A000290 -id:A000290 - cp's OEIS frontend

A033453 "INVERT" transform of squares A000290.

1, 5, 18, 63, 221, 776, 2725, 9569, 33602, 117995, 414345, 1454992, 5109273, 17941453, 63002258, 221235399, 776878533, 2728045592, 9579660701, 33639430153, 118126444802, 414806579603, 1456612858961, 5114964721440, 17961439747441, 63072442405845, 221481854849938, 777743974335503, 2731084630047981

Offset: 0

N. J. A. Sloane

Number of compositions of n+1 whose parts equal to q can be of q^2 kinds. Example: a(1)=5 because we have (2),(2'),(2"),(2'") and (1,1). Row sums of A105495. - Emeric Deutsch, Apr 10 2005

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2,1).

Cf. A105495.

read transforms; [seq(n^2,n=1..50)]; INVERT(%);

nn=20;a=(x+x^2)/(1-x)^3;Drop[CoefficientList[Series[1/(1-a),{x,0,nn}],x],1]  (* Geoffrey Critzer, Aug 31 2012*)

Vec((1 + x) / (1 - 4*x + 2*x^2 - x^3) + O(x^30)) \\ Colin Barker, Mar 19 2019

G.f.: (1 + x) / (1 - 4*x + 2*x^2 - x^3).

a(n) = 4*a(n-1) - 2*a(n-2) + a(n-3) for n>2. - Colin Barker, Mar 19 2019

A056992 Digital roots of square numbers A000290.

Original entry on oeis.org

1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9

Offset: 1

Eric W. Weisstein

Cyclic with a period of nine. Note that (7, 9, 4, 1, 9, 1, 4, 9, 7) is palindromic.
a(n) is also the decimal expansion of 499264730/333333333. - Enrique Pérez Herrero, Jul 28 2009
a(n) is also the digital root of A002477(n). - Enrique Pérez Herrero, Dec 20 2009
First comment above by Enrique Pérez Herrero and his formula below together give the following identity: 1+Sum_{n>=2}(1+9*((n^2-1)/9-floor((n^2-1)/9)))/10^(n-1) = 499264730/333333333 = 1.49779419149779419149779419... - Alexander R. Povolotsky, Jun 14 2012

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Square Number.

Cf. A000290, A002477, A010888, A057147, A056991.

a056992 = a010888 . a000290  -- Reinhard Zumkeller, Mar 19 2014

DigitalRoot[n_Integer?NonNegative] := 1 + 9*FractionalPart[(n - 1)/9] A056992[n_]:=DigitalRoot[n^2] (* Enrique Pérez Herrero, Dec 20 2009 *)
Table[FixedPoint[Total[IntegerDigits[#]]&,n^2],{n,90}] (* Zak Seidov, Jun 13 2015 *)
PadRight[{},120,{1,4,9,7,7,9,4,1,9}] (* Harvey P. Dale, Apr 16 2022 *)

a(n) = 1+9*{(n^2-1)/9}, where the symbol {} means fractional part. - Enrique Pérez Herrero, Dec 20 2009
a(n) = 3(1 + cos(2n*Pi/3) + cos(4n*Pi/3)) + mod(3n^4+3n^6+4n^8,9). - Ant King, Oct 07 2009
G.f.: x*(1+4*x+9*x^2+7*x^3+7*x^4+9*x^5+4*x^6+x^7+9*x^8)/((1-x)*(1+x+x^2)*(1+x^3+x^6)). - Ant King, Oct 20 2009
a(n) = A010888(A057147(n)). - Reinhard Zumkeller, Mar 19 2014

A033455 Convolution of nonzero squares A000290 with themselves.

Original entry on oeis.org

1, 8, 34, 104, 259, 560, 1092, 1968, 3333, 5368, 8294, 12376, 17927, 25312, 34952, 47328, 62985, 82536, 106666, 136136, 171787, 214544, 265420, 325520, 396045, 478296, 573678, 683704, 809999, 954304, 1118480, 1304512, 1514513, 1750728, 2015538, 2311464

Offset: 1

N. J. A. Sloane

Total area of all square regions from an n X n grid. E.g., at n = 3, there are nine individual squares, four 2 X 2's and one 3 X 3, total area 9 + 16 + 9 = 34, hence a(3) = 34. - Jon Perry, Jul 29 2003
If X is an n-set and Y and Z disjoint 2-subsets of X then a(n) is equal to the number of 7-subsets of X intersecting both Y and Z. - Milan Janjic, Aug 26 2007
Every fourth term is odd. However, there are no primes in the sequence. - Zak Seidov, Feb 28 2011
-120*a(n) is the real part of (n + n*i)*(n + 2 + n*i)*(n + (n + 2)i)*(n + 2+(n + 2)*i)*(n + 1 + (n + 1)*i), where i = sqrt(-1). - Jon Perry, Feb 05 2014
The previous formula rephrases the factorization of the 5th-order polynomial a(n) = (n+1)*((n+1)^4-1) = (n+1)*A123864(n+1) based on the factorization in A123865. - R. J. Mathar, Feb 08 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Abderrahim Arabi, Hacène Belbachir, Jean-Philippe Dubernard, Plateau Polycubes and Lateral Area, arXiv:1811.05707 [math.CO], 2018. See Column 1 Table 1 p. 8.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 2. - _N. J. A. Sloane_, Mar 23 2014
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000290, A001249, A123865, A219086.

List([1..40], n-> ((n+1)^5 -(n+1))/30); # G. C. Greubel, Jul 05 2019

[(n^5-n)/30: n in [2..41]]; // Vincenzo Librandi, Mar 24 2014

Table[(n^5 - n)/30, {n,2,41}] (* Vladimir Joseph Stephan Orlovsky, Feb 28 2011 *)
CoefficientList[Series[(1+x)^2/(1-x)^6, {x,0,40}], x] (* Vincenzo Librandi, Mar 24 2014 *)

vector(40, n, ((n+1)^5 -n-1)/30) \\ G. C. Greubel, Jul 05 2019

[((n+1)^5 -n-1)/30 for n in (1..40)] # G. C. Greubel, Jul 05 2019

a(n-1) = n*(n^4 - 1)/30 = A061167(n)/30. - Henry Bottomley, Apr 18 2001
G.f.: x*(1+x)^2/(1-x)^6. - Philippe Deléham, Feb 21 2012
a(n) = Sum_{k=1..n+1} k^2*(n+1-k)^2. - Kolosov Petro, Feb 07 2019
E.g.f.: x*(30 +90*x +65*x^2 +15*x^3 +x^4)*exp(x)/30. - G. C. Greubel, Jul 05 2019

More terms from Vincenzo Librandi, Mar 24 2014

A318512 Denominators (in their lowest terms) of the sequence whose Dirichlet convolution with itself yields squares (A000290), or equally A064549.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 8, 1, 2, 1, 2, 1, 4, 1, 2, 4, 2, 1, 4, 1, 2, 1, 8, 1, 16, 1, 2, 2, 2, 1, 4, 1, 4, 4, 2, 1, 4, 1, 2, 2, 2, 1, 16, 1, 2, 1, 8, 4, 4, 1, 2, 8, 4, 1, 4, 1, 2, 2, 2, 1, 16, 1, 4, 2, 2, 1, 4, 2, 2, 2, 2, 1, 16, 1, 4, 2, 2, 1, 128, 1, 2, 2, 4, 1, 4, 1, 2, 8, 4, 1, 4, 1, 4, 1, 2, 4, 16, 4, 2, 2, 2, 1, 8

Offset: 1

Antti Karttunen, Aug 30 2018

These are also denominators (in their lowest terms) for the sequence whose Dirichlet convolution with itself yields A064549, n * Product_{primes p|n} p.
From Antti Karttunen, Sep 02 2018: (Start)
Proof for the above claim:
This sequence is defined as the denominator (given in the lowest terms) of rational valued function r(1) = 1, r(n) = (1/2) * (A000290(n) - Sum_{d|n, d>1, d 1. Define sequence Ay(n) as the denominator of function s(n), with otherwise similar definition, but with A064549 in place of A000290. Let Ay(n) be the denominator of s(n), reduced also into the lowest terms. (Corresponding numerators are A318649 and A318511 respectively. Note that the denominators in both cases must always be of the form 2^k, with k >= 0).
  By applying the distributive property of Dirichlet Convolution [which says that for any completely multiplicative function f, it doesn't matter whether one multiplies the result of convolution afterwards, or whether one multiplies the operands separately before convolution: f(g * g) = (fg) * (fg)], with A000027 in the role of f in both cases, one obtains a pair of equations:
    A318649(n)     A318681(n)     n*A299149(n)
    ----------  =  ----------  =  ------------
    A318512(n)     A299150(n)      A299150(n)
  and
    A318511(n)     A318680(n)     n*A318653(n)
    ----------  =  ----------  =  ------------
       Ay(n)       A299150(n)      A299150(n)
  where the leftmost ratios are reduced into their lowest terms.
  Sequence A318656 gives the 2-adic valuation of ratio A318649(n)/A318512(n), and because there are no even terms neither in A299149 nor in A318653, it also gives the 2-adic valuation of the latter ratio. As A318511/Ay is given in the lowest terms (not both of A318511(n) and Ay(n) can be even at same n), this implies that Ay must indeed be identical to A318512, and furthermore that A318655(n) = A007814(A318649(n)) = A007814(A318511(n)).
(End)

Antti Karttunen, Table of n, a(n) for n = 1..16384
Wikipedia, Dirichlet convolution

Cf. A000290, A064549, A046644, A299150, A318513, A318651, A318653, A318654, A318655, A318656, A318658, A318680, A318681.

Cf. A318649, A318511 (numerators).

f[1] = 1; f[n_] := f[n] = 1/2 (n*Times @@ FactorInteger[n][[All, 1]] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]); Table[Denominator[f[n]], {n, 1, 100}] (* Vaclav Kotesovec, May 10 2025 *)

up_to = 65537;
A064549(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2]++); factorback(f); };
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA064549(n)));
A318512(n) = denominator(v318511_12[n]);

for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-p^2*X)^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 09 2025

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A000290(n) - Sum_{d|n, d>1, d 1. [Equally, one could use A064549 in place of A000290.]
a(n) = 2^A318513(n).
a(n) = A046644(n)/A318651(n).
a(2n-1) = A046644(2n-1) = A318658(2n-1), for all n >= 1.

The main definition changed, more formulas added by Antti Karttunen, Aug 31 2018

A077425 a(n) == 1 (mod 4) (see A016813), but not a square (i.e., not in A000290).

Original entry on oeis.org

5, 13, 17, 21, 29, 33, 37, 41, 45, 53, 57, 61, 65, 69, 73, 77, 85, 89, 93, 97, 101, 105, 109, 113, 117, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 173, 177, 181, 185, 189, 193, 197, 201, 205, 209, 213, 217, 221, 229, 233, 237, 241, 245, 249, 253, 257

Offset: 1

Wolfdieter Lang, Nov 29 2002

The Pell equation x^2 - a(n)*y^2 = +4 has infinitely many (integer) solutions (see A077428 and A078355).
These are the odd numbers in A079896. The even ones are 4*A000037. - Wolfdieter Lang, Sep 15 2015
First differences: 8, 4, 4, 8, 4, 4, 4, 4, 8, 4, 4, 4, 4, 4, 4, 8, ... , only 4's and 8's?. - Paul Curtz, Apr 11 2019
Yes. There are only 4's and 8's. Proof: Only multiples of 4 may appear. The 4's correspond to successive composite in A016813, whereas an 8 corresponds to a square. A greater multiple of 4 would imply to have at least 2 consecutive squares in A016813, which is not possible since 2 consecutive squares cannot have a difference of 4. That sequence of 4's and 8's can be obtained with A010052 (without the 1st term) where the 0's are replaced with 4's and 1's replaced with 8's. - Michel Marcus, Apr 16 2019

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
S. R. Finch, Class number theory [Cached copy, with permission of the author]
A. M. Legendre, Expression les plus simples des formules Ly^2+Myz+Nz^2 où M est impair pour toutes les valeurs de B = M^2-4LN depuis B=5 jusqu'à B=305, Essai sur la Théorie des Nombres An VI, Table II. [_Paul Curtz_, Apr 11 2019]

Intersection of A016813 and A000037.

Cf. A077426, A079896.

A077425 := proc(n::integer) local resul,i ; resul := 5 ; i := 1 ; while i < n do resul := resul+4 ; while issqr(resul) do resul := resul+4 ; od ; i:= i+1 ; od ; RETURN(resul) ; end proc:
seq(A077425(n),n=1..31) ; # R. J. Mathar, Apr 25 2006

Select[Range[5,300,4],!IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Dec 05 2012 *)

[n | n <- vector(100,n,4*n+1), !issquare(n)] \\ Charles R Greathouse IV, Mar 11 2014

list(lim)=my(v=List()); for(s=2,sqrtint((lim\=1)+1), forstep(n=s^2 + if(s%2,4,1), min((s+1)^2-1,lim), 4, listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Nov 04 2021

from operator import sub
from sympy import integer_nthroot
def A077425(n): return n+sub(*integer_nthroot(n,2))<<2|1 # Chai Wah Wu, Oct 01 2024

More terms from Max Alekseyev, Mar 03 2010

A033462 Exponential (or "EXP") transform of squares A000290.

Original entry on oeis.org

1, 1, 5, 22, 125, 836, 6277, 52396, 479593, 4757680, 50738921, 577894604, 6990138685, 89376020800, 1203182518189, 16995248375116, 251135780602193, 3871961504546624, 62141329025501905, 1035979079450355532, 17907209511611407141, 320387246623657457056, 5924125441456047522005

Offset: 0

N. J. A. Sloane

nonn

a(n) is the number of ways to select an ordered pair from each equivalence class in each equivalence relation on {1,2,...,n}. - Geoffrey Critzer, Oct 03 2011

Alois P. Heinz, Table of n, a(n) for n = 0..500

Column k=2 of A279636.

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j^2*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 30 2016

Range[0,20]! CoefficientList[Series[Exp[Exp[x](x+x^2)],{x,0,20}],x]
Table[Sum[BellY[n, k, Range[n]^2], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

N=33;  x='x+O('x^N);
egf=exp(x*(1+x)*exp(x));
Vec(serlaplace(egf))
/* Joerg Arndt, Sep 15 2012 */

E.g.f.: exp(exp(x)*(x+x^2)).

A318649 Numerators of the sequence whose Dirichlet convolution with itself yields squares, A000290.

Original entry on oeis.org

1, 2, 9, 6, 25, 9, 49, 20, 243, 25, 121, 27, 169, 49, 225, 70, 289, 243, 361, 75, 441, 121, 529, 90, 1875, 169, 3645, 147, 841, 225, 961, 252, 1089, 289, 1225, 729, 1369, 361, 1521, 250, 1681, 441, 1849, 363, 6075, 529, 2209, 315, 7203, 1875, 2601, 507, 2809, 3645, 3025, 490, 3249, 841, 3481, 675, 3721, 961, 11907, 924, 4225, 1089

Offset: 1

Antti Karttunen, Aug 31 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A000290, A318512 (denominators).

Cf. also A046643, A299149, A318511, A318651, A318654 (gives the positions of even terms), A318655 (the 2-adic valuation).

up_to = 65537;
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA318649(n) = numerator(v318649_aux[n]);

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-p^2*X)^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 09 2025

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * ((n^2) - Sum_{d|n, d>1, d 1.
a(n) = n*A318512(n)*A299149(n)/A299150(n).
Sum_{k=1..n} A318649(k) / A318512(k) ~ n^3/(3*sqrt(Pi*log(n))) * (1 + (1 - 3*gamma/2) / (6*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 09 2025

A160711 Cyclops squares: squares (A000290) that are also cyclops numbers (A134808).

Original entry on oeis.org

0, 11025, 19044, 21025, 24025, 32041, 38025, 42025, 47089, 51076, 58081, 59049, 65025, 66049, 67081, 75076, 87025, 93025, 1110916, 1140624, 1170724, 1190281, 1240996, 1270129, 1290496, 1340964, 1350244, 1380625, 1420864, 1430416

Offset: 1

Omar E. Pol, Jun 08 2009

			19044 is in the sequence because it is a square (138^2) and is also a cyclops number (odd number of digits, middle digit is the only zero).
11025 is in the sequence because it is a square (105^2) and is also a cyclops number (odd number of digits, middle digit is the only zero). - _Michael B. Porter_, Jul 09 2016

Kenny Lau, Table of n, a(n) for n = 1..20001

Cf. A000290, A134808, A134809, A134148, A138131, A153806, A160712, A160717, A239828.

Select[Range[0, 1200]^2, And[OddQ@ Length@ #, #[[Ceiling[Length[#]/2]]] == 0, Count[#, 0] == 1] &@ IntegerDigits@ # &] (* Michael De Vlieger, Jul 08 2016 *)
cnQ[n_]:=Module[{len=IntegerLength[n]},OddQ[len]&&DigitCount[n,10,0]==1 && IntegerDigits[n][[(len+1)/2]]==0]; Join[{0},Select[Range[1200]^2,cnQ]] (* Harvey P. Dale, Mar 19 2018 *)

A259167 Positive octagonal numbers (A000567) that are squares (A000290) divided by 2.

Original entry on oeis.org

8, 78408, 752875208, 7229107670408, 69413891098384008, 666512175097575576008, 6399849835873029582446408, 61451357457540654953074835208, 590055927907455532986394985222408, 5665716958316030570194709695030728008, 54402213643694597627554069505290065112008

Offset: 1

Colin Barker, Jun 19 2015

Intersection of A000567 and A001105. - Michel Marcus, Jun 20 2015

			8 is in the sequence because 8 is the 2nd octagonal number, and 2*8 is the 4th square.

Colin Barker, Table of n, a(n) for n = 1..251
Index entries for linear recurrences with constant coefficients, signature (9603,-9603,1).

Cf. A000290, A000567, A001105, A259156-A259166.

I:=[8, 78408, 752875208]; [n le 3 select I[n] else 9603*Self(n-1)-9603*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 20 2015

LinearRecurrence[{9603, -9603, 1}, {8, 78408, 752875208}, 20] (* Vincenzo Librandi, Jun 20 2015 *)

Vec(-8*x*(x^2+198*x+1)/((x-1)*(x^2-9602*x+1)) + O(x^20))

G.f.: -8*x*(x^2+198*x+1) / ((x-1)*(x^2-9602*x+1)).

A033464 Logarithmic (or "LOG") transform of squares A000290.

Original entry on oeis.org

1, 3, -1, -26, 29, 756, -1793, -45744, 189513, 4700260, -30515629, -730341600, 6948349069, 159130156836, -2123506814505, -46081244842304, 838034409016721, 17029766318842692, -414549408916313189, -7774211453384941440, 251026027696302116181, 4263756050277024153028

Offset: 0

N. J. A. Sloane

sign

Alois P. Heinz, Table of n, a(n) for n = 0..435

Cf. A000290, A033462.

logtr:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1 else p(n)- add(k *binomial(n,k) *p(n-k) *b(k), k=1..n-1)/n fi end; n->b(n+1) end: a:= logtr(n-> n^2): seq(a(n), n=0..25); # Alois P. Heinz, Sep 14 2008

With[{nn=30},CoefficientList[Series[(Exp[x](1+3x+x^2))/(1+Exp[x]x(1+x)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 03 2019 *)

E.g.f.: exp(x)*(1 + 3*x + x^2)/(1 + exp(x)*x*(1 + x)). - Ilya Gutkovskiy, Mar 06 2018

cp's OEIS Frontend

A033453 "INVERT" transform of squares A000290.

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A056992 Digital roots of square numbers A000290.

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A033455 Convolution of nonzero squares A000290 with themselves.

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A318512 Denominators (in their lowest terms) of the sequence whose Dirichlet convolution with itself yields squares (A000290), or equally A064549.

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A077425 a(n) == 1 (mod 4) (see A016813), but not a square (i.e., not in A000290).

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A033462 Exponential (or "EXP") transform of squares A000290.

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A318649 Numerators of the sequence whose Dirichlet convolution with itself yields squares, A000290.

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