OFFSET
1,3
COMMENTS
For n > 15, a(n) = A006431(n-1). - Thomas Ordowski, Nov 18 2012
REFERENCES
J. H. Conway, The Sensual (Quadratic) Form, M.A.A., 1997, p. 140.
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. 302.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, Theorem 3, pp. 74-75.
LINKS
T. D. Noe, Table of n, a(n) for n = 1..100
Brennan Benfield and Oliver Lippard, Integers that are not the sum of positive powers, arXiv:2404.08193 [math.NT], 2024. See p. 2.
Pierre de la Harpe, Lagrange et la variation des théorèmes, Images des Mathématiques, CNRS, 2014.
Index entries for linear recurrences with constant coefficients, signature (0,0,4).
FORMULA
Consists of the numbers 0, 1, 3, 5, 9, 11, 17, 29, 41, 2*4^m, 6*4^m and 14*4^m (m >= 0). Compare A123069.
From 224 on, a(n) = 4*a(n-3).
Numbers n such that A025428(n) = 0.
G.f.: x^2*(36*x^16 + 32*x^15 + 60*x^14 + 55*x^13 + 36*x^12 + 27*x^11 + 20*x^10 + 19*x^9 + 18*x^8 + 13*x^7 + 11*x^6 + 4*x^5 + 2*x^4 - x^3 - 3*x^2 - 2*x - 1)/(4*x^3 - 1). - Chai Wah Wu, Jul 09 2022
MATHEMATICA
q=22; lst={}; Do[Do[Do[Do[z=a^2+b^2+c^2+d^2; If[z<=q^2+3, AppendTo[lst, z]], {d, q}], {c, q}], {b, q}], {a, q}]; lst1=Union@lst lst={}; Do[AppendTo[lst, n], {n, q^2+3}]; lst2=lst Complement[lst2, lst1] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)
Join[{0, 1, 2, 3, 5, 6, 8, 9, 11, 14, 17, 24, 29, 32, 41}, LinearRecurrence[{0, 0, 4}, {56, 96, 128}, 30]] (* Jean-François Alcover, Feb 09 2016 *)
PROG
(PARI) for(n=1, 224, if(sum(a=1, n, sum(b=1, a, sum(c=1, b, sum(d=1, c, if(a^2+b^2+c^2+d^2-n, 0, 1)))))==0, print1(n, ", ")))
(PARI) {a(n)=if( n<2, 0, n<16, [1, 2, 3, 5, 6, 8, 9, 11, 14, 17, 24, 29, 32, 41][n-1], [4, 7, 12][n%3+1] * 2^(n\3*2-7))}; /* Michael Somos, Apr 23 2006 */
(PARI) is(n)=my(k=if(n, n/4^valuation(n, 4), 2)); k==2 || k==6 || k==14 || setsearch([0, 1, 3, 5, 9, 11, 17, 29, 41], n) \\ Charles R Greathouse IV, Sep 03 2014
(Python)
from itertools import count, islice
def A000534_gen(startvalue=0): # generator of terms >= startvalue
return filter(lambda n:n in {0, 1, 3, 5, 9, 11, 17, 29, 41} or n>>((~n&n-1).bit_length()&-2) in {2, 6, 14}, count(max(startvalue, 0)))
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved