = T(0,1) = T(0,2) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A146523 Binomial transform of A010685.

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A141495 a(n) = 3*a(n-1) for n>2; a(0)=1, a(1)=3, a(2)=7.

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A129687 A129686 * A007318.

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