A053381 -id:A053381 - cp's OEIS frontend

A220466 a((2n-1)2^p) = 4^p(n-1) + 2^(p-1)(1+2^p), p >= 0 and n >= 1.

1, 3, 2, 10, 3, 7, 4, 36, 5, 11, 6, 26, 7, 15, 8, 136, 9, 19, 10, 42, 11, 23, 12, 100, 13, 27, 14, 58, 15, 31, 16, 528, 17, 35, 18, 74, 19, 39, 20, 164, 21, 43, 22, 90, 23, 47, 24, 392, 25, 51, 26, 106, 27, 55, 28, 228, 29, 59, 30, 122, 31, 63, 32, 2080, 33, 67, 34, 138, 35

Offset: 1

Johannes W. Meijer, Dec 24 2012

The a(n) appeared in the analysis of A220002, a sequence related to the Catalan numbers.
The first Maple program makes use of a program by Peter Luschny for the calculation of the a(n) values. The second Maple program shows that this sequence has a beautiful internal structure, see the first formula, while the third Maple program makes optimal use of this internal structure for the fast calculation of a(n) values for large n.
The cross references lead to sequences that have the same internal structure as this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ralf Stephan, Some divide-and-conquer sequences with simple ordinary generating functions, The OEIS, Jan 01 2004.

Cf. A000027 (the natural numbers), A000120 (1's-counting sequence), A000265 (remove 2's from n), A001316 (Gould's sequence), A001511 (the ruler function), A003484 (Hurwitz-Radon numbers), A003602 (a fractal sequence), A006519 (highest power of 2 dividing n), A007814 (binary carry sequence), A010060 (Thue-Morse sequence), A014577 (dragon curve), A014707 (dragon curve), A025480 (nim-values), A026741, A035263 (first Feigenbaum symbolic sequence), A037227, A038712, A048460, A048896, A051176, A053381 (smooth nowhere-zero vector fields), A055975 (Gray code related), A059134, A060789, A060819, A065916, A082392, A085296, A086799, A088837, A089265, A090739, A091512, A091519, A096268, A100892, A103391, A105321 (a fractal sequence), A109168 (a continued fraction), A117973, A129760, A151930, A153733, A160467, A162728, A181988, A182241, A191488 (a companion to Gould's sequence), A193365, A220466 (this sequence).

-- Following Ralf Stephan's recurrence:
import Data.List (transpose)
a220466 n = a006519_list !! (n-1)
a220466_list = 1 : concat
   (transpose [zipWith (-) (map (* 4) a220466_list) a006519_list, [2..]])
-- Reinhard Zumkeller, Aug 31 2014

# First Maple program
a := n -> 2^padic[ordp](n, 2)*(n+1)/2 : seq(a(n), n=1..69); # Peter Luschny, Dec 24 2012
# Second Maple program
nmax:=69: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := 4^p*(n-1)  + 2^(p-1)*(1+2^p) od: od: seq(a(n), n=1..nmax);
# Third Maple program
nmax:=69: for p from 0 to ceil(simplify(log[2](nmax))) do n:=2^p: n1:=1: while n <= nmax do a(n) := 4^p*(n1-1)+2^(p-1)*(1+2^p): n:=n+2^(p+1): n1:= n1+1: od: od:  seq(a(n), n=1..nmax);

A220466 = Module[{n, p}, p = IntegerExponent[#, 2]; n = (#/2^p + 1)/2; 4^p*(n - 1) + 2^(p - 1)*(1 + 2^p)] &; Array[A220466, 50] (* JungHwan Min, Aug 22 2016 *)

a(n)=if(n%2,n\2+1,4*a(n/2)-2^valuation(n/2,2)) \\ Ralf Stephan, Dec 17 2013

a((2*n-1)*2^p) = 4^p*(n-1) + 2^(p-1)*(1+2^p), p >= 0 and n >= 1. Observe that a(2^p) = A007582(p).
a(n) = ((n+1)/2)*(A060818(n)/A060818(n-1))
a(n) = (-1/64)*(q(n+1)/q(n))/(2*n+1) with q(n) = (-1)^(n+1)*2^(4*n-5)*(2*n)!*A060818(n-1) or q(n) = (1/8)*A220002(n-1)*1/(A098597(2*n-1)/A046161(2*n))*1/(A008991(n-1)/A008992(n-1))
Recurrence: a(2n) = 4a(n) - 2^A007814(n), a(2n+1) = n+1. - Ralf Stephan, Dec 17 2013

A001676 Number of h-cobordism classes of smooth homotopy n-spheres.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 28, 2, 8, 6, 992, 1, 3, 2, 16256, 2, 16, 16, 523264, 24, 8, 4, 69524373504, 2, 4, 12, 67100672, 2, 3, 3, 7767211311104, 8, 32, 32, 3014494287036416, 6, 24, 120, 2303837503821447168, 192, 32, 96, 341653284209033216, 8, 11520, 48, 798366828940770681028608, 32, 12, 24, 11852230872517975212032, 24, 32, 8, 91678339751618435453288448, 1, 8, 4, 1986677733776616536315084668928, 4, 1, 24, 284423744326342962334231917756416

Offset: 1

N. J. A. Sloane

For n not equal to 4 (and possibly for all n) this is the number of oriented diffeomorphism classes of differentiable structures on the n-sphere.
a(3) = 1 follows now that the Poincaré conjecture has been proved.
a(n) for n != 4 is the order of S_n, the n-th group in Tables 1 and 2 (explained in Further Details p. 807) of Milnor 2011.
The sequence is essentially given in the rightmost column of tables 1 and 2 in Isaksen, Wang & Xu (2020). It corrects some errors in earlier work. - Andrey Zabolotskiy, Nov 27 2022

S. O. Kochman, Stable homotopy groups of spheres. A computer-assisted approach. Lecture Notes in Mathematics, 1423. Springer-Verlag, Berlin, 1990. 330 pp. ISBN: 3-540-52468-1. [Math. Rev. 91j:55016]
S. O. Kochman and M. E. Mahowald, On the computation of stable stems. The Cech Centennial (Boston, MA, 1993), 299-316, Contemp. Math., 181, Amer. Math. Soc., Providence, RI, 1995. [Math. Rev. 96j:55018]
J. P. Levine, Lectures on groups of homotopy spheres. In Algebraic and geometric topology (New Brunswick, NJ, 1983), 62-95, Lecture Notes in Math., 1126, Springer, Berlin, 1985.
J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 285.
S. P. Novikov ed., Topology I, Encyc. of Math. Sci., vol. 12.
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).
H. Whitney, The work of John W. Milnor, pp. 48-50 of Proc. Internat. Congress Mathematicians, Stockholm, 1962.

Andrey Zabolotskiy, Table of n, a(n) for n = 1..83 using data from Isaksen, Wang & Xu (2023).
Tom Copeland, The Kervaire-Milnor formula
Brady Haran and Ciprian Manolescu, The Puzzling Fourth Dimension (and exotic shapes), Numberphile video (2022).
Kevin Hartnett, An Old Conjecture Falls, Making Spheres a Lot More Complicated, Quanta Magazine, Aug 22 2023.
A. Hatcher, Stable Homotopy Groups of Spheres
Daniel C. Isaksen, Guozhen Wang and Zhouli Xu, Stable homotopy groups of spheres, PNAS, 117 (2020), 24757-24763.
Daniel C. Isaksen, Guozhen Wang and Zhouli Xu, Stable homotopy groups of spheres: from dimension 0 to 90, Publications mathématiques de l'IHÉS, 137 (2023), 107-243.
M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres: I, Ann. of Math. (2) 77 1963 504-537.
S. S. Khare, On Abel Prize 2011 to John Willard Milnor, Math. Student, 82 (2013), 247-279. [dead link]
Alexander Kupers, Lectures on diffeomorphism groups of manifolds, Version Apr 28 2018.
J. W. Milnor, On manifolds homeomorphic to the 7-sphere, Ann. of Math. 64 (1956), 399-405.
John W. Milnor, Differential Topology Forty-six Years Later, Notices Amer. Math. Soc. 58 (2011), 804-809.
John W. Milnor, Spheres, Abel Prize lecture (video), 2011.
G. D. Rizell, J. D. Evans, Exotic spheres and the topology of symplectomorphism groups, J. Topol. 8 (2015) 586-602
Anthony Saint-Criq, What is so exotic about dimension four?, Univ. de Toulouse (France 2022).
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Eric Weisstein's World of Mathematics, Exotic Sphere.
Wikipedia, Exotic sphere

Cf. A053381, A057617, A048648, A187595, A187717, A189995, A191783, A228689, A228690, A228691, A228692.

More terms from Paul Muljadi, Mar 17 2011
Further terms from Jonathan Sondow, Jun 16 2011
The terms a(56), a(57), a(63) corrected by Andrey Zabolotskiy, Nov 27 2022

A003484 Radon function, also called Hurwitz-Radon numbers.

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 9, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 10, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 9, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 12, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 9, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 10, 1, 2, 1, 4, 1, 2

Offset: 1

N. J. A. Sloane

This sequence and A006519 (greatest power of 2 dividing n) are very similar, the difference being all zeros except for every 16th term (see A101119 for nonzero differences). - Simon Plouffe, Dec 02 2004
For all n congruent to 2^k (mod 2^(k+1)), a(n) is the same. Therefore, for any natural number m, the list of the first 2^m - 1 terms is palindromic. - Ivan N. Ianakiev, Jul 21 2019
Named after the Austrian mathematician Johann Radon (1887-1956) and the German mathematician Adolf Hurwitz (1859-1919). - Amiram Eldar, Jun 15 2021

			G.f. = x + 2*x^2 + x^3 + 4*x^4 + x^5 + 2*x^6 + x^7 + 8*x^8 + x^9 + ...

T. Y. Lam, The Algebraic Theory of Quadratic Forms. Benjamin, Reading, MA, 1973, p. 131.
Takashi Ono, Variations on a Theme of Euler, Plenum, NY, 1994, p. 192.
A. R. Rajwade, Squares, Camb. Univ. Press, London Math. Soc. Lecture Notes Series 171, 1993; see p. 127.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
J. Frank Adams, Vector fields on spheres, Topology, Vol. 1 (1962), pp. 63-65.
J. Frank Adams, Vector fields on spheres, Bull. Amer. Math. Soc., Vol. 68 (1962), pp. 39-41.
J. Frank Adams, Vector fields on spheres, Annals of Math., Vol. 75 (1962), pp. 603-632.
J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II.
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., Vol. 307 (2003), pp. 3-29.
Adolf Hurwitz, Uber die Komposition der quadratischen formen, Math. Annalen, Vol. 88 (1923), pp. 1-25.
Michel A. Kervaire, Non-parallelizability of the sphere for n > 7, Proc. Nat. Acad. Sci. USA, Vol. 44, No. 3 (1958), pp. 280-283.
John Milnor, Some consequences of a theorem of Bott, Annals of Mathematics, Second Series, Vol. 68, No. 2 (1958), pp. 444-449.
Johann Radon, Lineare scharen orthogonaler matrizen,Abh. Math. Sem. Univ. Hamburg, Vol. 1 (1922), pp. 1-14.
Daniel B. Shapiro, Letter to N. J. A. Sloane, 1974.
Index entries for "core" sequences.

See A053381 for a closely related sequence.

Cf. A003485, A006519, A007814, A101119.

a003484 n = 2 * e + cycle [1,0,0,2] !! e  where e = a007814 n
-- Reinhard Zumkeller, Mar 11 2012

readlib(ifactors): for n from 1 to 150 do if n mod 2 = 1 then printf(`%d,`,1) fi: if n mod 2 = 0 then m := ifactors(n)[2][1][2]: if m mod 4 = 0 then printf(`%d,`,2*m+1) fi: if m mod 4 = 1 then printf(`%d,`,2*m) fi: if m mod 4 = 2 then printf(`%d,`,2*m) fi: if m mod 4 = 3 then printf(`%d,`,2*m+2) fi: fi: od: # James Sellers, Dec 07 2000
nmax:=102; A003485 := proc(n): A003485(n) := ceil((n+1)/4) + ceil(n/4) + 2*ceil((n-1)/4) + 4*ceil((n-2)/4) end: A029837 := n -> ceil(simplify(log[2](n))): for p from 0 to A029837(nmax) do for n from 1 to ceil(nmax/(p+2)) do A003484((2*n-1)*2^p):= A003485(p): od: od: seq(A003484(n), n=1..nmax); # Johannes W. Meijer, Jun 07 2011, Dec 15 2012

a[n_] := 8*Quotient[IntegerExponent[n, 2], 4] + 2^Mod[IntegerExponent[n, 2], 4]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Sep 08 2011, after Paul D. Hanna *)

a(n)=8*(valuation(n,2)\4)+2^(valuation(n,2)%4) /* Paul D. Hanna, Dec 02 2004 */

def A003484(n): return (((m:=(~n&n-1).bit_length())&-4)<<1)+(1<<(m&3)) # Chai Wah Wu, Jul 09 2022

a(n) = A003485(A007814(n)).
If n=2^(4*b+c)*d, 0<=c<=3, d odd, then a(n) = 8*b + 2^c.
If n=2^m*d, d odd, then a(n) = 2*m+1 if m=0 mod 4, a(n) = 2*m if m=1 or 2 mod 4, a(n) = 2*m+2 (otherwise, i.e., if m=3 mod 4).
Multiplicative with a(p^e) = 2e + a_(e mod 4) if p = 2; 1 if p > 2; where a = (1, 0, 0, 2). - David W. Wilson, Aug 01 2001
Dirichlet g.f. zeta(s) *(1-1/2^s)* {7*2^(-4*s) +1 +2^(3-3*s) +3*2^(1-5*s) +2^(1-s) +2^(2-6*s) +2^(2-2*s) }/ (1-2^(-4*s))^2. - R. J. Mathar, Mar 04 2011
a(A005408(n))=1; a(2*n) = A209675(n); a(A016825(n))=2; a(A017113(n))=4; a(A051062(n))=8. - Reinhard Zumkeller, Mar 11 2012
a((2*n-1)*2^p) = A003485(p), p >=0. - Johannes W. Meijer, Jun 07 2011, Dec 15 2012
Lambert series g.f. Sum_(k >=0) q^(2^(4*k))/(1-q^(2^(4*k))) +q^(2^(4*k+1))/(1-q^(2^(4*k+1))) +2*q^(2^(4*k+2))/(1-q^(2^(4*k+2))) +4*q^(2^(4*k+3))/(1-q^(2^(4*k+3))). - Mamuka Jibladze, Dec 07 2016
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 8/3. - Amiram Eldar, Oct 22 2022

More terms from Larry Reeves (larryr(AT)acm.org), Mar 20 2000

A209675 Radon function at even positions: a(n) = A003484(2*n).

Original entry on oeis.org

2, 4, 2, 8, 2, 4, 2, 9, 2, 4, 2, 8, 2, 4, 2, 10, 2, 4, 2, 8, 2, 4, 2, 9, 2, 4, 2, 8, 2, 4, 2, 12, 2, 4, 2, 8, 2, 4, 2, 9, 2, 4, 2, 8, 2, 4, 2, 10, 2, 4, 2, 8, 2, 4, 2, 9, 2, 4, 2, 8, 2, 4, 2, 16, 2, 4, 2, 8, 2, 4, 2, 9, 2, 4, 2, 8, 2, 4, 2, 10, 2, 4, 2, 8, 2

Offset: 1

Reinhard Zumkeller, Mar 11 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A003484, A003485, A005408, A016825, A017113, A051062, A053381.

Haskell
```
a209675 = a003484 . (* 2)
```

a[n_] := 8*Floor[(e = IntegerExponent[n, 2] + 1)/4] + 2^Mod[e, 4]; Array[a, 100] (* Amiram Eldar, Nov 29 2022 *)

a(n) = A053381(n-1) + 1.
a(n) > 1.
a(A005408(n)) = 2; a(A016825(n)) = 4; a(A017113(n)) = 8; a(A051062(n)) = 9.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4/3. - Amiram Eldar, Nov 29 2022

A047530 Numbers that are congruent to {0, 1, 3, 7} mod 8.

Original entry on oeis.org

0, 1, 3, 7, 8, 9, 11, 15, 16, 17, 19, 23, 24, 25, 27, 31, 32, 33, 35, 39, 40, 41, 43, 47, 48, 49, 51, 55, 56, 57, 59, 63, 64, 65, 67, 71, 72, 73, 75, 79, 80, 81, 83, 87, 88, 89, 91, 95, 96, 97, 99, 103, 104, 105, 107, 111, 112, 113, 115, 119, 120, 121, 123

Offset: 1

N. J. A. Sloane

Numbers n such that the n-th homotopy group of the topological group O(oo) does not vanish [see Baez]. Cf. A195679.

The a(n+1) determine the maximal number of linearly independent smooth nowhere zero vector fields on a (2n+1)-sphere, see A053381. - Johannes W. Meijer, Jun 07 2011

John C. Baez, The Octonions, Bull. Amer. Math. Soc., Vol. 39, No. 2 (2002), pp. 145-205; Errata, ibid., Vol. 42, No. 2 (2005), p. 213; alternative link.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A008621. - Johannes W. Meijer, Jun 07 2011

Cf. A047470, A047522, A053381, A195679.

A047530 := proc(n): ceil(n/4) + 2*ceil((n-1)/4) + 4*ceil((n-2)/4) + ceil((n-3)/4) end: seq(A047530(n), n=0..47); # Johannes W. Meijer, Jun 07 2011
A047530:=n->(8*n-9+I^(2*n)+(2+I)*I^(-n)+(2-I)*I^n)/4: seq(A047530(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Table[(8n-9+I^(2n)+(2+I)*I^(-n)+(2-I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *)

a(n)=n>>2<<3+[-1,0,1,3][n%4+1] \\ Charles R Greathouse IV, Jun 09 2011

From Johannes W. Meijer, Jun 07 2011: (Start)
a(n) = ceiling(n/4) + 2*ceiling((n-1)/4) + 4*ceiling((n-2)/4) + ceiling((n-3)/4).
a(n+1) = A053381(2^p). (End)
G.f.: x^2*(1+2*x+4*x^2+x^3) / ((1+x)*(x^2+1)*(x-1)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = a(n-1)+a(n-4)-a(n-5) for n>5.
a(n) = (8n-9+i^(2n)+(2+i)*i^(-n)+(2-i)*i^n)/4, where i=sqrt(-1).
a(2n) = A047522(n), a(2n-1) = A047470(n). (End)
E.g.f.: (2 + sin(x) + 2*cos(x) + (4*x - 5)*sinh(x) + 4*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 21 2016
Sum_{n>=2} (-1)^n/a(n) = (8-3*sqrt(2))*log(2)/16 + 3*sqrt(2)*log(2+sqrt(2))/8 - (sqrt(2)-1)*Pi/16. - Amiram Eldar, Dec 20 2021

More terms from Wesley Ivan Hurt, May 21 2016

cp's OEIS Frontend

A220466 a((2*n-1)*2^p) = 4^p*(n-1) + 2^(p-1)*(1+2^p), p >= 0 and n >= 1.

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A001676 Number of h-cobordism classes of smooth homotopy n-spheres.

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A003484 Radon function, also called Hurwitz-Radon numbers.

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Crossrefs

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Haskell

Maple

Mathematica

PARI

Python

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Extensions

A209675 Radon function at even positions: a(n) = A003484(2*n).

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Crossrefs

Programs

Haskell

Mathematica

Formula

A047530 Numbers that are congruent to {0, 1, 3, 7} mod 8.

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Author

Keywords

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Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A220466 a((2n-1)2^p) = 4^p(n-1) + 2^(p-1)(1+2^p), p >= 0 and n >= 1.