A209675 -id:A209675 - cp's OEIS frontend

A051062 a(n) = 16*n + 8.

8, 24, 40, 56, 72, 88, 104, 120, 136, 152, 168, 184, 200, 216, 232, 248, 264, 280, 296, 312, 328, 344, 360, 376, 392, 408, 424, 440, 456, 472, 488, 504, 520, 536, 552, 568, 584, 600, 616, 632, 648, 664, 680, 696, 712, 728, 744, 760, 776, 792, 808, 824, 840

Offset: 0

N. J. A. Sloane, Gary W. Adamson, Dec 11 1999

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0(97).
n such that 32 is the largest power of 2 dividing A003629(k)^n-1 for any k. - Benoit Cloitre, Mar 23 2002
Continued fraction expansion of tanh(1/8). - Benoit Cloitre, Dec 17 2002
If Y and Z are 2-blocks of a (4n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 28 2007
General form: (q*n+x)*q x=+1; q=2=A016825, q=3=A017197, q=4=A119413, ... x=-1; q=3=A017233, q=4=A098502, ... x=+2; q=4=A051062, ... - Vladimir Joseph Stephan Orlovsky, Feb 16 2009
a(n)*n+1 = (4n+1)^2 and a(n)*(n+1)+1 = (4n+3)^2 are both perfect squares. - Carmine Suriano, Jun 01 2014
For all positive integers n, there are infinitely many positive integers k such that k*n + 1 and k*(n+1) + 1 are both perfect squares. Except for 8, all the numbers of this sequence are the smallest integers k which are solutions for getting two perfect squares. Example: a(1) = 24 and 24 * 1 + 1 = 25 = 5^2, then 24 * (1+1) + 1 = 49 = 7^2. [Reference AMM] - Bernard Schott, Sep 24 2017
Numbers k such that 3^k + 1 is divisible by 17*193. - Bruno Berselli, Aug 22 2018

Letter from Gary W. Adamson concerning Prouhet-Thue-Morse sequence, Nov 11 1999.

Mia Boudreau, Table of n, a(n) for n = 0..10000
Mihaly Bencze, Problem 11508, The American Mathematical Monthly, Vol. 117, N° 5, May 2010, p. 459.
Milan Janjić, Two Enumerative Functions. [Wayback Machine link]
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005408, A008598, A017113, A106839, A139098, A244978.
Cf. A003484, A007814, A037227, A118413, A209675.
Cf. A003629, A016825, A017197, A017233, A098502, A119413.

[16*n+8: n in [0..50]]; // Wesley Ivan Hurt, Jun 01 2014

A051062:=n->16*n+8; seq(A051062(n), n=0..50); # Wesley Ivan Hurt, Jun 01 2014

Range[8, 1000, 16] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
Table[16n+8, {n,0,50}] (* Wesley Ivan Hurt, Jun 01 2014 *)
LinearRecurrence[{2,-1},{8,24},60] (* or *) NestList[#+16&,8,60] (* Harvey P. Dale, Aug 18 2019 *)

a(n)=16*n+8 \\ Charles R Greathouse IV, May 09 2016

a(n) = A118413(n+1,4) for n>3. - Reinhard Zumkeller, Apr 27 2006
a(n) = 32*n - a(n-1) for n>0, a(0)=8. - Vincenzo Librandi, Aug 06 2010
A003484(a(n)) = 8; A209675(a(n)) = 9. - Reinhard Zumkeller, Mar 11 2012
A007814(a(n)) = 3; A037227(a(n)) = 7. - Reinhard Zumkeller, Jun 30 2012
a(-1 - n) = - a(n). - Michael Somos, Jun 02 2014
Sum_{n>=0} (-1)^n/a(n) = Pi/32 (A244978). - Amiram Eldar, Feb 28 2023
From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 8*(1+x)/(1-x)^2.
E.g.f.: 8*exp(x)*(1 + 2*x).
a(n) = 8*A005408(n) = A008598(n) + 8 = A139098(n+1) - A139098(n).
a(n) = 4*A016825(n) = 2*A017113(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=0} (1 - (-1)^n/a(n)) = sqrt(2)*sin(7*Pi/32).
Product_{n>=0} (1 + (-1)^n/a(n)) = sqrt(2)*cos(7*Pi/32). (End)

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

A003485 Hurwitz-Radon function at powers of 2.

Original entry on oeis.org

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, 124

Offset: 0

N. J. A. Sloane

Positive integers that are congruent to {0, 1, 2, 4} mod 8. - Michael Somos, Dec 12 2023

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 9*x^4 + 10*x^5 + 12*x^6+ 16*x^7 + ... - _Michael Somos_, Dec 12 2023

T. Y. Lam, The Algebraic Theory of Quadratic Forms. Benjamin, Reading, MA, 1973, p. 131.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
V. Ovsienko and Serge Tabachnikov, Affine Hopf fibration, arXiv preprint arXiv:1511.08894 [math.AT], 2015.
V. Ovsienko and Serge Tabachnikov, Hopf fibrations and Hurwitz-Radon numbers, Math. Intell. 38 (2016) 11-18
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
D. B. Shapiro, Letter to N. J. A. Sloane, 1974
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A003484, A047507.
Essentially the same as A047466.
Cf. A008621. - Johannes W. Meijer, Jun 07 2011
Cf. A209675.

a003485 n = a003485_list !! n
a003485_list = 1 : 2 : 4 : 8 : 9 : zipWith (+)
   (drop 4 a003485_list) (zipWith (-) (tail a003485_list) a003485_list)
-- Reinhard Zumkeller, Mar 11 2012

A003485:= proc(n): ceil((n+1)/4) + ceil((n)/4) + 2*ceil((n-1)/4) + 4*ceil((n-2)/4) end: seq(A003485(n), n=0..62); # Johannes W. Meijer, Jun 07 2011

CoefficientList[Series[(1+x+2x^2+4x^3)/((1-x)(1-x^4)),{x,0,70}],x] (* or *) LinearRecurrence[{1,0,0,1,-1},{1,2,4,8,9},71] (* Harvey P. Dale, Jun 13 2011 *)
a[ n_] := 2*n + Max[0, 2-Mod[n-3, 4]]; (* Michael Somos, Dec 12 2023 *)

{a(n) = 2*n + max(0, 2 - (n-3)%4)}; /* Michael Somos, Dec 12 2023 */

G.f.: (1 + x + 2*x^2 + 4*x^3) / ((1-x)*(1-x^4)). - Simon Plouffe in his 1992 dissertation
a(n) = ceiling((n+1)/4) + ceiling((n)/4) + 2*ceiling((n-1)/4) + 4*ceiling((n-2)/4). - Johannes W. Meijer, Jun 07 2011
a(n) = a(n-1) + a(n-4) - a(n-5); a(0)=1, a(1)=2, a(2)=4, a(3)=8, a(4)=9. - Harvey P. Dale, Jun 13 2011
a(n) = -A047507(-n) = a(n+4) - 8 for all n in Z. - Michael Somos, Dec 12 2023

A053381 Maximal number of linearly independent smooth nowhere-zero vector fields on a (2n+1)-sphere.

Original entry on oeis.org

1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 9, 1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 11, 1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 9, 1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 9, 1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 11, 1, 3, 1, 7, 1, 3

Offset: 0

Warren D. Smith, Jan 06 2000

The corresponding terms for a 2n-sphere are all 0 ("you can't comb the hair on a billiard ball"). The "3" and "7" come from the quaternions and octonions.
b(n) = a(n-1): b(2^e) = ((e+1) idiv 4) + 2^((e+1) mod 4) - 1, b(p^e) = 1, p>2. - Christian G. Bower, May 18 2005
a(n-1) is multiplicative. - Christian G. Bower, Jun 03 2005

T. D. Noe, Table of n, a(n) for n = 0..10000
J. Frank Adams, Vector fields on spheres, Topology, 1 (1962), 63-65.
J. Frank Adams, Vector fields on spheres, Bull. Amer. Math. Soc. 68 (1962) 39-41.
J. Frank Adams, Vector fields on spheres, Annals of Math. 75 (1962) 603-632.
A. Hurwitz, Uber die Komposition der quadratischen formen, Math. Annalen 88 (1923) 1-25.
M. Kervaire, Non-parallelizability of the sphere for n > 7, Proc. Nat. Acad. Sci. USA 44 (1958) 280-283.
J. Milnor, Some consequences of a theorem of Bott, Annals Math. 68 (1958) 444-449.
J. Radon, Lineare Scharen Orthogonaler Matrizen, Abh. Math. Sem. Univ. Hamburg 1 (1922) 1-14.

For another version see A003484. Cf. A189995, A001676.

Cf. A047530, A220466.

int MaxLinInd(int n){ /* Returns max # linearly indep smooth nowhere zero * vector fields on S^{n-1}, n=1,2,... */ int b,c,d,rho; b = 0; while((n & 1)==0){ n /= 2; b++; } c = b & 3; d = (b - c)/4; rho = (1 << c) + 8*d; return( rho - 1); }

int MaxLinInd(int n) { int b = _builtin_ctz(n); return (1<<b%4) + b/4*8 - 1; } /* _Jeremy Tan, Apr 09 2021 */

with(numtheory): for n from 1 to 601 by 2 do c := irem(ifactors(n+1)[2,1,2],4): d := iquo(ifactors(n+1)[2,1,2],4): printf(`%d,`, 2^c+8*d-1) od:
nmax:=101: A047530 := proc(n): ceil(n/4) + 2*ceil((n-1)/4) + 4*ceil((n-2)/4) + ceil((n-3)/4) end: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 0 to ceil(nmax/(p+2))+1 do A053381((2*n+1)*2^p-1) := A047530(p+1): od: od: seq(A053381(n), n=0..nmax); # Johannes W. Meijer, Jun 07 2011, revised Jan 29 2013

a[n_] := Module[{b, c, d, rho, n0}, n0 = 2*n; b = 0; While[BitAnd[n0, 1] == 0, n0 /= 2; b++]; c = BitAnd[b, 3]; d = (b - c)/4; rho = 2^c + 8*d; Return[rho - 1]]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, May 16 2013, translated from C *)

Let f(n) be the number of linearly independent smooth nowhere-zero vector fields on an n-sphere. Then f(n) = 2^c + 8d - 1 where n+1 = (2a+1) 2^b and b = c+4d and 0 <= c <= 3. f(n) = 0 if n is even.
a((2*n+1)*2^p-1) = A047530(p+1), p >= 0 and n >= 0. a(2*n) = 1, n >= 0, and a(2^p-1) = A047530(p+1), p >= 0. - Johannes W. Meijer, Jun 07 2011
a(n) = A209675(n+1) - 1. - Reinhard Zumkeller, Mar 11 2012
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 10/3. - Amiram Eldar, Nov 29 2022

More terms from James Sellers, Jun 01 2000

A209776 Triangle of coefficients of polynomials v(n,x) jointly generated with A209773; see the Formula section.

Original entry on oeis.org

1, 3, 2, 5, 8, 4, 9, 22, 22, 8, 15, 52, 78, 56, 16, 25, 112, 226, 242, 136, 32, 41, 228, 580, 828, 692, 320, 64, 67, 446, 1374, 2456, 2726, 1872, 736, 128, 109, 848, 3074, 6612, 9158, 8336, 4864, 1664, 256, 177, 1578, 6590, 16590, 27564, 31250

Offset: 1

Clark Kimberling, Mar 15 2012

Alternating row sums: 1,1,1,1,1,1,1,1,...

For a discussion and guide to related arrays, see A208510.

			First five rows:
1
3....2
5....8....4
9....22...22...8
15...52...78...56...16
First three polynomials v(n,x): 1, 3 + 2x , 5 + 8x + 4x^2.

Cf. A209675, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209775 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209776 *)

u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=(x+1)*u(n-1,x)+2x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

cp's OEIS Frontend

A051062 a(n) = 16*n + 8.

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A003484 Radon function, also called Hurwitz-Radon numbers.

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A003485 Hurwitz-Radon function at powers of 2.

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A053381 Maximal number of linearly independent smooth nowhere-zero vector fields on a (2n+1)-sphere.

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A209776 Triangle of coefficients of polynomials v(n,x) jointly generated with A209773; see the Formula section.

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