A055039 -id:A055039 - cp's OEIS frontend

A124169 Numbers in A055039 but not in A044075.

238, 494, 750, 910, 926, 942, 952, 958, 1006, 1262, 1518, 1774, 1934, 1950, 1966, 1976, 1982, 2030, 2286, 2542, 2798, 2958, 2974, 2990, 3000, 3006, 3054, 3310, 3566, 3598, 3614, 3630, 3640, 3646, 3662, 3678, 3694, 3704, 3710, 3726, 3742, 3758, 3768

Offset: 1

N. J. A. Sloane, Dec 04 2006

Numbers whose base 4 representation ends in 3,2 followed by some number of zeros and includes at least one other 3,2. This means that, in spite of the initial terms, asymptotically most members of A055039 are not in A044075. In fact, A055039 has density 1/12, while A044075 has density zero. - Franklin T. Adams-Watters, Dec 04 2006

R. J. Mathar, Table of n, a(n) for n=1..3449

{ hex14(n)=local(m);m=n;while(m,if(m%16==14,return(1));m\=16);0 }
for(n=1,10^4, k=n; while(k%4==0,k\=4); if(k%16!=14,next); if(hex14(k-14)||hex14((k-14)\4),print1(n,",")) ) \\ Max Alekseyev, Dec 04 2006

More terms from Max Alekseyev and Klaus Brockhaus, Dec 04 2006

A004215 Numbers that are the sum of 4 but no fewer nonzero squares.

Original entry on oeis.org

7, 15, 23, 28, 31, 39, 47, 55, 60, 63, 71, 79, 87, 92, 95, 103, 111, 112, 119, 124, 127, 135, 143, 151, 156, 159, 167, 175, 183, 188, 191, 199, 207, 215, 220, 223, 231, 239, 240, 247, 252, 255, 263, 271, 279, 284, 287, 295, 303, 311, 316, 319, 327, 335, 343

Offset: 1

N. J. A. Sloane and J. H. Conway

Lagrange's theorem tells us that each positive integer can be written as a sum of four squares.
If n is in the sequence and k is an odd positive integer then n^k is in the sequence because n^k is of the form 4^i(8j+7). - Farideh Firoozbakht, Nov 23 2006
Numbers whose cubes do not have a partition as a sum of 3 squares. a(n)^3 = A134738(n). - Artur Jasinski, Nov 07 2007
A002828(a(n)) = 4; A025427(a(n)) > 0. - Reinhard Zumkeller, Feb 26 2015
There are infinitely many adjacent pairs (for example, 128n + 111 and 128n + 112 for any n), but never a triple of consecutive integers. - Ivan Neretin, Aug 17 2017
These numbers are called "forbidden numbers" in crystallography: for a cubic crystal, no reflection with index hkl such that h^2 + k^2 + l^2 = a(n) appears in the crystal's diffraction pattern. - A. Timothy Royappa, Aug 11 2021

			15 is in the sequence because it is the sum of four squares, namely, 3^2 + 2^2 + 1^2 + 1^2, and it can't be expressed as the sum of fewer squares.
16 is not in the sequence, because, although it can be expressed as 2^2 + 2^2 + 2^2 + 2^2, it can also be expressed as 4^2.

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 261.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 12.
E. Poznanski, 1901. Pierwiastki pierwotne liczb pierwszych. Warszawa, pp. 1-63.
W. Sierpiński, 1925. Teorja Liczb. pp. 1-410 (p. 125).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, entry 4181.

T. D. Noe, Table of n, a(n) for n = 1..10000
David S. Bettes, Letter to N. J. A. Sloane, Nov 05 1976
Richard T. Bumby, Sums Of Four Squares
International Union of Crystallography, Cubic structures.
Shuo Li, The characteristic sequence of the integers that are the sum of two squares is not morphic, arXiv:2404.08822 [math.NT], 2024.
Louis J. Mordell, A new Waring's problem with squares of linear forms, Quart. J. Math., 1 (1930), 276-288 (see p. 283).
Saburô Uchiyama, A five-square theorem, Publ. Res. Math. Sci., Vol 13, Number 1 (1977), 301-305.
Steve Waterman, Missing numbers formula
Eric Weisstein's World of Mathematics, Square Number
Wikipedia, Lagrange's four-square theorem.
Index entries for sequences related to sums of squares

Complement of A000378.
Cf. A002828, A055039, A072401, A125084, A134738, A134739, A055045, A055046, A234000.
Cf. A000118 (ways to write n as sum of 4 squares), A025427.

a004215 n = a004215_list !! (n-1)
a004215_list = filter ((== 4) . a002828) [1..]
-- Reinhard Zumkeller, Feb 26 2015

N:= 1000: # to get all terms <= N
{seq(seq(4^i * (8*j + 7), j = 0 .. floor((N/4^i - 7)/8)), i = 0 .. floor(log[4](N)))}; # Robert Israel, Sep 02 2014

Sort[Flatten[Table[4^i(8j + 7), {i, 0, 2}, {j, 0, 42}]]] (* Alonso del Arte, Jul 05 2005 *)
Select[Range[120], Mod[ #/4^IntegerExponent[ #, 4], 8] == 7 &] (* Ant King, Oct 14 2010 *)

isA004215(n)={ local(fouri,j) ; fouri=1 ; while( n >=7*fouri, if( n % fouri ==0, j= n/fouri -7 ; if( j % 8 ==0, return(1) ) ; ) ; fouri *= 4 ; ) ; return(0) ; } { for(n=1,400, if(isA004215(n), print1(n,",") ; ) ; ) ; } \\ R. J. Mathar, Nov 22 2006

isA004215(n)= n\4^valuation(n,4)%8==7 \\ M. F. Hasler, Mar 18 2011

def valuation(n, b):
    v = 0
    while n > 1 and n%b == 0: n //= b; v += 1
    return v
def ok(n): return n//4**valuation(n, 4)%8 == 7 # after M. F. Hasler
print(list(filter(ok, range(344)))) # Michael S. Branicky, Jul 15 2021

from itertools import count, islice
def A004215_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:not (m:=(~n&n-1).bit_length())&1 and (n>>m)&7==7,count(max(startvalue,1)))
A004215_list = list(islice(A004215_gen(),30)) # Chai Wah Wu, Jul 09 2022

def A004215(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(((x>>(i<<1))-7>>3)+1 for i in range(x.bit_length()>>1))
    return bisection(f,n,n) # Chai Wah Wu, Feb 14 2025

a(n) = A055039(n)/2. - Ray Chandler, Jan 30 2009
Numbers of the form 4^i*(8*j+7), i >= 0, j >= 0. [A.-M. Legendre & C. F. Gauss]
Products of the form A000302(i)*A004771(j), i, j >= 0. - R. J. Mathar, Nov 29 2006
a(n) = 6*n + O(log(n)). - Charles R Greathouse IV, Dec 19 2013
Conjecture: The number of terms < 2^n is A023105(n) - 2. - Tilman Neumann, Sep 20 2020

More terms from Arlin Anderson (starship1(AT)gmail.com)

Additional comments from Jud McCranie, Mar 19 2000

A004015 Theta series of face-centered cubic (f.c.c.) lattice.

Original entry on oeis.org

1, 12, 6, 24, 12, 24, 8, 48, 6, 36, 24, 24, 24, 72, 0, 48, 12, 48, 30, 72, 24, 48, 24, 48, 8, 84, 24, 96, 48, 24, 0, 96, 6, 96, 48, 48, 36, 120, 24, 48, 24, 48, 48, 120, 24, 120, 0, 96, 24, 108, 30, 48, 72, 72, 32, 144, 0, 96, 72, 72, 48, 120, 0, 144, 12, 48, 48, 168, 48, 96

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 + 12*x + 6*x^2 + 24*x^3 + 12*x^4 + 24*x^5 + 8*x^6 + 48*x^7 + 6*x^8 + ...
G.f. = 1 + 12*q^2 + 6*q^4 + 24*q^6 + 12*q^8 + 24*q^10 + 8*q^12 + 48*q^14 + 6*q^16 + ...
From _Michael Somos_, Jan 05 2012: (Start)
a(2) = 6 since (1, -1, -1) is a solution to x^2 + y^2 + z^2 + x*y + x*z + y*z = 2 and the other 5 solutions are permutations and negations of this one.
a(2) = 6 since (1, 1, -1, -1) is a solution to x + y + z + w = 0 and x^2 + y^2 + z^2 + w^2 = 4 and the other 5 solutions are permutations of this one. (End)

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 113.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 263.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.
L. V. Woodcock, Nature, Jan 09 1997, pp. 141-143.

N. J. A. Sloane, Table of n, a(n) for n = 0..9999
Charles L. Hohn, a(0) to a(15), animated
G. Nebe and N. J. A. Sloane, Home page for this lattice
N. J. A. Sloane, A portion of the f.c.c. lattice packing.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Theta Series
Index entries for sequences related to f.c.c. lattice

Cf. A004013, A005875, A005901, A045828. A055039 gives the positions of the 0's in this sequence.

Cf. A000007, A000122, A004016, A008444, A008445, A008446, A008447, A008448, A008449 (Theta series of lattices A_0, A_1, A_2, A_4, ...)

L := Lattice("A",3); A := ThetaSeries(L, 140); A; /* Michael Somos, Nov 13 2014 */

A := Basis( ModularForms( Gamma1(8), 3/2), 70); A[1] + 12*A[2] + 6*A[3] + 24*A[4]; /* Michael Somos, Sep 08 2018 */

maxd := 201: temp0 := trunc(evalf(sqrt(maxd)))+2: a := 0: for i from -temp0 to temp0 do a := a+q^( (i+1/2)^2): od: th2 := series(a,q,maxd); a := 0: for i from -temp0 to temp0 do a := a+q^(i^2): od: th3 := series(a,q,maxd); th4 := series(subs(q=-q, th3),q,maxd); series((1/2)*(th3^3+th4^3),q,200);

a[n_] := SquaresR[3, 2n]; Table[a[n], {n, 0, 69}] (* Jean-François Alcover, Jul 12 2012 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^3 + EllipticTheta[ 4, 0, q]^3) / 2, {q, 0, 2 n}]; (* Michael Somos, May 24 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2]^3 + 12 q QPochhammer[ q^4]^3 QPochhammer[ q^8]^2 / QPochhammer[ q^2]^2, {q, 0, n}]; (* Michael Somos, Nov 13 2014 *)
SquaresR[3,2*Range[0,70]] (* Harvey P. Dale, Jun 01 2015 *)

{a(n) = if( n<0, 0, n*=2; polcoeff( sum( k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n))^3, n))}; /* Michael Somos, Oct 25 2006 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A)^5 / eta(x^2 + A)^2 / eta(x^8 + A)^2)^3 + 12 * x * eta(x^4 + A)^3 * eta(x^8 + A)^2 / eta(x^2 + A)^2, n))}; /* Michael Somos, May 17 2008 */

{a(n) = if( n<1, n==0, 2 * qfrep( [2, 1, 1; 1, 2, 1; 1, 1, 2], n, 1)[n])}; /* Michael Somos, Jan 02 2012 */

from math import prod, isqrt
from sympy import factorint
def A004018(n): return prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in factorint(n).items())<<2 if n else 1
def A004015(n): return A004018(m:=n<<1)+(sum(A004018(m-k**2) for k in range(1,isqrt(m)+1))<<1) # Chai Wah Wu, Feb 24 2025

Expansion of phi(q^2)^3 + 12 * q * phi(q^2) * psi(q^4)^2 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Oct 25 2006
Expansion of (phi(q)^3 + phi(-q)^3) / 2 in powers of q^2 where phi() is a Ramanujan theta function. - Michael Somos, Oct 25 2006
Expansion of b(q) * phi(q^18) + c(q^3) * phi(q^2) in powers of q^3 where b(), c() are cubic AGM theta functions and phi() is a Ramanujan theta function. - Michael Somos, Oct 25 2006
Expansion of (theta_3(q)^3 + theta_4(q)^3) / 2 in powers of q^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2^(7/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A004013.
a(n) = A005875(2*n).
G.f.: Sum_{i, j, k in Z} x^( i*i + j*j + k*k + i*j + i*k + j*k ). - Michael Somos, Jan 02 2012
From Michael Somos, Jan 05 2012: (Start)
Number of integer solutions to x^2 + y^2 + z^2 + x*y + x*z + y*z = n.
Number of integer solutions to x + y + z even and x^2 + y^2 + z^2 = 2 * n.
Number of integer solutions to x + y + z + w = 0 and x^2 + y^2 + z^2 + w^2 = 2 * n. (End)
a(2*n) = A005875(n). a(2*n+1) = 12 * A045828(n). - Michael Somos, Dec 28 2017

A293654 Integers not represented by cyclotomic binary forms.

Original entry on oeis.org

1, 2, 6, 14, 15, 22, 23, 24, 30, 33, 35, 38, 42, 44, 46, 47, 51, 54, 56, 59, 60, 62, 66, 69, 70, 71, 77, 78, 83, 86, 87, 88, 92, 94, 95, 96, 99, 102, 105, 107, 110, 114, 115, 118, 119, 120, 123, 126, 131, 132, 134, 135, 138, 140, 141, 142, 143, 150

Offset: 1

Michel Waldschmidt, Feb 16 2018

nonn

Possibly a supersequence of A055039. - C. S. Davis, May 10 2025

Michel Waldschmidt, Table of n, a(n) for n = 1..249
Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Complement of A296095.

For n>2, subsequence of A383785, following from Proposition 6.2 of Fouvry et al.

g := 1;
for m from 1 to 1000 do
    for n from 3 to 50 do
        for x from -50 to 50 do
            for y from -50 to 50 do
                if (F[n] = m, max(abs(x), abs(y)) > 1
                then r[g] := m; m := m+1; n := 3;
                     x := -50; y := -50; g := g+1
                fi;
od; od; od; od;
for t to 519 do print(r[{t}] = r[t]) od;
s[1] := 1; s[2] := 2; g := 2;
for i from 1 to 518 do
    for j from r[i]+1 to r[i+1]-1 do
        g := g+1; s[g] := j
od; od;
for t to 481 do s[t] od;

isA296095[n_] := If[n<3, Return[False], logn = Log[n]^1.161; K = Floor[ 5.383*logn]; M = Floor[2*(n/3)^(1/2)]; k = 3; While[True, If[k == 7, K = Ceiling[4.864*logn]; M = Ceiling[2*(n/11)^(1/4)]]; For[y = 2, y <= M, y++, p[z_] = y^EulerPhi[k]*Cyclotomic[k, z]; For[x = 1, x <= y, x++, If[n == p[x/y], Return[True]]]]; k++; If[k>K, Break[]]]; Return[False]];
Select[Range[150], !isA296095[#]&] (* Jean-François Alcover, Jun 21 2018, after Peter Luschny *)

def A293654list(upto):
    return [n for n in (1..upto) if not isA296095(n)]
print(A293654list(150)) # Peter Luschny, Feb 25 2018

A119870 Number of vertices of the root-n Waterman polyhedron.

Original entry on oeis.org

12, 6, 24, 12, 24, 32, 48, 54, 36, 24, 48, 24, 72, 72, 48, 60, 48, 54, 72, 72, 72, 72, 48, 56, 132, 96, 120, 96, 72, 72, 96, 102, 96, 96, 120, 84, 120, 144, 96, 72, 120, 72, 168, 168, 120, 120, 144, 168, 108, 126, 168, 72, 144, 152, 144, 144, 192, 120, 144, 144

Offset: 1

Hugo Pfoertner, May 26 2006

nonn

The root-n Waterman polyhedron is the convex hull of the intersection of a closed ball of radius sqrt(2*n) with the lattice of sphere-center points of a cubic close packing. [Probably the f.c.c. lattice is intended here. - N. J. A. Sloane, Aug 09 2006]

The basic sphere center series of Waterman polyhedra is obtained by choosing a sphere center as the center of the closed ball. Other choices are possible. An example is given in A119874 ... A119878. For n in A055039 no lattice points are hit; the corresponding polyhedra are the same as for n-1.

Cf. A119870, A119875 [vertices of void-centered Waterman polyhedron].

Cf. A055039 [missing polyhedra]. Properties of Waterman polyhedra: A119870 [vertices], A119871 [faces], A119872 [edges], A119873 [volume]. Waterman polyhedra with different center: A119874, A119875, A119876, A119877, A119878.

A119869 Sizes of successive clusters in f.c.c. lattice centered at a lattice point.

Original entry on oeis.org

1, 13, 19, 43, 55, 79, 87, 135, 141, 177, 201, 225, 249, 321, 321, 369, 381, 429, 459, 531, 555, 603, 627, 675, 683, 767, 791, 887, 935, 959, 959, 1055, 1061, 1157, 1205, 1253, 1289, 1409, 1433, 1481, 1505, 1553, 1601, 1721, 1745, 1865, 1865, 1961, 1985, 2093, 2123

Offset: 0

Hugo Pfoertner, May 26 2006

N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.

N. J. A. Sloane, Table of n, a(n) for n = 0..9999
Paul Bourke, Waterman Polyhedra.
Paul Bourke, On-line generator
Martin Kraus, Live Graphics3d
Mirek Majewski, Dedicated commands in MuPAD
Wouter Meeussen, ConvexHull3D package & demo-file.
Mark Newbold, Waterman Polyhedra. CCPOLY Java Applet.
Steve Waterman, Waterman Polyhedron.
Steve Waterman, Missing numbers formula

Cf. A004015, A004215, A055039.

maxd:=20001: read format: temp0:=trunc(evalf(sqrt(maxd)))+2: a:=0: for i from -temp0 to temp0 do a:=a+q^( (i+1/2)^2): od: th2:=series(a,q,maxd): a:=0: for i from -temp0 to temp0 do a:=a+q^(i^2): od: th3:=series(a,q,maxd): th4:=series(subs(q=-q,th3),q,maxd):
t1:=series((th3^3+th4^3)/2,q,maxd): t1:=series(subs(q=sqrt(q),t1),q,floor(maxd/2)): t2:=seriestolist(t1): t4:=0; for n from 1 to nops(t2) do t4:=t4+t2[n]; lprint(n-1, t4); od: # N. J. A. Sloane, Aug 09 2006

a[n_] := Sum[SquaresR[3, 2k], {k, 0, n}]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 12 2012, after formula *)
Accumulate[SquaresR[3,2*Range[0,70]]] (* Harvey P. Dale, Jun 01 2015 *)

Partial sums of A004015, which has an explicit generating function.

Edited by N. J. A. Sloane, Aug 09 2006

Additional links from Steve Waterman, Nov 26 2006

A000401 Numbers of form x^2 + y^2 + 2*z^2.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76

Offset: 1

N. J. A. Sloane

Numbers represented by quadratic form with Gram matrix [ 1, 0, 0; 0, 1, 0; 0, 0, 2 ].
These are the numbers not of the form 4^k*(16*m + 14). [Dickson] - Everett W. Howe, May 18 2008
The asymptotic density of this sequence is 11/12. - Amiram Eldar, Mar 29 2025

Wacław Sierpiński, Elementary Theory of Numbers, (Ed. A. Schinzel), North-Holland, 1988, see Exercise 4 on p. 395.

T. D. Noe, Table of n, a(n) for n = 1..1000
L. E. Dickson, Integers represented by positive ternary quadratic forms, Bull. Amer. Math. Soc. 33 (1927), 63-70.
Gabriele Nebe and N. J. A. Sloane, Home page for this lattice.

Complement of A055039.

L := [seq(0,i=1..1)]: for x from 0 to 20 do for y from 0 to 20 do for z from 0 to 20 do if member(x^2+y^2+2*z^2, L)=false then L := [op(L), x^2+y^2+2*z^2] fi: od: od: od: L2 := sort(L): for i from 1 to 100 do printf(`%d,`,L2[i]) od:

q=16;imax=q^2;Select[Union[Flatten[Table[x^2+y^2+2*z^2,{z,0,q},{y,0,q},{x,0,q}]]],#<=imax&] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)
Select[Range[0, 100], Mod[# / 4^IntegerExponent[#, 4], 16] != 14 &] (* Amiram Eldar, Mar 29 2025 *)

def A000401(n):
    def f(x): return n-1+sum(((x>>i)-7>>3)+1 for i in range(1,x.bit_length(),2))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 24 2025

More terms from James Sellers, May 31 2000

A044075 Numbers k such that the string 3,2 occurs in the base-4 representation of k but not of k-1.

Original entry on oeis.org

14, 30, 46, 56, 62, 78, 94, 110, 120, 126, 142, 158, 174, 184, 190, 206, 222, 224, 248, 254, 270, 286, 302, 312, 318, 334, 350, 366, 376, 382, 398, 414, 430, 440, 446, 462, 478, 480, 504, 510, 526, 542, 558, 568, 574, 590, 606

Offset: 1

Clark Kimberling

Numbers whose base-4 representation ends in 3,2 followed by some number of zeros and includes no other 3,2. - Franklin T. Adams-Watters, Dec 04 2006
Not the same as A055039 - see A124169.
A 4-automatic set: membership is determined by comparing the base-4 representation of the number to the regular expression /[012]*(3+([01][012]*)?)*320*/. - Charles R Greathouse IV, Feb 11 2012 [corrected by Pontus von Brömssen, Jan 12 2019]
Alternatively, numbers whose base-4 representation is in the language generated by the regular expression /([012]|3*[01])*3+20*/. - Pontus von Brömssen, Jan 17 2019

R. J. Mathar, Table of n, a(n) for n = 1..10000
Index entries for 2-automatic sequences.

has32 := proc(n) local shft : shft := n : while shft > 0 do if shft mod 16 = 14 then RETURN(true) ; fi : shft := floor(shft/4) : od : RETURN(false) ; end: isA044075 := proc(n) if has32(n) and not has32(n-1) then return(true): else return(false) : fi : end: n := 1 : a := 1 : while n <= 10000 do while not isA044075(a) do a := a+1 : od : printf("%d %d ",n,a) : a := a+1 : n := n+1 : od : # R. J. Mathar, Dec 07 2006

Flatten[Position[Partition[Table[If[MemberQ[Partition[IntegerDigits[n, 4], 2, 1], {3, 2}], 1, 0], {n, 1000}], 2, 1], {0, 1}]] + 1 (* Vincenzo Librandi, Aug 19 2015 *)

cp's OEIS Frontend

A124169 Numbers in A055039 but not in A044075.

Views

Author

Keywords

Comments

Links

Programs

PARI

Extensions

A004215 Numbers that are the sum of 4 but no fewer nonzero squares.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Python

Python

Python

Formula

Extensions

A004015 Theta series of face-centered cubic (f.c.c.) lattice.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Magma

Maple

Mathematica

PARI

PARI

PARI

Python

Formula

A293654 Integers not represented by cyclotomic binary forms.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Sage

A119870 Number of vertices of the root-n Waterman polyhedron.

Views

Author

Keywords

Comments

Crossrefs

A119869 Sizes of successive clusters in f.c.c. lattice centered at a lattice point.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A000401 Numbers of form x^2 + y^2 + 2*z^2.

Views