A001650 -id:A001650 - cp's OEIS frontend

A192455 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n*A(-x)^A001650(n+1), where A001650 is defined by "n appears n times (n odd).".

1, 1, 2, 7, 27, 112, 492, 2249, 10580, 50885, 249067, 1236602, 6212563, 31523293, 161317863, 831615320, 4314659345, 22512421092, 118052038100, 621825506334, 3288597601727, 17455485596492, 92958082866815, 496535775228131, 2659574264906443

Offset: 0

Paul D. Hanna, Jul 14 2011

nonn

Compare the g.f. to a g.f. C(x) of the Catalan numbers: 1 = Sum_{n>=0} x^n*C(-x)^(2*n+1).

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 27*x^4 + 112*x^5 + 492*x^6 +...
The g.f. satisfies:
1 = A(-x) + x*A(-x)^3 + x^2*A(-x)^3 + x^3*A(-x)^3 + x^4*A(-x)^5 + x^5*A(-x)^5 + x^6*A(-x)^5 + x^7*A(-x)^5 + x^8*A(-x)^5 + x^9*A(-x)^7 +...+ x^n*A(-x)^A001650(n+1) +...
where A001650 begins: [1, 3,3,3, 5,5,5,5,5, 7,7,7,7,7,7,7, 9,...].
The g.f. also satisfies:
1-x = (1-x)*A(-x) + x*(1-x^3)*A(-x)^3 + x^4*(1-x^5)*A(-x)^5 + x^9*(1-x^7)*A(-x)^7 + x^16*(1-x^9)*A(-x)^9 +...

Cf. A193039, A193040, A193050, A001650.

{a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(sum(m=1, #A, (-x)^m*Ser(A)^(1+2*sqrtint(m-1)) ), #A)); if(n<0, 0, A[n+1])}

G.f. satisfies: 1-x = Sum_{n>=1} x^(n^2) * (1-x^(2*n-1)) * A(-x)^(2*n-1).

A000122 Expansion of Jacobi theta function theta_3(x) = Sum_{m =-oo..oo} x^(m^2) (number of integer solutions to k^2 = n).

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (the present sequence), psi(q) (A010054), chi(q) (A000700).
Theta series of the one-dimensional lattice Z.
Also, essentially the same as the theta series of the one-dimensional lattices A_1, A*_1, D_1, D*_1.
Number of ways of writing n as a square.
Closely related: theta_4(x) = Sum_{m = -oo..oo} (-x)^(m^2). See A002448.
Number 6 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

			G.f. = 1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + 2*q^25 + 2*q^36 + 2*q^49 + 2*q^64 + 2*q^81 + ...

Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, Exercise 1, p. 91.
Richard Bellman, A Brief Introduction to Theta Functions, Dover, 2013.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 64.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 104, [5n].
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 93, Eq. (34.1); p. 78, Eq. (32.22).
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 133.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Sixth Edition, Clarendon Press, Oxford, 2009, Theorem 352, p. 372.
J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques, Vol. 2, Gauthier-Villars, Paris, 1902; Chelsea, NY, 1972, see p. 27.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1963, p. 464.

T. D. Noe, Table of n, a(n) for n = 0..10000
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
M. D. Hirschhorn and J. A. Sellers, A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four, Article 14.9.6, Journal of Integer Sequences, Vol. 17 (2014).
K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

1st column of A286815. - Seiichi Manyama, May 27 2017
Row d=1 of A122141.
Cf. A002448 (theta_4). Partial sums give A001650.
Cf. A010052, A010054, A089801.
Cf. A000007, A004015, A004016, A008444, A008445, A008446, A008447, A008448, A008449 (Theta series of lattices A_0, A_3, A_2, A_4, ...).

using Nemo
function JacobiTheta3(len, r)
    R, x = PolynomialRing(ZZ, "x")
    e = theta_qexp(r, len, x)
    [fmpz(coeff(e, j)) for j in 0:len - 1] end
A000122List(len) = JacobiTheta3(len, 1)
A000122List(105) |> println # Peter Luschny, Mar 12 2018

Basis( ModularForms( Gamma0(4), 1/2), 100) [1]; /* Michael Somos, Jun 10 2014 */

L := Lattice("A",1); A := ThetaSeries(L, 20); A; /* Michael Somos, Nov 13 2014 */

add(x^(m^2),m=-10..10): seq(coeff(%,x,n), n=0..100);
# alternative
A000122 := proc(n)
    if n = 0 then
        1;
    elif issqr(n) then
        2;
    else
        0 ;
    end if;
end proc:
seq(A000122(n),n=0..100) ; # R. J. Mathar, Feb 22 2021

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q], {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
CoefficientList[ Sum[ x^(m^2), {m, -(n=10), n} ], x ]
SquaresR[1, Range[0, 104]] (* Robert G. Wilson v, Jul 16 2014 *)
QP = QPochhammer; s = QP[q^2]^5/(QP[q]*QP[q^4])^2 + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 24 2015 *)
(4 QPochhammer[q^2]/QPochhammer[-1,-q]^2 + O[q]^101)[[3]] (* Vladimir Reshetnikov, Sep 16 2016 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A))^2, n))}; /* Michael Somos, Mar 14 2011 */

{a(n) = issquare(n) * 2 -(n==0)}; /* Michael Somos, Jun 17 1999 */

from sympy.ntheory.primetest import is_square
def A000122(n): return is_square(n)<<1 if n else 1 # Chai Wah Wu, May 17 2023

Q = DiagonalQuadraticForm(ZZ, [1])
Q.representation_number_list(105) # Peter Luschny, Jun 20 2014

Expansion of eta(q^2)^5 / (eta(q)*eta(q^4))^2 in powers of q.
Euler transform of period 4 sequence [2, -3, 2, -1, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 - v^2 + 2 * w * (w - u). - Michael Somos, Jul 20 2004
G.f. A(x) satisfies 0 = f(A(x), A(x^3), A(x^9)) where f(u, v, w) = w^4 - v^4 + w * (u - w)^3. - Michael Somos, May 11 2019
G.f.: Sum_{m=-oo..oo} x^(m^2);
a(0) = 1; for n > 0, a(n) = 0 unless n is a square when a(n) = 2.
G.f.: Product_{k>0} (1 - x^(2*k))*(1 + x^(2*k-1))^2.
G.f.: s(2)^5/(s(1)^2*s(4)^2), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
The Jacobi triple product identity states that for |x| < 1, z != 0, Product_{n>0} {(1-x^(2n))(1+x^(2n-1)z)(1+x^(2n-1)/z)} = Sum_{n=-inf..inf} x^(n^2)*z^n. Set z=1 to get theta_3(x).
For n > 0, a(n) = 2*(floor(sqrt(n))-floor(sqrt(n-1))). - Mikael Aaltonen, Jan 17 2015
G.f. is a period 1 Fourier series which satisfies f(-1/(4 t)) = 2^(1/2) (t/i)^(1/2) f(t) where q = exp(2 Pi i t). - Michael Somos, May 05 2016
a(n) = A000132(n)(mod 4). - John M. Campbell, Jul 07 2016
a(n) = (2/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017
a(n) = 2 * A010052(n) if n>0. a(3*n + 1) = 2 * A089801(n). a(3*n + 2) = 0. a(4*n) = a(n). a(4*n + 2) = a(4*n + 3) = 0. a(8*n + 1) = 2 * A010054(n). - Michael Somos, May 11 2019
Dirichlet g.f.: 2*zeta(2s). - Francois Oger, Oct 26 2019 [Corrected by Sean A. Irvine, Nov 26 2024]
G.f. appears to equal exp( 2*Sum_{n >= 0} x^(2*n+1)/((2*n+1)*(1 + x^(2*n+1))) ). - Peter Bala, Dec 23 2021
From Peter Bala, Sep 27 2023: (Start)
G.f. A(x) satisfies A(x)*A(-x) = A(-x^2)^2.
A(x) = Sum_{n >= 1} x^(n-1)*Product_{k >= n} 1 - (-x)^k.
A(x)^2 = 1 + 4*Sum_{n >= 1} (-1)^(n+1)*x^(2*n-1)/(1 - x^(2*n-1)), which gives the number of representations of an integer as a sum of two squares. See, for example, Fine, 26.63.
A(x) = 1 + 2*Sum_{n >= 1} x^(n*(n+1)/2) * ( Product_{k = 1..n-1} 1 + x^k ) /( Product_{k = 1..n} 1 + x^(2*k) ). See Fine, equation 14.43. (End)

A122510 Array T(d,n) = number of integer lattice points inside the d-dimensional hypersphere of radius sqrt(n), read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 1, 7, 9, 3, 1, 9, 19, 9, 5, 1, 11, 33, 27, 13, 5, 1, 13, 51, 65, 33, 21, 5, 1, 15, 73, 131, 89, 57, 21, 5, 1, 17, 99, 233, 221, 137, 81, 21, 5, 1, 19, 129, 379, 485, 333, 233, 81, 25, 7, 1, 21, 163, 577, 953, 797, 573, 297, 93, 29, 7, 1, 23, 201, 835, 1713, 1793

Offset: 1

R. J. Mathar, Oct 29 2006, Oct 31 2006

Number of solutions to sum_(i=1,..,d) x[i]^2 <= n, x[i] in Z.

			T(2,2)=9 counts 1 pair (0,0) with sum 0, 4 pairs (-1,0),(1,0),(0,-1),(0,1) with sum 1 and 4 pairs (-1,-1),(-1,1),(1,1),(1,-1) with sum 2.
Array T(d,n) with rows d=1,2,3... and columns n=0,1,2,3.. reads
  1  3   3    3    5     5     5     5      5      7      7
  1  5   9    9   13    21    21    21     25     29     37
  1  7  19   27   33    57    81    81     93    123    147
  1  9  33   65   89   137   233   297    321    425    569
  1 11  51  131  221   333   573   893   1093   1343   1903
  1 13  73  233  485   797  1341  2301   3321   4197   5757
  1 15  99  379  953  1793  3081  5449   8893  12435  16859
  1 17 129  577 1713  3729  6865 12369  21697  33809  47921
  1 19 163  835 2869  7189 14581 27253  49861  84663 129303
  1 21 201 1161 4541 12965 29285 58085 110105 198765 327829

Index entries for sequences related to sums of squares

Rows d=1..10 give A001650, A057655, A117609, A046895, A175360, A175361, A341396, A341397, A341398, A341399.

Cf. A005408 (column 1), A058331 (column 2), A161712 (column 3), A055426 (column 4), A055427 (column 9)

T := proc(d,n) local i,cnts ; cnts := 0 ; for i from -trunc(sqrt(n)) to trunc(sqrt(n)) do if n-i^2 >= 0 then if d > 1 then cnts := cnts+T(d-1,n-i^2) ; else cnts := cnts+1 ; fi ; fi ; od ; RETURN(cnts) ; end: for diag from 1 to 14 do for n from 0 to diag-1 do d := diag-n ; printf("%d,",T(d,n)) ; od ; od;

t[d_, n_] := t[d, n] = t[d, n-1] + SquaresR[d, n]; t[d_, 0] = 1; Table[t[d-n, n], {d, 1, 12}, {n, 0, d-1}] // Flatten (* Jean-François Alcover, Jun 13 2013 *)

Recurrence along rows: T(d,n)=T(d,n-1)+A122141(d,n) for n>=1; T(d,n)=sum_{i=0..n} A122141(d,i). Recurrence along columns: cf. A123937.

A131507 2n+1 appears n+1 times.

Original entry on oeis.org

1, 3, 3, 5, 5, 5, 7, 7, 7, 7, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 15, 15, 15, 15, 15, 15, 15, 15, 17, 17, 17, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23

Offset: 0

Paul Curtz, Aug 13 2007

Sum of terms of row n is (n+1)*(2n+1) = A000384(n+1). - Michel Marcus, Feb 02 2014

Where records occur give A000217. - Omar E. Pol, Nov 05 2015

			As a triangle, the sequence starts:
1;
3, 3;
5, 5, 5;
7, 7, 7, 7;
9, 9, 9, 9, 9;
...

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A000217, A002024, A003056, A005408.

Cf. A001650.

a131507 n k = a131507_tabl !! n !! k
a131507_row n = a131507_tabl !! n
a131507_tabl = zipWith ($) (map replicate [1..]) [1, 3 ..]
a131507_list = concat a131507_tabl
-- Reinhard Zumkeller, Jul 12 2014, Mar 18 2011
(Chipmunk BASIC v3.6.4(b8)) # http://www.nicholson.com/rhn/basic/
for n=1 to 23 step 2
for j=1 to n  step 2
print str$(n)+", ";
next j : next n : print
end
# Jeremy Gardiner, Feb 02 2014

seq(2*floor(sqrt(2*n+1)+1/2)-1, n=0..70); # Ridouane Oudra, Oct 20 2019

Table[2 n + 1, {n, 0, 11}, {n + 1}] // Flatten (* Michael De Vlieger, Nov 05 2015 *)

from math import isqrt
def A131507(n): return (k:=isqrt(m:=n+1<<1))+(m>k*(k+1))-1<<1|1 # Chai Wah Wu, Nov 04 2024

a(n) = 2*floor(sqrt(2n+1)+1/2) - 1. - Ridouane Oudra, Oct 20 2019

A193832 Irregular triangle read by rows in which row n lists 2n-1 copies of 2n-1 and n copies of 2n, for n >= 1.

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14

Offset: 1

Omar E. Pol, Aug 22 2011

Sequence of successive positive integers k in which if k is odd then k appears k times, otherwise if k is even then k appears k/2 times.
Note that an arrangement of the blocks of this sequence shows the growth of the generalized pentagonal numbers A001318 (see example).
The sums of each block give the positive integers of A129194: 1, 2, 9, 8, 25, 18, 49,...
Partial sums of A080995. - Paolo P. Lava, Aug 23 2011.
Concatenations of rows of triangles A001650 and A111650; also, seen as a flat list, the row lengths of triangle A260672 and the first differences of its row sums (cf. A260706). - Reinhard Zumkeller, Nov 17 2015
Also a(n) = number of squares in the arithmetic progression {24k + 1: 0 <= k <= n-1} [Granville]. - N. J. A. Sloane, Dec 13 2017

			a) If written as a triangle the initial rows are
  1, 2,
  3, 3, 3, 4, 4,
  5, 5, 5, 5, 5, 6, 6, 6,
  7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8,
  9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10,
  ...
Row sums give A126587.
b) An application using the blocks of this sequence: the illustration of the growth of an arrangement which represents the generalized pentagonal numbers A001318. For example; the first 9 positive initial terms: 1, 2, 5, 7, 12, 15, 22, 26, 35.
.
.         9
.       8 9
.     8 7 9
.   8 6 7 9
. 8 6 5 7 9
. 6 4 5 7 9
. 4 3 5 7 9
. 2 3 5 7 9
. 1 3 5 7 9
...

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
Andrew Granville, Squares in arithmetic progressions and infinitely many primes, arXiv:1708.06951 [math.NT], 2017.
Andrew Granville, Squares in arithmetic progressions and infinitely many primes, The American Mathematical Monthly, 124.10 (2017): 951-954. See p. 952.

Cf. A001318, A080995, A126587, A129194, A221671, A221672.

Cf. A001650, A111650, A260672, A260706.

a193832 n k = a193832_tabf !! (n-1) !! (k-1)
a193832_row n = a193832_tabf !! (n-1)
a193832_tabf = zipWith (++) a001650_tabf a111650_tabl
a193832' n = a193832_list !! (n - 1)
a193832_list = concat a193832_tabf
-- Reinhard Zumkeller, Nov 15 2015

Array[Join @@ MapIndexed[ConstantArray[#, #/(1 + Boole[First@ #2 == 2])] &, {2 # - 1, 2 #}] &, 7] // Flatten (* or *)
Table[If[k <= 2 n - 1, 2 n - 1, 2 n], {n, 7}, {k, 3 n - 1}] // Flatten (* Michael De Vlieger, Dec 14 2017 *)

a(n) = sqrt(8n/3) plus or minus 1 [Granville] - N. J. A. Sloane, Dec 13 2017

If 8 <= n <= 52, then a(n-1) < a(n) if and only if n is in A221672. - Jonathan Sondow, Dec 14 2017

Edited by N. J. A. Sloane, Dec 13 2017

A001670 k appears k times (k even).

Original entry on oeis.org

2, 2, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A001650, A000194.

Equals A130829(n) - 1.

a = @(n) 2*floor((sqrt(4*n-3)+1)/2); % Néstor Jofré, Apr 24 2017

[2*Round(Sqrt(n)): n in [1..70]]; // Vincenzo Librandi, Jun 23 2011

seq(2*n $ 2*n, n = 1 .. 10); # Robert Israel, Jan 14 2015

a[1]=2, a[2]=2, a[n_]:=a[n]=a[n-a[n-2]]+2 (* Branko Curgus, May 11 2010 *)
Flatten[Table[Table[n,{n}],{n,2,16,2}]] (* Harvey P. Dale, May 31 2012 *)

a(n)=round(sqrt(n))<<1 \\ Charles R Greathouse IV, Jun 23 2011

from math import isqrt
def A001670(n): return (m:=isqrt(n))+int((n-m*(m+1)<<2)>=1)<<1 # Chai Wah Wu, Jul 29 2022

a(n) = 2*floor(1/2 + sqrt(n)). - Antonio Esposito, Jan 21 2002; corrected by Branko Curgus, May 11 2010
With a different offset: g.f. = Sum_{j>=0} 2*x^(j^2+i)/(1-x). - Ralf Stephan, Mar 11 2003
From Branko Curgus, May 11 2010: (Start)
a(n) = a(n - a(n-2)) + 2; a(1)=2, a(2)=2.
a(n) = 2*round(sqrt(n)). (End)
G.f.: x^(3/4)*theta_2(0,x)/(1-x) where theta_2 is the second Jacobi theta function. - Robert Israel, Jan 14 2015
a(n) = 2*floor((sqrt(4*n-3)+1)/2). - Néstor Jofré, Apr 24 2017

Offset changed from 2 to 1 by Vincenzo Librandi, Jun 23 2011

A341397 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_8)^2 <= n.

Original entry on oeis.org

1, 17, 129, 577, 1713, 3729, 6865, 12369, 21697, 33809, 47921, 69233, 101041, 136209, 174737, 231185, 306049, 384673, 469457, 579217, 722353, 876465, 1025649, 1220337, 1481521, 1733537, 1979713, 2306753, 2697537, 3087777, 3482913, 3959585, 4558737, 5155473

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A000143.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sums of squares

Cf. A000122, A000143, A001650, A046895, A055407, A055414, A057655, A117609, A122510, A175360, A175361, A302860, A341396, A341398, A341399.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+2*add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 8)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..33);  # Alois P. Heinz, Feb 10 2021

nmax = 33; CoefficientList[Series[EllipticTheta[3, 0, x]^8/(1 - x), {x, 0, nmax}], x]
Table[SquaresR[8, n], {n, 0, 33}] // Accumulate

from math import prod
from sympy import factorint
def A341397(n): return (sum((prod((p**(3*(e+1))-(1 if p&1 else 15))//(p**3-1) for p, e in factorint(m).items()) for m in range(1,n+1)))<<4)+1 # Chai Wah Wu, Jun 21 2024

G.f.: theta_3(x)^8 / (1 - x).

a(n^2) = A055414(n).

A111650 2n appears n times (n>0).

Original entry on oeis.org

2, 4, 4, 6, 6, 6, 8, 8, 8, 8, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 14, 14, 14, 14, 14, 14, 14, 16, 16, 16, 16, 16, 16, 16, 16, 18, 18, 18, 18, 18, 18, 18, 18, 18, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 24, 24, 24, 24, 24, 24, 24

Offset: 1

Jonathan Vos Post, Aug 12 2005

Seen as a triangle read by rows: T(n,k) = 2*n, 1<=k<=n. - Reinhard Zumkeller, Mar 18 2011

Reinhard Zumkeller, Rows n = 1..128 of triangle, flattened

Cf. A000194, A111651, A111652.

Cf. A001650, A193832, A002024.

a111650 n k = a111650_tabl !! (n-1) !! (k-1)
a111650_row n = a111650_tabl !! (n-1)
a111650_tabl = iterate (\xs@(x:_) -> map (+ 2) (x:xs)) [2]
a111650_list = concat a111650_tabl
-- Reinhard Zumkeller, Nov 14 2015, Mar 18 2011

Table[Table[2n,n],{n,12}]//Flatten (* Harvey P. Dale, Apr 21 2018 *)

from math import isqrt
def A111650(n): return isqrt(n<<3)+1&-2 # Chai Wah Wu, Jun 06 2025

a(n) = 2*A002024(n). - Chai Wah Wu, Jun 06 2025

A302860 a(n) = [x^n] theta_3(x)^n/(1 - x), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 3, 9, 27, 89, 333, 1341, 5449, 21697, 84663, 327829, 1275739, 5020457, 19964623, 79883141, 320317827, 1284656385, 5152761033, 20686311261, 83182322509, 335110196569, 1352277390001, 5463873556381, 22097867887045, 89441286136465, 362277846495883, 1468465431530457

Offset: 0

Ilya Gutkovskiy, Apr 14 2018

nonn

a(n) = number of integer lattice points inside the n-dimensional hypersphere of radius sqrt(n).

Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

Main diagonal of A122510.

Cf. A000122, A001650, A046895, A057655, A066535, A117609, A302861, A066536, A166952.

Table[SeriesCoefficient[EllipticTheta[3, 0, x]^n/(1 - x), {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, -n, n}]^n, {x, 0, n}], {n, 0, 26}]

a(n) = A122510(n,n).

a(n) ~ c / (sqrt(n) * r^n), where r = 0.241970723224463308846762732757915397312... (= radius of convergence A166952) and c = 0.716940866073606328... - Vaclav Kotesovec, Apr 14 2018

A341396 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_7)^2 <= n.

Original entry on oeis.org

1, 15, 99, 379, 953, 1793, 3081, 5449, 8893, 12435, 16859, 24419, 33659, 42115, 53203, 69779, 88273, 106081, 125821, 153541, 187981, 217437, 248741, 298469, 351277, 394691, 446939, 515259, 589307, 657683, 728803, 828259, 939223, 1029159, 1124023, 1260103

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A008451.

Index entries for sequences related to sums of squares

Cf. A000122, A001650, A008451, A046895, A055406, A055413, A057655, A117609, A122510, A175360, A175361, A302860, A341397, A341398, A341399.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+2*add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 7)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 10 2021

nmax = 35; CoefficientList[Series[EllipticTheta[3, 0, x]^7/(1 - x), {x, 0, nmax}], x]
Table[SquaresR[7, n], {n, 0, 35}] // Accumulate

my(q='q+O('q^(55))); Vec((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^7/(1-q)) \\ Joerg Arndt, Jun 21 2024

G.f.: theta_3(x)^7 / (1 - x).

a(n^2) = A055413(n).

cp's OEIS Frontend

A192455 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n*A(-x)^A001650(n+1), where A001650 is defined by "n appears n times (n odd).".

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A000122 Expansion of Jacobi theta function theta_3(x) = Sum_{m =-oo..oo} x^(m^2) (number of integer solutions to k^2 = n).

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A122510 Array T(d,n) = number of integer lattice points inside the d-dimensional hypersphere of radius sqrt(n), read by ascending antidiagonals.

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A131507 2n+1 appears n+1 times.

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A193832 Irregular triangle read by rows in which row n lists 2n-1 copies of 2n-1 and n copies of 2n, for n >= 1.

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A001670 k appears k times (k even).

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