A033431 -id:A033431 - cp's OEIS frontend

A000404 Numbers that are the sum of 2 nonzero squares.

2, 5, 8, 10, 13, 17, 18, 20, 25, 26, 29, 32, 34, 37, 40, 41, 45, 50, 52, 53, 58, 61, 65, 68, 72, 73, 74, 80, 82, 85, 89, 90, 97, 98, 100, 101, 104, 106, 109, 113, 116, 117, 122, 125, 128, 130, 136, 137, 145, 146, 148, 149, 153, 157, 160, 162, 164, 169, 170, 173, 178

Offset: 1

N. J. A. Sloane and J. H. Conway

From the formula it is easy to see that if k is in this sequence, then so are all odd powers of k. - T. D. Noe, Jan 13 2009
Also numbers whose cubes are the sum of two nonzero squares. - Joe Namnath and Lawrence Sze
A line perpendicular to y=mx has its first integral y-intercept at a^2+b^2. The remaining ones for that slope are multiples of that primitive value. - Larry J Zimmermann, Aug 19 2010
The primes in this sequence are sequence A002313.
Complement of A018825; A025426(a(n)) > 0; A063725(a(n)) > 0. - Reinhard Zumkeller, Aug 16 2011
If the two squares are not equal, then any power is still in the sequence: if k = x^2 + y^2 with x != y, then k^2 = (x^2-y^2)^2 + (2xy)^2 and k^3 = (x(x^2-3y^2))^2 + (y(3x^2-y^2))^2, etc. - Carmine Suriano, Jul 13 2012
There are never more than 3 consecutive terms that differ by 1. Triples of consecutive terms that differ by 1 occur infinitely many times, for example, 2(k^2 + k)^2, (k^2 - 1)^2 + (k^2 + 2 k)^2, and (k^2 + k - 1)^2 + (k^2 + k + 1)^2 for any integer k > 1. - Ivan Neretin, Mar 16 2017 [Corrected by Jerzy R Borysowicz, Apr 14 2017]
Number of terms less than 10^k, k=1,2,3,...: 3, 34, 308, 2690, 23873, 215907, 1984228, ... - Muniru A Asiru, Feb 01 2018
The squares in this sequence are the squares of the so-called hypotenuse numbers A009003. - M. F. Hasler, Jun 20 2025

			25 = 3^2 + 4^2, therefore 25 is a term. Note that also 25^3 = 15625 = 44^2 + 117^2, therefore 15625 is a term.

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.
GCHQ, The GCHQ Puzzle Book, Penguin, 2016. See page 103.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 75, Theorem 4, with Theorem 2, p. 15.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, p. 219, th. 251, 252.
Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.

T. D. Noe, Table of n, a(n) for n = 1..10000
J. M. De Koninck and V. Ouellet, On the n-th element of a set of positive integers, Annales Univ. Sci. Budapest Sect. Comput. 44 (2015), 153-164. See 2. on p. 162.
Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.
Joshua Harrington, Lenny Jones, and Alicia Lamarche, Representing integers as the sum of two squares in the ring Z_n, arXiv:1404.0187 [math.NT], 2014.
David Rabahy, Google Sheets
G. Xiao, Two squares.
Reinhard Zumkeller, Illustration for A084888 and A000404.
Index entries for sequences related to sums of squares

A001481 gives another version (allowing for zero squares).
Cf. A004431 (2 distinct squares), A063725 (number of representations), A024509 (numbers with multiplicity), A025284, A018825. Also A050803, A050801, A001105, A033431, A084888, A000578, A000290, A057961, A232499, A007692.
Cf. A003325 (analog for cubes), A003336 (analog for 4th powers).
Cf. A009003 (square roots of the squares in this sequence).
Column k=2 of A336725.
Cf. A355237, A355238.

P:=List([1..10^4],i->i^2);;
A000404 := Set(Flat(List(P, i->List(P, j -> i+j)))); # Muniru A Asiru, Feb 01 2018

import Data.List (findIndices)
a000404 n = a000404_list !! (n-1)
a000404_list = findIndices (> 0) a025426_list
-- Reinhard Zumkeller, Aug 16 2011

lst:=[]; for n in [1..178] do f:=Factorization(n); if IsSquare(n) then for m in [1..#f] do d:=f[m]; if d[1] mod 4 eq 1 then Append(~lst, n); break; end if; end for; else t:=0; for m in [1..#f] do d:=f[m]; if d[1] mod 4 eq 3 and d[2] mod 2 eq 1 then t:=1; break; end if; end for; if t eq 0 then Append(~lst, n); end if; end if; end for; lst; // Arkadiusz Wesolowski, Feb 16 2017

nMax:=178: A:={}: for i to floor(sqrt(nMax)) do for j to floor(sqrt(nMax)) do if i^2+j^2 <= nMax then A := `union`(A, {i^2+j^2}) else  end if end do end do: A; # Emeric Deutsch, Jan 02 2017

nMax=1000; n2=Floor[Sqrt[nMax-1]]; Union[Flatten[Table[a^2+b^2, {a,n2}, {b,a,Floor[Sqrt[nMax-a^2]]}]]]
Select[Range@ 200, Length[PowersRepresentations[#, 2, 2] /. {0, } -> Nothing] > 0 &] (* _Michael De Vlieger, Mar 24 2016 *)
Module[{upto=200},Select[Union[Total/@Tuples[Range[Sqrt[upto]]^2,2]],#<= upto&]] (* Harvey P. Dale, Sep 18 2021 *)

is_A000404(n)= for( i=1,#n=factor(n)~%4, n[1,i]==3 && n[2,i]%2 && return); n && ( vecmin(n[1,])==1 || (n[1,1]==2 && n[2,1]%2)) \\ M. F. Hasler, Feb 07 2009

list(lim)=my(v=List(),x2); lim\=1; for(x=1,sqrtint(lim-1), x2=x^2; for(y=1,sqrtint(lim-x2), listput(v,x2+y^2))); Set(v) \\ Charles R Greathouse IV, Apr 30 2016

from itertools import count, islice
from sympy import factorint
def A000404_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        c = False
        for p in (f:=factorint(n)):
            if (q:= p & 3)==3 and f[p]&1:
                break
            elif q == 1:
                c = True
        else:
            if c or f.get(2,0)&1:
                yield n
A000404_list = list(islice(A000404_gen(),30)) # Chai Wah Wu, Jul 01 2022

Let k = 2^t * p_1^a_1 * p_2^a_2 * ... * p_r^a_r * q_1^b_1 * q_2^b_2 * ... * q_s^b_s with t >= 0, a_i >= 0 for i=1..r, where p_i == 1 (mod 4) for i=1..r and q_j == -1 (mod 4) for j=1..s. Then k is a term iff 1) b_j == 0 (mod 2) for j=1..s and 2) r > 0 or t == 1 (mod 2) (or both).
From Charles R Greathouse IV, Nov 18 2022: (Start)
a(n) ~ k*n*sqrt(log n), where k = 1.3085... = 1/A064533.
There are B(x) = (x/sqrt(log x)) * (K + B2/log x + O(1/log^2 x)) terms of this sequence up to x, where K = A064533 and B2 = A227158. (End)

Edited by Ralf Stephan, Nov 15 2004
Typo in formula corrected by M. F. Hasler, Feb 07 2009
Erroneous Mathematica program fixed by T. D. Noe, Aug 07 2009
PARI code fixed for versions > 2.5 by M. F. Hasler, Jan 01 2013

A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.

Original entry on oeis.org

1, 4, 2, 10, 7, 3, 20, 16, 10, 4, 35, 30, 22, 13, 5, 56, 50, 40, 28, 16, 6, 84, 77, 65, 50, 34, 19, 7, 120, 112, 98, 80, 60, 40, 22, 8, 165, 156, 140, 119, 95, 70, 46, 25, 9, 220, 210, 192, 168, 140, 110, 80, 52, 28, 10, 286, 275, 255, 228, 196, 161, 125, 90

Offset: 1

Clark Kimberling, Jun 14 2012

Principal diagonal: A002412.
Antidiagonal sums: A002415.
Row 1:  (1,2,3,...)**(1,2,3,...) = A000292.
Row 2:  (1,2,3,...)**(2,3,4,...) = A005581.
Row 3:  (1,2,3,...)**(3,4,5,...) = A006503.
Row 4:  (1,2,3,...)**(4,5,6,...) = A060488.
Row 5:  (1,2,3,...)**(5,6,7,...) = A096941.
Row 6:  (1,2,3,...)**(6,7,8,...) = A096957.
...
In general, the convolution of two infinite sequences is defined from the convolution of two n-tuples:  let X(n) = (x(1),...,x(n)) and Y(n)=(y(1),...,y(n)); then X(n)**Y(n) = x(1)*y(n)+x(2)*y(n-1)+...+x(n)*y(1); this sum is the n-th term in the convolution of infinite sequences:(x(1),...,x(n),...)**(y(1),...,y(n),...), for all n>=1.
...
In the following guide to related arrays and sequences, row n of each array T(n,k) is the convolution b**c of the sequences b(h) and c(h+n-1). The principal diagonal is given by T(n,n) and the n-th antidiagonal sum by S(n). In some cases, T(n,n) or S(n) differs in offset from the listed sequence.
b(h)........ c(h)........ T(n,k) .. T(n,n) .. S(n)
h .......... h .......... A213500 . A002412 . A002415
h .......... h^2 ........ A212891 . A213436 . A024166
h^2 ........ h .......... A213503 . A117066 . A033455
h^2 ........ h^2 ........ A213505 . A213546 . A213547
h .......... h*(h+1)/2 .. A213548 . A213549 . A051836
h*(h+1)/2 .. h .......... A213550 . A002418 . A005585
h*(h+1)/2 .. h*(h+1)/2 .. A213551 . A213552 . A051923
h .......... h^3 ........ A213553 . A213554 . A101089
h^3 ........ h .......... A213555 . A213556 . A213547
h^3 ........ h^3 ........ A213558 . A213559 . A213560
h^2 ........ h*(h+1)/2 .. A213561 . A213562 . A213563
h*(h+1)/2 .. h^2 ........ A213564 . A213565 . A101094
2^(h-1) .... h .......... A213568 . A213569 . A047520
2^(h-1) .... h^2 ........ A213573 . A213574 . A213575
h .......... Fibo(h) .... A213576 . A213577 . A213578
Fibo(h) .... h .......... A213579 . A213580 . A053808
Fibo(h) .... Fibo(h) .... A067418 . A027991 . A067988
Fibo(h+1) .. h .......... A213584 . A213585 . A213586
Fibo(n+1) .. Fibo(h+1) .. A213587 . A213588 . A213589
h^2 ........ Fibo(h) .... A213590 . A213504 . A213557
Fibo(h) .... h^2 ........ A213566 . A213567 . A213570
h .......... -1+2^h ..... A213571 . A213572 . A213581
-1+2^h ..... h .......... A213582 . A213583 . A156928
-1+2^h ..... -1+2^h ..... A213747 . A213748 . A213749
h .......... 2*h-1 ...... A213750 . A007585 . A002417
2*h-1 ...... h .......... A213751 . A051662 . A006325
2*h-1 ...... 2*h-1 ...... A213752 . A100157 . A071238
2*h-1 ...... -1+2^h ..... A213753 . A213754 . A213755
-1+2^h ..... 2*h-1 ...... A213756 . A213757 . A213758
2^(n-1) .... 2*h-1 ...... A213762 . A213763 . A213764
2*h-1 ...... Fibo(h) .... A213765 . A213766 . A213767
Fibo(h) .... 2*h-1 ...... A213768 . A213769 . A213770
Fibo(h+1) .. 2*h-1 ...... A213774 . A213775 . A213776
Fibo(h) .... Fibo(h+1) .. A213777 . A001870 . A152881
h .......... 1+[h/2] .... A213778 . A213779 . A213780
1+[h/2] .... h .......... A213781 . A213782 . A005712
1+[h/2] .... [(h+1)/2] .. A213783 . A213759 . A213760
h .......... 3*h-2 ...... A213761 . A172073 . A002419
3*h-2 ...... h .......... A213771 . A213772 . A132117
3*h-2 ...... 3*h-2 ...... A213773 . A214092 . A213818
h .......... 3*h-1 ...... A213819 . A213820 . A153978
3*h-1 ...... h .......... A213821 . A033431 . A176060
3*h-1 ...... 3*h-1 ...... A213822 . A213823 . A213824
3*h-1 ...... 3*h-2 ...... A213825 . A213826 . A213827
3*h-2 ...... 3*h-1 ...... A213828 . A213829 . A213830
2*h-1 ...... 3*h-2 ...... A213831 . A213832 . A212560
3*h-2 ...... 2*h-1 ...... A213833 . A130748 . A213834
h .......... 4*h-3 ...... A213835 . A172078 . A051797
4*h-3 ...... h .......... A213836 . A213837 . A071238
4*h-3 ...... 2*h-1 ...... A213838 . A213839 . A213840
2*h-1 ...... 4*h-3 ...... A213841 . A213842 . A213843
2*h-1 ...... 4*h-1 ...... A213844 . A213845 . A213846
4*h-1 ...... 2*h-1 ...... A213847 . A213848 . A180324
[(h+1)/2] .. [(h+1)/2] .. A213849 . A049778 . A213850
h .......... C(2*h-2,h-1) A213853
...
Suppose that u = (u(n)) and v = (v(n)) are sequences having generating functions U(x) and V(x), respectively. Then the convolution u**v has generating function U(x)*V(x). Accordingly, if u and v are homogeneous linear recurrence sequences, then every row of the convolution array T satisfies the same homogeneous linear recurrence equation, which can be easily obtained from the denominator of U(x)*V(x). Also, every column of T has the same homogeneous linear recurrence as v.

			Northwest corner (the array is read by southwest falling antidiagonals):
  1,  4, 10, 20,  35,  56,  84, ...
  2,  7, 16, 30,  50,  77, 112, ...
  3, 10, 22, 40,  65,  98, 140, ...
  4, 13, 28, 50,  80, 119, 168, ...
  5, 16, 34, 60,  95, 140, 196, ...
  6, 19, 40, 70, 110, 161, 224, ...
T(6,1) = (1)**(6) = 6;
T(6,2) = (1,2)**(6,7) = 1*7+2*6 = 19;
T(6,3) = (1,2,3)**(6,7,8) = 1*8+2*7+3*6 = 40.

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A000027.

b[n_] := n; c[n_] := n
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213500 *)

t(n,k) = sum(i=0, k - 1, (k - i) * (n + i));
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(t(k,n - k + 1),", ");); print(););};
tabl(12) \\ Indranil Ghosh, Mar 26 2017

def t(n, k): return sum((k - i) * (n + i) for i in range(k))
for n in range(1, 13):
    print([t(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 26 2017

T(n,k) = 4*T(n,k-1) - 6*T(n,k-2) + 4*T(n,k-3) - T(n,k-4).
T(n,k) = 2*T(n-1,k) - T(n-2,k).
G.f. for row n:  x*(n - (n - 1)*x)/(1 - x)^4.

A117642 a(n) = 3*n^3.

Original entry on oeis.org

0, 3, 24, 81, 192, 375, 648, 1029, 1536, 2187, 3000, 3993, 5184, 6591, 8232, 10125, 12288, 14739, 17496, 20577, 24000, 27783, 31944, 36501, 41472, 46875, 52728, 59049, 65856, 73167, 81000, 89373, 98304, 107811, 117912, 128625, 139968, 151959, 164616

Offset: 0

Roger L. Bagula, Apr 10 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578, A002117, A033431.

[3*n^3: n in [0..40]]; // Vincenzo Librandi, Jun 26 2011

seq(3*n^3, n=0..38); # Nathaniel Johnston, Jun 26 2011

3*Range[0, 35]^3 (* Alonso del Arte, Oct 04 2013 *)

a(n)=3*n^3 \\ Charles R Greathouse IV, Oct 12 2017

a(n+1) = a(n) + 9*n^2 + 9*n + 3 with a(0) = 0. - Jean-Bernard François, Oct 04 2013
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, May 27 2021
From Amiram Eldar, Jan 10 2023: (Start)
Sum_{n>=1} 1/a(n) = zeta(3)/3 = A386403.
Sum_{n>=1} (-1)^(n+1)/a(n) = zeta(3)/4. (End)
a(n) = n*(2*A000217(n) + A000217(2*n-1)). - Charlie Marion, Mar 01 2024

Edited by N. J. A. Sloane, Apr 30 2006

A098547 a(n) = n^3 + n^2 + 1.

Original entry on oeis.org

1, 3, 13, 37, 81, 151, 253, 393, 577, 811, 1101, 1453, 1873, 2367, 2941, 3601, 4353, 5203, 6157, 7221, 8401, 9703, 11133, 12697, 14401, 16251, 18253, 20413, 22737, 25231, 27901, 30753, 33793, 37027, 40461, 44101, 47953, 52023, 56317, 60841, 65601, 70603, 75853

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Oct 26 2004

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578, A001093, A011379, A027444, A033431, A033562, A034262, A053698, A061317, A066023, A071568.

Cf. A000217, A081423.

[(n^3+n^2+1): n in [1..60]]; // Vincenzo Librandi, Apr 06 2011

with(combinat): seq(fibonacci(3,n)+n^3, n=0..40); # Zerinvary Lajos, May 25 2008

Table[n^3+n^2+1,{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Mar 09 2011 *)

Vec(-(x^3-7*x^2+x-1)/(x-1)^4 + O(x^100)) \\ Colin Barker, Aug 29 2014

From Colin Barker, Aug 29 2014: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: (1 - x + 7*x^2 - x^3)/(1-x)^4. (End)
a(n) = A081423(n) + A000217(n-1). - Bruce J. Nicholson, Jan 06 2019
E.g.f.: exp(x)*(1 + 2*x + 4*x^2 + x^3). - Elmo R. Oliveira, Apr 20 2025

A244725 a(n) = 5*n^3.

Original entry on oeis.org

0, 5, 40, 135, 320, 625, 1080, 1715, 2560, 3645, 5000, 6655, 8640, 10985, 13720, 16875, 20480, 24565, 29160, 34295, 40000, 46305, 53240, 60835, 69120, 78125, 87880, 98415, 109760, 121945, 135000, 148955, 163840, 179685, 196520, 214375, 233280, 253265

Offset: 0

Vincenzo Librandi, Jul 05 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. similar sequences of the type k*n^3: A000578 (k=1), A033431 (k=2), A117642 (k=3), A033430 (k=4), this sequence (k=5), A244726 (k=6), A244727 (k=7), A016743 (k=8), A244728 (k=9), A244729 (k=10), A016767 (k=27), A016803 (k=64), A016851 (k=125), A016911 (k=216), A016983 (k=343), A017067 (k=512), A017163 (k=729), A017271 (k=1000), A017391 (k=1331), A017523 (k=1728).

Magma
```
[5*n^3: n in [0..40]];
```

I:=[0,5,40,135]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]];

Table[5 n^3, {n, 0, 40}] (* or *) CoefficientList[Series[5 x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 40}], x]

a(n)=5*n^3 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 5*x*(1 + 4*x + x^2)/(1 - x)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3.

A131474 a(n) = ceiling(n/2)*ceiling(n^2/2).

Original entry on oeis.org

0, 1, 2, 10, 16, 39, 54, 100, 128, 205, 250, 366, 432, 595, 686, 904, 1024, 1305, 1458, 1810, 2000, 2431, 2662, 3180, 3456, 4069, 4394, 5110, 5488, 6315, 6750, 7696, 8192, 9265, 9826, 11034, 11664, 13015, 13718, 15220, 16000, 17661

Offset: 0

Mohammad K. Azarian, Jul 27 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A000982, A004526, A008619, A033431, A059722.

[Ceiling(n/2)*Ceiling(n^2/2) : n in [0..50]]; // Vincenzo Librandi, Oct 01 2011

Table[Ceiling[n/2]*Ceiling[n^2/2], {n, 0, 50}] (* Paolo Xausa, Aug 21 2024 *)

a(n)=ceil(n/2)*ceil(n^2/2) \\ Charles R Greathouse IV, Jun 11 2015

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: x*(1 + x + 5*x^2 + 2*x^4 + 3*x^3)/((1+x)^3*(x-1)^4).
a(n) = A000982(n)*A004526(n+1). (End)
a(2*n) = A033431(n). - Bernard Schott, Mar 29 2020
a(2*n-1) = A059722(n). - Michel Marcus, Mar 29 2020

A099721 a(n) = n^2(2n+1).

Original entry on oeis.org

0, 3, 20, 63, 144, 275, 468, 735, 1088, 1539, 2100, 2783, 3600, 4563, 5684, 6975, 8448, 10115, 11988, 14079, 16400, 18963, 21780, 24863, 28224, 31875, 35828, 40095, 44688, 49619, 54900, 60543, 66560, 72963, 79764, 86975, 94608, 102675, 111188, 120159, 129600

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Nov 07 2004

For a right triangle with sides of lengths 8*n^3 + 12*n^2 + 8*n + 2, 4*n^4 + 8*n^3 + 4*n^2, and 4*n^4 + 8*n^3 + 12*n^2 + 8*n + 2, dividing the area by the perimeter gives a(n). - J. M. Bergot, Jul 30 2013
This sequence is the difference between the centered icosahedral (or cuboctahedral) numbers (A005902(n)) and the centered octagonal pyramidal numbers (A000447(n+1)). - Peter M. Chema, Jan 09 2016
a(n) is the sum of the integers in the closed interval (n-1)*n to n*(n+1). - J. M. Bergot, Apr 19 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Row n=3 of A229079.
Cf. A000578, A001093, A011379, A015237, A027444, A033431, A033562, A034262, A053698, A061317, A066023, A071568, A098547.
Cf. A000290, A000447, A002378, A005408, A005902, A024196.

[n^2*(2*n+1): n in [0..50]]; // Vincenzo Librandi, May 01 2011

A099721 := proc(n) n^2*(2*n+1) ; end proc:
seq(A099721(n),n=0..10) ;

a[n_]:=2*n^3+n^2; (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)
LinearRecurrence[{4,-6,4,-1},{0,3,20,63},40] (* Harvey P. Dale, Aug 19 2022 *)

a(n) = ceil(sum(i=n^2-(n-1), n^2+(n-1), if(!issquare(4*i+1), (2*i+1+sqrt(4*i+1))/2, 0))); \\ Michel Marcus, Nov 14 2014, after Richard R. Forberg

G.f.: x*(3 + 8*x + x^2)/(x-1)^4.
a(n) = A024196(n) - A024196(n-1). - Philippe Deléham, May 07 2012
a(n) = ceiling(Sum_{i=n^2-(n-1)..n^2+(n-1)} s(i)), for n > 0 and integer i, where s(i) are the real solutions to x = i + sqrt(x), and the summation range excludes the integer solutions which occur where i is an oblong number (A002378). The fractional portion of the summation converges to 2/3 for large n. If s(i) is replaced with i, then the summation equals n^2*(2*n-1) = A015237. - Richard R. Forberg, Oct 15 2014
a(n) = A005902(n) - A000447(n+1). - Peter M. Chema, Jan 09 2016
From Amiram Eldar, May 17 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/6 + 4*log(2) - 4.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 - Pi - 2*log(2) + 4. (End)
From Elmo R. Oliveira, Aug 08 2025: (Start)
E.g.f.: x*(1 + 2*x)*(3 + x)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = A000290(n)*A005408(n). (End)

A332821 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 1 (mod 3).

Original entry on oeis.org

2, 5, 9, 11, 12, 16, 17, 21, 23, 28, 30, 31, 39, 40, 41, 47, 49, 52, 54, 57, 59, 66, 67, 70, 72, 73, 75, 76, 83, 87, 88, 91, 96, 97, 100, 102, 103, 109, 111, 116, 126, 127, 128, 129, 130, 133, 135, 136, 137, 138, 148, 149, 154, 157, 159, 165, 167, 168, 169, 172, 175, 179, 180, 183, 184, 186, 190, 191, 197, 203, 211, 212

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

nonn

The positive integers are partitioned between A332820, this sequence and A332822.
For each prime p, the terms include exactly one of p and p^2. The primes alternate between this sequence and A332822. This sequence has the primes with odd indexes, those in A031368.
The terms are the even numbers in A332822 halved. The terms are also the numbers m such that 5m is in A332822, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332820, and so on for alternate primes: 7, 13, 19 etc.
The numbers that are half of the even terms of this sequence are in A332820, which consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332822, which consists exactly of those numbers. For larger primes, an alternating pattern applies as described in the previous paragraph.
If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, the resulting number is in A332822, which consists entirely of those numbers.
The product of any 2 terms of this sequence is in A332822, the product of any 3 terms is in A332820, and the product of a term of A332820 and a term of this sequence is in this sequence. So if a number k is present, k^2 is in A332822, k^3 is in A332820, and k^4 is in this sequence.
If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.

Positions of ones in A332823; equivalently, numbers in row 3k+1 of A277905 for some k >= 0.
Cf. A048675, A332820, A332822.
Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
Subsequences: intersection of A026478 and A066208, A031368 (prime terms), A033431\{0}, A052934\{1}, A069486, A099800, A167747\{1}, A244725\{0}, A244728\{0}, A338911 (semiprime terms).

Select[Range@ 212, Mod[Total@ #, 3] == 1 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]] &] (* Michael De Vlieger, Mar 15 2020 *)

isA332821(n) =  { my(f = factor(n)); (1==((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3)); };

{a(n) : n >= 1} = {2 * A332820(k) : k >= 1} U {A003961(A332822(k)) : k >= 1}.

{a(n) : n >= 1} = {A332822(k)^2 : k >= 1} U {A331590(2, A332820(k)) : k >= 1}.

A323723 a(n) = (-2 - (-1)^n(-2 + n) + n + 2n^3)/4.

Original entry on oeis.org

0, 0, 4, 14, 32, 64, 108, 174, 256, 368, 500, 670, 864, 1104, 1372, 1694, 2048, 2464, 2916, 3438, 4000, 4640, 5324, 6094, 6912, 7824, 8788, 9854, 10976, 12208, 13500, 14910, 16384, 17984, 19652, 21454, 23328, 25344, 27436, 29678, 32000, 34480, 37044, 39774

Offset: 0

Stefano Spezia, Jan 25 2019

For n > 1, a(n) is the subdiagonal sum of the matrix M(n) whose permanent is A322277(n).

All the terms of this sequence are even numbers (A005843).

Stefano Spezia, Table of n, a(n) for n = 0..10000
Christian Krause, LODA, an assembly language, a computational model and a tool for mining integer sequences
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A000982, A004526, A005843, A033431, A033999, A317614, A322277, A323724, A325655.

Flat(List([0..50], n -> (-2-(-1)^n*(-2+n)+n+2*n^3)/4));

[(-2-(-1)^n*(-2+n)+n+2*n^3)/4: n in [0..50]];

a:=n->(-2 - (-1)^n*(-2 + n) + n + 2*n^3)/4: seq(a(n), n=0..50);

a[n_]:=(6 + n + n^3 + 12 Floor[1/2 (-3 + n)] + 4 Floor[1/2 (-3 + n)]^2 - 2 (1 + n) Floor[1/2 (-1 + n)])/2; Array[a,50,0]

makelist((-2-(-1)^n*(-2+n)+n+2*n^3)/4, n, 0, 50);

PARI
```
a(n) = (-2-(-1)^n*(-2+n)+n+2*n^3)/4;
```

[(-2-(-1)**n*(-2+n)+n+2*n**3)/4 for n in range(50)]

O.g.f.: 2*x^2*(2 + 3*x + x^3)/((1 - x)^4*(1 + x)^2).
E.g.f.: (1/4)*exp(-x)*(2 + x)*(1 + exp(2*x)*(-1 + 2*x + 2* x^2)).
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n > 5.
a(n) = (6 + n + n^3 + 12*floor((n - 3)/2) + 4*floor((n - 3)/2)^2 - 2*(1 + n)*floor((n - 1)/2))/2.
a(n) = (-2 - A033999(n)*(-2 + n) + n + A033431(n))/4.
a(n) = n^3/2 for even n; a(n) = (n - 1)*(n^2 + n + 2)/2 otherwise. - Bruno Berselli, Feb 06 2019
a(n) = 2*A004526(n*A000982(n)). [Found by Christian Krause's LODA miner] - Stefano Spezia, Dec 12 2021

A185869 (Odd,even)-polka dot array in the natural number array A000027; read by antidiagonals.

Original entry on oeis.org

2, 7, 9, 16, 18, 20, 29, 31, 33, 35, 46, 48, 50, 52, 54, 67, 69, 71, 73, 75, 77, 92, 94, 96, 98, 100, 102, 104, 121, 123, 125, 127, 129, 131, 133, 135, 154, 156, 158, 160, 162, 164, 166, 168, 170, 191, 193, 195, 197, 199, 201, 203, 205, 207, 209, 232, 234, 236, 238, 240, 242, 244, 246, 248, 250, 252, 277, 279, 281, 283, 285, 287, 289, 291, 293, 295, 297, 299, 326, 328, 330, 332, 334, 336, 338, 340, 342, 344, 346, 348, 350, 379, 381, 383, 385, 387, 389, 391, 393, 395, 397, 399, 401, 403, 405

Offset: 1

Clark Kimberling, Feb 05 2011

This is the second of four polka dot arrays; see A185868.
row 1: A130883;
row 2: A100037;
row 3: A100038;
row 4: A100039;
col 1: A014107;
col 2: A033537;
col 3: A100040;
col 4: A100041;
diag (2,18,...): A077591;
diag (7,31,...): A157914;
diag (16,48,...): A035008;
diag (29,69,...): A108928;
antidiagonal sums: A033431;
antidiagonal sums: 2*(1^3, 2^3, 3^3, 4^3,...) = 2*A000578.
A060432(n) + n is odd if and only if n is in this sequence. - Peter Kagey, Feb 03 2016

			Northwest corner:
  2....7....16...29...46
  9....18...31...48...69
  20...33...50...71...96
  35...52...73...98...127

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A000027 (as an array), A060432, A185868, A185870, A185871.

a185869 n = a185869_list !! (n - 1)
a185869_list = scanl (+) 2 $ a' 1
  where  a' n = 2 * n + 3 : replicate n 2 ++ a' (n + 1)
-- Peter Kagey, Sep 02 2016

f[n_,k_]:=2n-1+(2n+2k-3)(n+k-1);
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

from math import isqrt, comb
def A185869(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    x = n-comb(a,2)
    y = a-x+1
    return y*((y+(c:=x<<1)<<1)-5)+x*(c-3)+2 # Chai Wah Wu, Jun 18 2025

T(n,k) = 2n-1+(n+k-1)*(2n+2k-3), k>=1, n>=1.

cp's OEIS Frontend

A000404 Numbers that are the sum of 2 nonzero squares.

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A213500 Rectangular array T(n,k): (row n) = b**c, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and ** = convolution.

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A117642 a(n) = 3*n^3.

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A098547 a(n) = n^3 + n^2 + 1.

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A244725 a(n) = 5*n^3.

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A131474 a(n) = ceiling(n/2)*ceiling(n^2/2).

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A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.

A099721 a(n) = n^2(2n+1).

A323723 a(n) = (-2 - (-1)^n(-2 + n) + n + 2n^3)/4.