A000408 -id:A000408 - cp's OEIS frontend

A001974 Numbers that are the sum of 3 distinct squares, i.e., numbers of the form x^2 + y^2 + z^2 with 0 <= x < y < z.

5, 10, 13, 14, 17, 20, 21, 25, 26, 29, 30, 34, 35, 37, 38, 40, 41, 42, 45, 46, 49, 50, 52, 53, 54, 56, 58, 59, 61, 62, 65, 66, 68, 69, 70, 73, 74, 75, 77, 78, 80, 81, 82, 83, 84, 85, 86, 89, 90, 91, 93, 94, 97, 98, 100, 101, 104, 105, 106, 107, 109, 110, 113

Offset: 1

N. J. A. Sloane

Also: Numbers which are the sum of two or three distinct nonzero squares. - M. F. Hasler, Feb 03 2013

According to Halter-Koch (below), a number n is a sum of 3 squares, but not a sum of 3 distinct squares (i.e., is in A001974 but not A000408), if and only if it is of the form 4^j*s, where j >= 0 and s in {1, 2, 3, 6, 9, 11, 18, 19, 22, 27, 33, 43, 51, 57, 67, 99, 102, 123, 163, 177, 187, 267, 627, ?}, where ? denotes at most one unknown number that, if it exists, is > 5*10^10. - Jeffrey Shallit, Jan 15 2017

			5 = 0^2 + 1^2 + 2^2.

T. D. Noe, Table of n, a(n) for n=1..1000
Franz Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arithmetica 42 (1982), pp. 11-20.
Index entries for sequences related to sums of squares

Cf. A001983, A024803, A025339, A025442, A004432.

Cf. A004436 (complement).

r[n_] := Reduce[0 <= x < y < z && x^2 + y^2 + z^2 == n, {x, y, z}, Integers]; ok[n_] := r[n] =!= False; Select[ Range[113], ok] (* Jean-François Alcover, Dec 05 2011 *)

from itertools import combinations
def aupto(lim):
  s = filter(lambda x: x <= lim, (i*i for i in range(int(lim**.5)+2)))
  s3 = set(filter(lambda x: x<=lim, (sum(c) for c in combinations(s, 3))))
  return sorted(s3)
print(aupto(113)) # Michael S. Branicky, May 10 2021

A025321 Numbers that are the sum of 3 nonzero squares in exactly 1 way.

Original entry on oeis.org

3, 6, 9, 11, 12, 14, 17, 18, 19, 21, 22, 24, 26, 29, 30, 34, 35, 36, 42, 43, 44, 45, 46, 48, 49, 50, 53, 56, 61, 65, 67, 68, 70, 72, 73, 76, 78, 82, 84, 88, 91, 93, 96, 97, 104, 106, 109, 115, 116, 120, 133, 136, 140, 142, 144, 145, 157, 163, 168, 169, 172, 176, 180, 184, 190

Offset: 1

David W. Wilson

nonn

It appears that all terms have the form 4^i A094740(j) for some i and j. - T. D. Noe, Jun 06 2008

This is true, because A025427(4*n) = A025427(n) for all n. - Robert Israel, Mar 09 2016

Donovan Johnson, Table of n, a(n) for n = 1..605 (terms < 10^8; first 417 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Square Number.
Index entries for sequences related to sums of squares

Cf. A000408, A025427, A243148.

lim=20; nLst=Table[0, {lim^2}]; Do[n=a^2+b^2+c^2; If[n>0 && nT. D. Noe, Jun 06 2008 *)
b[n_, i_, k_, t_] := b[n, i, k, t] = If[n == 0, If[t == 0, 1, 0], If[i<1 || t<1, 0, b[n, i - 1, k, t] + If[i^2 > n, 0, b[n - i^2, i, k, t - 1]]]];
T[n_, k_] := b[n, Sqrt[n] // Floor, k, k];
Position[Table[T[n, 3], {n, 0, 200}], 1] - 1 // Flatten (* Jean-François Alcover, Nov 06 2020, after Alois P. Heinz in A243148 *)

is(n)=if(n<11, return(n>0 && n%3==0)); if(n%4==0, return(is(n/4))); my(w); for(i=sqrtint((n-1)\3)+1,sqrtint(n-2), my(t=n-i^2); for(j=sqrtint((t-1)\2)+1,min(sqrtint(t-1),i), if(issquare(t-j^2), w++>1 && return(0)))); w \\ Charles R Greathouse IV, Aug 05 2024

A243148(a(n),3) = 1. - Alois P. Heinz, Feb 25 2019

A051952 Numbers that are not a sum of 3 positive squares nor are of the form 4^a*(8b+7) and which are not multiples of 4.

Original entry on oeis.org

1, 2, 5, 10, 13, 25, 37, 58, 85, 130

Offset: 1

Eberhard R. Hilf, Dec 21 1999

The asymptotic eigenvalue spectrum of the Schroedinger equation for a free particle in a box in three dimensions is known only (that is: average level density and average degeneracy) if the a(n) are finite series.
It is not known whether 130 is the largest such number or if this is the start of an infinite series.
See Theorem 6 of Grosswald's book, p. 79: every positive integer n not of the form 4^a*(8*m+7), with a and m nonnegative integers [i.e., n is a sum of three squares, proved by Legendre (1798)] and not of the form 4^b*a(n), with b a nonnegative integer, n=1,...,10, and possibly one a(11) > 5*10^10, is a sum of three non-vanishing squares. See A004214 and A000408. In the F. Halter-Koch reference, p. 13, one finds a Korollar 1 (b) stating for positive integers n, not 0, 4, 7 modulo 8: n is not the sum of three positive coprime squares if and only if n = a(n), n=1,...,10, and possibly one more number a(11) >= 5*10^10. See A223731. - Wolfdieter Lang, Apr 04 2013

			Consider a(3)=5: 1^2 +1^2 +1^2=3, too low; 1^2+1^2+2^2=6, too high. 4^1=4 too low; 4^2=16 too high; (8*0+7)=7 too low, (8*1+7)= 15 too high; thus 5 is a member of this sequence.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 79 with p. 76.

H. P. Baltes and E. R. Hilf, Spectra of finite systems; BI-Verlag.
H.-P. Baltes, Peter K. J. Draxl and Eberhard R. Hilf, Quadratsummen und gewisse Randwertprobleme der Mathematischen Physik, Publications of the Small Systems Group Oldenburg, preprint, 1973.
H.-P. Baltes, Peter K. J. Draxl and Eberhard R. Hilf, Quadratsummen und gewisse Randwertprobleme der Mathematischen Physik, Journ. Reine Angewandte Mathematik, Vol. 268/269, 1974, 410-417.
P. K. J. Draxl, Sommes de deux carrés qui ne sont pas sommes de trois carrés strictement positifs, Mémoires de la Société Mathématique de France, 37 (1974), p. 53-53.
E. Grosswald, A. Calloway and J. Calloway, The representations of integers by three positive squares, Proc. Amer. Math. Soc. 10 (1959), 451-455. [Math. Rev. 21 #3376; E24-73 in Leveque's Reviews in Number Theory, Vol. 2, p. 290]
F. Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arith. 42 (1982) 11-20, p. 13.
Eberhard R. Hilf, Publications
Eberhard R. Hilf, Über den Oberflächenterm der Gesamtenergie der Atomkerne nach dem Fermigas-Modell, Diploma-thesis, Universität Frankfurt, Germany, 1963.
E. R. Hilf and H. P. Baltes, 130 and the cube spectrum, unpublished
E. R. Hilf, G. Suessmann, Surface Tension of nuclei according to the Fermi-gas-model, Physics Letters, Vol. 21, No. 6, p. 654-656, (1966).
Index entries for sequences related to sums of squares

nmax = 1000; amax = Ceiling[Log[nmax/7]/Log[4]]; notThreeSquaresQ[n_] := Reduce[0 < a <= b <= c && n == a^2 + b^2 + c^2, {a, b, c}, Integers] === False; notOfTheFormQ[n_, a_] := Reduce[n == 4^a*(8*b+7), b, Integers] === False; notOfTheFormQ[n_] := And @@ (notOfTheFormQ[n, #] & ) /@ Range[0, amax]; Select[Range[nmax], !Mod[#, 4] == 0 && notThreeSquaresQ[#] && notOfTheFormQ[#] & ](* Jean-François Alcover, Jun 12 2012 *)

Grosswald et al. reference from N. J. A. Sloane, Jun 07 2000

A223730 Multiplicities for representations of positive numbers n as primitive sums of three nonzero squares.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 2, 1, 1, 0, 0, 2, 0, 0, 2, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 0, 1, 2, 0, 0, 2, 0, 2, 0, 1, 2, 0, 0, 1, 3, 1, 0, 2, 1, 0, 0, 1, 2, 1, 0, 2, 1, 0, 0, 2, 1, 2, 0, 0, 3, 0, 0, 3, 2, 1, 0, 1, 2, 0, 0, 1, 2, 2, 0, 3, 2, 0, 0, 2, 1, 2, 0, 1, 3, 0, 0, 2, 3, 1, 0, 2, 2, 0, 0, 2, 2, 2, 0, 2, 2, 0, 0, 4, 0, 3, 0, 1, 4

Offset: 1

Wolfdieter Lang, Apr 04 2013

nonn

Primitive sums of three nonzero squares a^2 + b^2 + c^2, with positive integers a, b and c, satisfy gcd(a,b,c) = 1. (coprimality of the three squares).
a(n) gives the number of different representations (multiplicities) of the number n >= 1 as primitive sums of three nonzero squares. If a(n) = 0 there is no such representation for n. The numbers n with a(n) not vanishing are given in A223731. The ones with a(n) = 1, 2 and 3 are in A223732, A223733 and A223734, respectively.
For the multiplicities of the positive numbers as sums of three nonzero squares see A025427. The numbers with A025427(n) >= 1 are given in A000408.
A corollary in the Halter-Koch reference (Korollar 1. (b) on p. 13) states for the positive numbers n, not 0, 4, 7 (mod 8) [otherwise n cannot be a primitive sum of three nonzero squares; see p. 11, the r_3(n) formula]:  n is not the sum of three positive coprime squares if and only if n is from the set T := {1, 2, 5, 10, 13, 25, 37, 58, 85, 130, ?}, with ? possibly a number >= 5*10^10 . Therefore a(n) = 0 if and only if n >= 1 is of the form mentioned in this corollary: i)  0, 4, 7 (mod 8) or ii) in the set T.
For representations of n as a sum of three nonzero squares see the Grosswald reference, Theorem 7, p. 79. There also the above mentioned set T appears and for the Conjecture it is assumed that the extra eleventh member of T is absent.

			a(12) = 0 because the only representation of 12 as a sum of three nonzero squares is given by [2,2,2], i.e., 12 = 2^2 + 2^2 + 2^2, but this is not a primitive sum because gcd(2,2,2) = 2, not 1. Such a situation appears for n = 12, 24, 36, 44, 48, 56, 68, 72, 76, 84, 88, 96, ... For these numbers A025427(n) = 1 and a(n) = 0.
a(27) = 1 because the only primitive representation of 27 as a sum of three nonzero squares is denoted by [1,1,5]. The representation [3,3,3] is not primitive.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
F. Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arith. 42 (1982) 11-20.

Cf. A223731, A025427 (non-primitive case), A223732, A223733, A223734.

with(numtheory):
b:= proc(n, i, t, s) option remember;
      `if`(n=0, `if`(t=0 and s={}, 1, 0), `if`(i=1, `if`(t=n, 1, 0),
      `if`(t*i^2xn, 0, b(n-i^2, i, t-1, `if`(s={1}, factorset(i),
       s intersect factorset(i)))))))
    end:
a:= n-> b(n, isqrt(n), 3, {1}):
seq(a(n), n=1..200);  # Alois P. Heinz, Apr 06 2013

a[n_] := Select[ PowersRepresentations[n, 3, 2], Times @@ # != 0 && GCD @@ # == 1 &] // Length; Table[a[n], {n, 1, 134}] (* Jean-François Alcover, Jun 21 2013 *)

a(n) = 0 if there is no representation of n as a primitive sum of three nonzero squares. a(n) = k >= 1 if there are k distinct such representations for n.

A237707 Number of unit cubes, aligned with a three-dimensional Cartesian mesh, completely within the first octant of a sphere centered at the origin, ordered by increasing radius.

Original entry on oeis.org

1, 4, 7, 10, 11, 17, 20, 23, 26, 32, 35, 38, 44, 48, 54, 60, 66, 69, 75, 78, 87, 96, 102, 105, 108, 114, 120, 121, 127, 133, 139, 145, 157, 163, 169, 178, 184, 196, 202, 214, 217, 220, 232, 238, 241, 244, 256, 263, 266, 278, 284, 296, 299, 308, 314, 329, 332

Offset: 1

Rajan Murthy, Feb 11 2014

nonn

			When the radius of the sphere reaches 3^(1/2), one cube is completely within the sphere. When the radius reaches 6^(1/2), four cubes are completely within the sphere.

Rajan Murthy, Table of n, a(n) for n = 1..200
Rajan Murthy, Table of n, a(n), and squared radius for n = 1..200
Rajan Murthy, Scilab program for this sequence
Charles R Greathouse IV, Illustration of this sequence

The radii corresponding to the terms are given by the square roots of A000408 starting with squared radius 3.
Cf. A232499 (2-dimensional analog).
Partial sums of A014465 and A063691 (but then with repeated terms omitted).

(* Illustrates the sequence *)
Cube[x_,y_,z_]:=Cuboid[{x-1,y-1,z-1},{x,y,z}]
Cubes[r_]:=Cube@@#&/@Select[Flatten[Table[{x,y,z},{x,1,r},{y,1,r},{z,1,r}],2],Norm[#]<=r&]
Draw[r_]:=Graphics3D[Union[Cubes[r],{{Green, Opacity[0.3], Sphere[{0,0,0},r]}}],PlotRange->{{0,r},{0,r},{0,r}},ViewPoint->{r,3r/4,3r/5}];
Draw/@Sqrt/@{3,6,9,11,12,14} (* Charles R Greathouse IV, Mar 12 2014 *)

Scilab
```
// See Murthy link.
```

a(n) ~ (Pi*sqrt(30)/25)*n^(3/2). - Charles R Greathouse IV, Mar 14 2014

Duplicate terms deleted by Rajan Murthy, Mar 06 2014

Terms a(36) and beyond added from b-file by Andrew Howroyd, Feb 27 2018

A003386 Numbers that are the sum of 8 nonzero 8th powers.

Original entry on oeis.org

8, 263, 518, 773, 1028, 1283, 1538, 1793, 2048, 6568, 6823, 7078, 7333, 7588, 7843, 8098, 8353, 13128, 13383, 13638, 13893, 14148, 14403, 14658, 19688, 19943, 20198, 20453, 20708, 20963, 26248, 26503, 26758, 27013, 27268, 32808, 33063, 33318, 33573

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
9534597 is in the sequence as 9534597 = 2^8 + 3^8 + 3^8 + 3^8 + 5^8 + 6^8 + 6^8 + 7^8.
13209988 is in the sequence as 13209988 = 1^8 + 1^8 + 2^8 + 2^8 + 2^8 + 6^8 + 7^8 + 7^8.
19046628 is in the sequence as 19046628 = 2^8 + 2^8 + 3^8 + 4^8 + 6^8 + 7^8 + 7^8 + 7^8. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

A###### (x, y): Numbers that are the form of x nonzero y-th powers.

Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

M = 92646056; m = M^(1/8) // Ceiling; Reap[
For[a = 1, a <= m, a++, For[b = a, b <= m, b++, For[c = b, c <= m, c++,
For[d = c, d <= m, d++, For[e = d, e <= m, e++, For[f = e, f <= m, f++,
For[g = f, g <= m, g++, For[h = g, h <= m, h++,
s = a^8 + b^8 + c^8 + d^8 + e^8 + f^8 + g^8 + h^8;
If[s <= M, Sow[s]]]]]]]]]]][[2, 1]] // Union (* Jean-François Alcover, Dec 01 2020 *)

b-file checked by R. J. Mathar, Aug 01 2020

Incorrect program removed by David A. Corneth, Aug 01 2020

A014465 A063691 without zeros.

Original entry on oeis.org

1, 3, 3, 3, 1, 6, 3, 3, 3, 6, 3, 3, 6, 4, 6, 6, 6, 3, 6, 3, 9, 9, 6, 3, 3, 6, 6, 1, 6, 6, 6, 6, 12, 6, 6, 9, 6, 12, 6, 12, 3, 3, 12, 6, 3, 3, 12, 7, 3, 12, 6, 12, 3, 9, 6, 15, 3, 15, 12, 6, 6, 12, 3, 3, 12, 9, 18, 6, 6, 12, 6, 9, 4, 6, 18, 9, 12, 6, 6, 12, 9, 6, 9, 12, 6, 12, 18, 18, 15, 6, 6, 21, 3, 9, 12, 9, 6, 12

Offset: 1

A. Timothy Royappa, 1997; entry revised Jun 13 2003

Let b(n) = n-th number of form x^2 + y^2 + z^2, x,y,z >= 1 (A000408); a(n) = number of solutions (x,y,z) to x^2 + y^2 + z^2 = b(n).
The a(n) are also the degeneracies of the energy levels E(n) in the three-dimensional cubic "particle-in-a-box" model in elementary quantum mechanics. - A. Timothy Royappa, Jan 09 2009
Continuously increase the radius of a sphere centered at the origin. Whenever the number of entire unit cubes that fit into one quadrant of the sphere increases (cf. A237707), list the number of additional cubes. - M. F. Hasler, Jun 25 2022

			b(1) = 3 = 1^2 + 1^2 + 1^2 (1 way), so a(1) = 1;
b(2) = 6 = 2^2 + 1^2 + 1^2 (3 ways), so a(2) = 3; etc.

G. M. Barrow, Physical Chemistry (6th ed.), McGraw-Hill, 1996, p. 69.

Daniel Leary, Table of n, a(n) for n = 1..8283

First différences of A237707.

for(n=1,200,r=sqrtint(n);s=0;for(i=1,r,si=i*i;for(j=1,r,sj=j*j;for(k=1,r,if(si+sj+k*k==n,s=s+1))));if(s,print1(s,","))) /* Ralf Stephan, Aug 31 2013 */

More terms and better name from Ralf Stephan, Aug 31 2013

A018820 Numbers k that are the sum of m nonzero squares for all 1 <= m <= k - 14.

Original entry on oeis.org

169, 225, 289, 625, 676, 841, 900, 1156, 1225, 1369, 1521, 1681, 2025, 2500, 2601, 2704, 2809, 3025, 3364, 3600, 3721, 4225, 4624, 4900, 5329, 5476, 5625, 6084, 6724, 7225, 7569, 7921, 8100, 8281, 9025, 9409, 10000, 10201, 10404, 10816, 11025, 11236

Offset: 1

David W. Wilson

nonn

Intersection of A000290, A000404 and A000408. - Zak Seidov, Nov 12 2013
A square k^2 is the sum of m positive squares for all 1 <= m <= k^2 - 14 iff k^2 is the sum of 2 and 3 positive squares (see A309778 and proof in Kuczma). - Bernard Schott, Aug 17 2019
Note that k is never the sum of k - 13 positive squares. - Jianing Song, Feb 09 2021

			169 is a term: 169 = 13^2 = 5^2 + 12^2 = 3^2 + 4^2 + 12^2 = 11^2 + 4^2 + 4^2 + 4^2 = 6^2 + 6^2 + 6^2 + 6^2 + 5^2 = 6^2 + 6^2 + 6^2 + 6^2 + 4^2 + 3^2 = ... = 3^2 + 2^2 + 2^2 + 1^2 + 1^2 + ... + 1^2 (sum of 155 positive squares, with 152 (1^2)'s), but 169 cannot be represented as the sum of 156 positive squares. - _Jianing Song_, Feb 09 2021

Marcin E. Kuczma, International Mathematical Olympiads, 1986-1999, The Mathematical Association of America, 2003, pages 76-79.

Zak Seidov and Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..100 from Zak Seidov)
Index entries for sequences related to sums of squares

Cf. A000290, A000404, A000408, A309778, A341329.

isA018820(n) = issquare(n) && isA341329(sqrtint(n)) \\ Jianing Song, Feb 09 2021, see A341329 for its program

a(n) = A341329(n)^2. - Jianing Song, Feb 09 2021

A166687 Numbers of the form x^2 + y^2 + 1, x, y integers.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 10, 11, 14, 17, 18, 19, 21, 26, 27, 30, 33, 35, 37, 38, 41, 42, 46, 50, 51, 53, 54, 59, 62, 65, 66, 69, 73, 74, 75, 81, 82, 83, 86, 90, 91, 98, 99, 101, 102, 105, 107, 110, 114, 117, 118, 122, 123, 126, 129, 131, 137, 138, 145, 146, 147, 149, 150, 154, 158, 161

Offset: 1

N. J. A. Sloane, Mar 05 2010

nonn

A001481 is the main entry for this sequence.

As Ng points out (Lemma 2.2), each prime divides some member of this sequence: 2 divides a(2) = 2, 3 divides a(3) = 3, 5 divides a(4) = 5, 7 divides a(9) = 14, etc. - Charles R Greathouse IV, Jan 04 2016

Robert Israel, Table of n, a(n) for n = 1..10000
Jia Hong Ray Ng, Quarternions and the four square theorem, 2008 Summer VIGRE Program for Undergraduates
Yu-Chen Sun and Hao Pan, The Green-Tao theorem for primes of the form x^2 + y^2 + 1, Monatshefte für Mathematik vol. 189 (2019), pp. 715-733. arXiv:1708.08629 [math.NT]

Cf. A000408, A000378, A005767, A169580, A166265, A079545.

N:= 1000: # to get all terms <= N
S:= {seq(seq(x^2+y^2+1,y=0..floor(sqrt(N-1-x^2))),x=0..floor(sqrt(N-1)))}:
sort(convert(S,list)); # Robert Israel, Jan 05 2016

Select[Range@ 162, Resolve[Exists[{x, y}, Reduce[# == x^2 + y^2 + 1, {x, y}, Integers]]] &] (* Michael De Vlieger, Jan 05 2016 *)

is(n)=my(f=factor(n-1)); for(i=1, #f~, if(f[i,1]%4==3 && f[i,2]%2, return(0))); 1 \\ Charles R Greathouse IV, Jan 04 2016

list(lim)=my(v=List(),t); lim\=1; for(m=0,sqrtint(lim-1), t=m^2+1; for(n=0, min(sqrtint(lim-t),m), listput(v,t+n^2))); Set(v) \\ Charles R Greathouse IV, Jan 05 2016

A010008 a(0) = 1, a(n) = 18*n^2 + 2 for n>0.

Original entry on oeis.org

1, 20, 74, 164, 290, 452, 650, 884, 1154, 1460, 1802, 2180, 2594, 3044, 3530, 4052, 4610, 5204, 5834, 6500, 7202, 7940, 8714, 9524, 10370, 11252, 12170, 13124, 14114, 15140, 16202, 17300, 18434, 19604, 20810, 22052, 23330, 24644, 25994, 27380, 28802, 30260

Offset: 0

N. J. A. Sloane

The identity (18*n^2+2)^2-(9*n^2+2)*(6*n)^2=4 can be written as a(n+1)^2-A010002(n+1)*A008588(n+1)^2=4. - Vincenzo Librandi, Feb 07 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

After 20, all terms are in A000408.

Cf. A206399.

[1] cat [18*n^2+2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015

Join[{1}, 18 Range[41]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {20, 74, 164}, 50]] (* Vincenzo Librandi, Aug 03 2015 *)

G.f.: (1+x)*(1+16*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
a(n) = (3*n-1)^2+(3*n+1)^2 = (n-1)^2+(n+1)^2+(4*n)^2 for n>0. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*18+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4+ (1/12)*Pi*coth(Pi/3) = 1.0853330948... - R. J. Mathar, May 07 2024
a(n) = 2*A247792(n), n>0. - R. J. Mathar, May 07 2024
a(n) = A069131(n)+A069131(n+1). - R. J. Mathar, May 07 2024

More terms from Bruno Berselli, Feb 06 2012

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