A003485 -id:A003485 - cp's OEIS frontend

A003484 Radon function, also called Hurwitz-Radon numbers.

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

A101120 Records in A101119, which forms the nonzero differences of A006519 and A003484.

Original entry on oeis.org

7, 22, 52, 112, 239, 494, 1004, 2024, 4071, 8166, 16356, 32736, 65503, 131038, 262108, 524248, 1048535, 2097110, 4194260, 8388560, 16777167, 33554382, 67108812, 134217672, 268435399, 536870854, 1073741764, 2147483584, 4294967231, 8589934526, 17179869116, 34359738296

Offset: 1

Simon Plouffe and Paul D. Hanna, Dec 02 2004

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

Cf. A003484, A006519, A101119, A101121, A101122.

LinearRecurrence[{3,-2,0,1,-3,2},{7,22,52,112,239,494},30] (* Harvey P. Dale, Jan 23 2023 *)

PARI
```
a(n)=2^(n+3)-2^((n-1)%4)-8*((n+3)\4)
```

def A101120(n): return (1<<(n+3))-(1<<((n-1)&3))-(((n+3)&-4)<<1) # Chai Wah Wu, Jul 10 2022

a(n) = A101119(2^(n-1)) for n>=1.
a(n) = 2^(n+3) - 2^((n-1)(mod 4)) - 8*floor((n+3)/4).
a(n) = 2^(n+3) - A003485(n+3). - Johannes W. Meijer, Oct 31 2012
From Chai Wah Wu, Apr 15 2017: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) + a(n-4) - 3*a(n-5) + 2*a(n-6) for n > 6.
G.f.: x*(-x - 7)/((x - 1)^2*(x + 1)*(2*x - 1)*(x^2 + 1)). (End)
E.g.f.: (exp(x)*(32*exp(x) - 8*x - 27) - 4*cos(x) - cosh(x) - 2*sin(x) + sinh(x))/4. - Stefano Spezia, Jun 06 2023

A047466 Numbers that are congruent to {0, 1, 2, 4} mod 8.

Original entry on oeis.org

0, 1, 2, 4, 8, 9, 10, 12, 16, 17, 18, 20, 24, 25, 26, 28, 32, 33, 34, 36, 40, 41, 42, 44, 48, 49, 50, 52, 56, 57, 58, 60, 64, 65, 66, 68, 72, 73, 74, 76, 80, 81, 82, 84, 88, 89, 90, 92, 96, 97, 98, 100, 104, 105, 106, 108, 112, 113, 114, 116, 120, 121, 122

Offset: 1

N. J. A. Sloane

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

Essentially the same as A003485.

Cf. A047461, A047467.

[n: n in [0..120] | n mod 8 in [0,1,2,4]];

A047466:=n->2*n-4+(3-I^(2*n))*(1-I^(n*(n+1)))/4: seq(A047466(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016

Select[Range[0,120], MemberQ[{0, 1, 2, 4}, Mod[#, 8]] &] (* or *) LinearRecurrence[{1, 0, 0, 1, -1}, {0, 1, 2, 4, 8}, 60] (* Bruno Berselli, Jul 18 2012 *)

makelist(2*n-4+(3-(-1)^n)*(1-%i^(n*(n+1)))/4,n,1,60);

concat(0, Vec((1+x+2*x^2+4*x^3)/((1+x)*(1+x^2)*(1-x)^2)+O(x^60))) (End)

G.f.: x^2*(1+x+2*x^2+4*x^3) / ( (1+x)*(1+x^2)*(1-x)^2 ). - R. J. Mathar, Oct 08 2011
a(n) = 2*n-4+(3-(-1)^n)*(1-i^(n*(n+1)))/4, where i=sqrt(-1). - Bruno Berselli, Jul 18 2012
From Wesley Ivan Hurt, Jun 01 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(2k) = A047461(k), a(2k-1) = A047467(k). (End)
Sum_{n>=2} (-1)^n/a(n) = (1+2*sqrt(2))*Pi/32 + (3+sqrt(2))*log(2)/16 - sqrt(2)*log(2-sqrt(2))/8. - Amiram Eldar, Dec 20 2021

A047507 Numbers that are congruent to {0, 4, 6, 7} mod 8.

Original entry on oeis.org

0, 4, 6, 7, 8, 12, 14, 15, 16, 20, 22, 23, 24, 28, 30, 31, 32, 36, 38, 39, 40, 44, 46, 47, 48, 52, 54, 55, 56, 60, 62, 63, 64, 68, 70, 71, 72, 76, 78, 79, 80, 84, 86, 87, 88, 92, 94, 95, 96, 100, 102, 103, 104, 108, 110, 111, 112, 116, 118, 119, 120, 124

Offset: 1

N. J. A. Sloane

			G.f. = 4*x^2 + 6*x^3 + 7*x^4 + 8*x^5 + 12*x^6 + 14*x^7 + 15*x^8 + 16*x^9 + ... - _Michael Somos_, Dec 12 2023

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A003485, A047451, A047535.

[n : n in [0..150] | n mod 8 in [0, 4, 6, 7]]; // Wesley Ivan Hurt, May 27 2016

A047507:=n->(8*n-3+I^(2*n)-(1+2*I)*I^(-n)-(1-2*I)*I^n)/4: seq(A047507(n), n=1..100); # Wesley Ivan Hurt, May 27 2016

Table[(8n-3+I^(2n)-(1+2*I)*I^(-n)-(1-2*I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 27 2016 *)
a[ n_] := 2*n - Max[0, 2 - Mod[1-n, 4]]; (* Michael Somos, Dec 12 2023 *)

{a(n) = 2*n - max(0, 2 - (1-n)%4)}; /* Michael Somos, Dec 12 2023 */

G.f.: x^2*(4+2*x+x^2+x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Nov 06 2015
From Wesley Ivan Hurt, May 27 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (8*n-3+i^(2*n)-(1+2*i)*i^(-n)-(1-2*i)*i^n)/4 where i=sqrt(-1).
a(2k) = A047535(k), a(2k-1) = A047451(k). (End)
E.g.f.: (2 - 2*sin(x) - cos(x) + (4*x - 2)*sinh(x) + (4*x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 27 2016
Sum_{n>=2} (-1)^n/a(n) = (6-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8 - sqrt(2)*Pi/16. - Amiram Eldar, Dec 23 2021
a(n) = -A003485(-n) = a(n+4) - 8 for all n in Z. - Michael Somos, Dec 12 2023

cp's OEIS Frontend

A003484 Radon function, also called Hurwitz-Radon numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A101120 Records in A101119, which forms the nonzero differences of A006519 and A003484.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A047466 Numbers that are congruent to {0, 1, 2, 4} mod 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

Formula

A047507 Numbers that are congruent to {0, 4, 6, 7} mod 8.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula