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Search: a001650 -id:a001650
Displaying 1-10 of 18 results found. page 1 2
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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).". +20
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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).
LINKS
Table of n, a(n) for n=0..24.
FORMULA
G.f. satisfies: 1-x = Sum_{n>=1} x^(n^2) * (1-x^(2*n-1)) * A(-x)^(2*n-1).
EXAMPLE
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 +...
PROG
(PARI) {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])}
CROSSREFS
Cf. A193039, A193040, A193050, A001650.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 14 2011
STATUS
approved
A000122 Expansion of Jacobi theta function theta_3(x) = Sum_{m =-oo..oo} x^(m^2) (number of integer solutions to k^2 = n). +10
1506
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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
REFERENCES
Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, Exercise 1, p. 91.
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 and E. M. Wright, Theorem 352, p. 282.
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.
LINKS
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
FORMULA
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) - 1. - Francois Oger, Oct 26 2019
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)
EXAMPLE
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 + ...
MAPLE
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
MATHEMATICA
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 *)
PROG
(PARI) {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 */
(PARI) {a(n) = issquare(n) * 2 -(n==0)}; /* Michael Somos, Jun 17 1999 */
(Magma) Basis( ModularForms( Gamma0(4), 1/2), 100) [1]; /* Michael Somos, Jun 10 2014 */
(Magma) L := Lattice("A", 1); A<q> := ThetaSeries(L, 20); A; /* Michael Somos, Nov 13 2014 */
(Sage)
Q = DiagonalQuadraticForm(ZZ, [1])
Q.representation_number_list(105) # Peter Luschny, Jun 20 2014
(Julia)
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
(Python)
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
CROSSREFS
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, ...).
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane
STATUS
approved
A122510 Array T(d,n) = number of integer lattice points inside the d-dimensional hypersphere of radius sqrt(n), read by ascending antidiagonals. +10
19
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Number of solutions to sum_(i=1,..,d) x[i]^2 <= n, x[i] in Z. T(1,n)=A001650(n+1); T(2,n)=A057655(n); T(3,n)=A117609(n); T(4,n)=A046895(n); T(d,1)=A005408(d); T(d,2)=A058331(d).
LINKS
Table of n, a(n) for n=1..72.
Index entries for sequences related to sums of squares
FORMULA
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.
EXAMPLE
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
MAPLE
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;
MATHEMATICA
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 *)
CROSSREFS
Rows d=1..10 give A001650, A057655, A117609, A046895, A175360, A175361, A341396, A341397, A341398, A341399.
KEYWORD
nonn,tabl
AUTHOR
R. J. Mathar, Oct 29 2006, Oct 31 2006
STATUS
approved
A131507 2n+1 appears n+1 times. +10
11
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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
LINKS
Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
FORMULA
a(n) = 2*floor(sqrt(2n+1)+1/2) - 1. - Ridouane Oudra, Oct 20 2019
EXAMPLE
As a triangle, the sequence starts:
1;
3, 3;
5, 5, 5;
7, 7, 7, 7;
9, 9, 9, 9, 9;
...
MAPLE
seq(2*floor(sqrt(2*n+1)+1/2)-1, n=0..70); # Ridouane Oudra, Oct 20 2019
MATHEMATICA
Table[2 n + 1, {n, 0, 11}, {n + 1}] // Flatten (* Michael De Vlieger, Nov 05 2015 *)
PROG
(Haskell)
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
CROSSREFS
Cf. A000217, A002024, A003056, A005408.
Cf. A001650.
KEYWORD
nonn,tabl
AUTHOR
Paul Curtz, Aug 13 2007
STATUS
approved
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. +10
8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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
LINKS
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.
FORMULA
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
EXAMPLE
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
...
MATHEMATICA
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 *)
PROG
(Haskell)
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
CROSSREFS
Cf. A001318, A080995, A126587, A129194, A221671, A221672.
Cf. A001650, A111650, A260672, A260706.
KEYWORD
nonn,easy,tabf
AUTHOR
Omar E. Pol, Aug 22 2011
EXTENSIONS
Edited by N. J. A. Sloane, Dec 13 2017
STATUS
approved
A001670 k appears k times (k even). +10
7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..10000
FORMULA
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
MAPLE
seq(2*n $ 2*n, n = 1 .. 10); # Robert Israel, Jan 14 2015
MATHEMATICA
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 *)
PROG
(Magma) [2*Round(Sqrt(n)): n in [1..70]]; // Vincenzo Librandi, Jun 23 2011
(PARI) a(n)=round(sqrt(n))<<1 \\ Charles R Greathouse IV, Jun 23 2011
(MATLAB) a = @(n) 2*floor((sqrt(4*n-3)+1)/2); % handle function // Néstor Jofré, Apr 24 2017
(Python)
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
CROSSREFS
Cf. A001650, A000194.
Equals A130829(n) - 1.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
EXTENSIONS
Offset changed from 2 to 1 by Vincenzo Librandi, Jun 23 2011
STATUS
approved
A341397 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_8)^2 <= n. +10
7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Partial sums of A000143.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sums of squares
FORMULA
G.f.: theta_3(x)^8 / (1 - x).
a(n^2) = A055414(n).
MAPLE
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
MATHEMATICA
nmax = 33; CoefficientList[Series[EllipticTheta[3, 0, x]^8/(1 - x), {x, 0, nmax}], x]
Table[SquaresR[8, n], {n, 0, 33}] // Accumulate
PROG
(Python)
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
CROSSREFS
Cf. A000122, A000143, A001650, A046895, A055407, A055414, A057655, A117609, A122510, A175360, A175361, A302860, A341396, A341398, A341399.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 10 2021
STATUS
approved
A111650 2n appears n times (n>0). +10
6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Seen as a triangle read by rows: T(n,k) = 2*n, 1<=k<=n. - Reinhard Zumkeller, Mar 18 2011
LINKS
Reinhard Zumkeller, Rows n = 1..128 of triangle, flattened
MATHEMATICA
Table[Table[2n, n], {n, 12}]//Flatten (* Harvey P. Dale, Apr 21 2018 *)
PROG
(Haskell)
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
CROSSREFS
Cf. A000194, A111651, A111652.
Cf. A001650, A193832.
KEYWORD
easy,nonn,tabl
AUTHOR
Jonathan Vos Post, Aug 12 2005
STATUS
approved
A302860 a(n) = [x^n] theta_3(x)^n/(1 - x), where theta_3() is the Jacobi theta function. +10
6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
a(n) = number of integer lattice points inside the n-dimensional hypersphere of radius sqrt(n).
LINKS
Table of n, a(n) for n=0..26.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares
FORMULA
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
MATHEMATICA
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}]
CROSSREFS
Main diagonal of A122510.
Cf. A000122, A001650, A046895, A057655, A066535, A117609, A302861, A066536, A166952.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 14 2018
STATUS
approved
A341396 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_7)^2 <= n. +10
6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Partial sums of A008451.
LINKS
Table of n, a(n) for n=0..35.
Index entries for sequences related to sums of squares
FORMULA
G.f.: theta_3(x)^7 / (1 - x).
a(n^2) = A055413(n).
MAPLE
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
MATHEMATICA
nmax = 35; CoefficientList[Series[EllipticTheta[3, 0, x]^7/(1 - x), {x, 0, nmax}], x]
Table[SquaresR[7, n], {n, 0, 35}] // Accumulate
PROG
(PARI) 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
CROSSREFS
Cf. A000122, A001650, A008451, A046895, A055406, A055413, A057655, A117609, A122510, A175360, A175361, A302860, A341397, A341398, A341399.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 10 2021
STATUS
approved
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Last modified August 31 23:29 EDT 2024. Contains 375575 sequences. (Running on oeis4.)