A000415 -id:A000415 - cp's OEIS frontend

A254371 Sum of cubes of the first n even numbers (A016743).

0, 8, 72, 288, 800, 1800, 3528, 6272, 10368, 16200, 24200, 34848, 48672, 66248, 88200, 115200, 147968, 187272, 233928, 288800, 352800, 426888, 512072, 609408, 720000, 845000, 985608, 1143072, 1318688, 1513800, 1729800, 1968128, 2230272, 2517768, 2832200, 3175200

Offset: 0

Luciano Ancora, Mar 16 2015

Property: for n >= 2, each (a(n), a(n)+1, a(n)+2) is a triple of consecutive terms that are the sum of two nonzero squares; precisely: a(n) = (n*(n + 1))^2 + (n*(n + 1))^2, a(n)+1 = (n^2+2n)^2 + (n^2-1)^2 and a(n)+2 = (n^2+n+1)^2 + (n^2+n-1)^2 (see Diophante link). - Bernard Schott, Oct 05 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Luciano Ancora, The Square Pyramidal Number and other figurate numbers, ch. 3.
Diophante, Une miniature avec trois entiers consécutifs (in French).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000537 (sum of first n cubes); A002593 (sum of first n odd cubes).
Cf. A060300 (2*a(n)).
First bisection of A105636; second bisection of A212892.
Cf. A000217, A000415, A002378, A035287, A163102.

List([0..35],n->2*(n*(n+1))^2); # Muniru A Asiru, Oct 24 2018

[2*n^2*(n+1)^2: n in [0..40]]; // Bruno Berselli, Mar 23 2015

A254371:=n->2*n^2*(n + 1)^2: seq(A254371(n), n=0..50); # Wesley Ivan Hurt, Apr 28 2017

Table[2 n^2 (n+1)^2, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 8, 72, 288, 800}, 40]
Accumulate[Range[0,80,2]^3] (* Harvey P. Dale, Jun 26 2017 *)

a(n)=sum(i=0, n, 8*i^3); \\ Michael B. Porter, Mar 16 2015

G.f.: 8*x*(1 + 4*x + x^2)/(1 - x)^5.
a(n) = 2*n^2*(n + 1)^2.
a(n) = 2*A035287(n+1) = 2*A002378(n)^2 = 8*A000217(n)^2. - Bruce J. Nicholson, Apr 23 2017
a(n) = 8*A000537(n). - Michel Marcus, Apr 23 2017
From Amiram Eldar, Aug 25 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/6 - 3/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3/2 - 2*log(2). (End)
From Elmo R. Oliveira, Aug 14 2025: (Start)
E.g.f.: 2*x*(2 + x)*(2 + 6*x + x^2)*exp(x).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 4*A163102(n) = A060300(n)/2. (End)

A073412 Lesser of three consecutive nonsquare integers each of which is the sum of two squares.

Original entry on oeis.org

72, 232, 520, 584, 800, 808, 1096, 1152, 1312, 1664, 1744, 1800, 1872, 1960, 2248, 2312, 2384, 2592, 2824, 3328, 3392, 3528, 4112, 4176, 4328, 5120, 5408, 5904, 6056, 6120, 6272, 6352, 6408, 6568, 6920, 8080, 8144, 8296, 8352, 8584, 8712, 9160, 9376

Offset: 1

Jason Earls, Aug 23 2002

nonn

a(n) == 0 mod 8. - Zak Seidov, Jan 26 2013
Is this sequence the same as A064715? - Zak Seidov, Jan 26 2013
This sequence is distinct from A064715 since it allows numbers equal to twice a square, like 72, 1152, 2592, 3528, etc. - Giovanni Resta, Jan 29 2013
This sequence lists lesser of three consecutive nonsquare integers each of which is the sum of two squares. So this sequence is a subsequence of A064716. - Altug Alkan, Jul 07 2016

			232 is here since 232 = 6^2 + 14^2; 233 = 8^2 + 13^2; 234 = 3^2 + 15^2 and 232, 233, 234 are all nonsquares.
288 is not a term because 288 = 12^2 + 12^2, 289 = 8^2 + 15^2, 290 = 1^2 + 17^2 but 289 is also square.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000404,A000415, A001481, A064716.

is415:= proc(n) local F;
  if issqr(n) then return false fi;
  F:= select(t -> t[1] mod 4 = 3, ifactors(n)[2]);
  andmap(t -> t[2]::even, F);
end proc:
Q:= select(is415, [seq(seq(8*i+j,j=0..2),i=1..2000)]):
Q[select(t -> Q[t+2]-Q[t]=2, [$1..nops(Q)-2])]; # Robert Israel, Mar 05 2018

nsQ[x_] := !IntegerQ[Sqrt[x]];
prQ[x_] := With[{pr = PowersRepresentations[x, 2, 2]}, pr != {} && AllTrue[pr[[1]], IntegerQ]];
selQ[x_] := nsQ[x] && nsQ[x+1] && nsQ[x+2] && prQ[x] && prQ[x+1] && prQ[x+2];
Select[8 Range[10000], selQ] (* Jean-François Alcover, Jun 11 2020 *)

isA001481(n) = my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]%2 && f[i, 1]%4==3, return(0))); 1;
isok(n) = isA001481(n) && isA001481(n+1) && isA001481(n+2) && !issquare(n) && !issquare(n+1);
lista(nn) = for(n=1, nn, if(isok(8*n), print1(8*n, ", "))); \\ Altug Alkan, Jul 07 2016

Edited by Robert Israel, Mar 05 2018

A172001 Nonsquare positive integers n such that Pell equation y^2 - n*x^2 = -1 has rational solutions but the norm of fundamental unit of quadratic field Q(sqrt(n)) is 1.

Original entry on oeis.org

34, 136, 146, 178, 194, 205, 221, 305, 306, 377, 386, 410, 466, 482, 505, 514, 544, 545, 562, 584, 674, 689, 706, 712, 745, 776, 793, 802, 820, 850, 866, 884, 890, 898, 905, 1154, 1186, 1202, 1205, 1220, 1224, 1234, 1282, 1314, 1345, 1346, 1394, 1405, 1469

Offset: 1

Max Alekseyev, Jan 21 2010

nonn

If the fundamental unit y0 + x0*sqrt(n) of Q(sqrt(n)) has norm -1, then (x0,y0) represents a rational solution to Pell equation y^2 - n*x^2 = -1. For n in this sequence, rational solutions exist but not delivered by the fundamental unit.

Set difference of A000415 and its subsequence A172000.
Set difference of A087643 and its subsequence A022544.
Squarefree terms form A031398.
Odd terms form A249052.
Cf. A031399, A137351.

A positive integer n is in this sequence iff its squarefree core A007913(n) belongs to A031398.

Edited by Max Alekseyev, Mar 09 2010

A263765 Minimum number of squares necessary to write n as a sum or difference of squares.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 1, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 1, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2

Offset: 0

Jean-Christophe Hervé, Oct 25 2015

nonn

This sequence is equivalent to A002828 (least number of squares that add up to n) for sums and differences of squares. Here the possible forms include not only sums of squares, but also differences like x^2 - y^2 or x^2 + y^2 - z^2.
a(n) <= A002828(n) which is <= 4 (Lagrange's "Four Squares theorem"). In fact, a(n) <= 3: numbers of the form 4k, 4k+1 or 4k+3 are equal to the difference of two squares, therefore a(n) <= 2 in this case, and a(4k+2) <= 3 because 4k+2 = 4k+1+1^2. More precisely, a(4k) = 1 or 2; a(4k+1) = 1 or 2; a(4k+2) = 2 or 3; a(4k+3) = 2.
If A002828(n) = 4, a(n) = 2 (see A004215); if A002828(n) = 3, a(n) = 2 or 3: this shows that the form x^2 + y^2 - z^2 is never necessary to write an integer with the minimum number of squares; and of course, if A002828 = 1 or 2, a(n) = A002828.

			a(6) = 3 because 6 = 2^2 + 1^2 + 1^2 and 6 is not the sum or the difference of two squares; a(28) = 2 because 28 = 8^2 - 6^2.

Jean-Christophe Hervé, Table of n, a(n) for n = 0..10000

Cf. A002828, A000290, A000415, A062316, A263715, A263737.

Using the partition of the natural numbers into A000290 (square numbers), A000415 (sum of 2 nonzero squares), A263737 (difference but not sum of 2 squares) and A062316 (neither the sum or difference of 2 squares), the sequence is completely defined by: a(A000290(n)) = 1, a(A000415(n)) = a(A263737(n)) = 2, a(A062316(n)) = 3.

A355769 Numbers k such that both k and k+1 can be written as the sum of two nonzero squares.

Original entry on oeis.org

17, 25, 40, 52, 72, 73, 89, 97, 100, 116, 136, 145, 148, 169, 180, 193, 225, 232, 233, 241, 244, 260, 288, 289, 292, 305, 313, 337, 369, 388, 400, 404, 409, 424, 449, 457, 481, 520, 521, 544, 548, 577, 584, 585, 592, 612, 625, 628, 640, 656, 673, 676, 697, 724

Offset: 1

Angad Singh, Jul 16 2022

nonn

The numbers in the sequence are useful in solving various second-degree Diophantine equations.

The identity (3n-12)^2 + (4n-12)^2 + 1 = (3n-8)^2 + (4n-15)^2 proves that there are infinitely many such numbers in this sequence.

			17 is a term since 17 = 4^2 + 1^2 and 17 + 1 = 18 = 3^2 + 3^2.
169 is a term since 169 = 5^2 + 12^2 and 169 + 1 = 170 = 1^2 + 13^2.

Cf. A000290, A000404, A000415.

is1(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)); \\ A000404
isok(k) = is1(k) && is1(k+1); \\ Michel Marcus, Jul 18 2022

A355778 Numbers k such that both k and k^2 + 2 can be written as the sum of two nonzero squares.

Original entry on oeis.org

40, 68, 72, 104, 148, 180, 320, 392, 468, 544, 612, 648, 720, 788, 832, 900, 936, 968, 1040, 1044, 1156, 1192, 1256, 1300, 1332, 1508, 1732, 1796, 1800, 1832, 1872, 1940, 2056, 2196, 2308, 2336, 2372, 2448, 2664, 2696, 2740, 2804, 2848, 2880, 3060, 3200, 3280

Offset: 1

Angad Singh, Jul 16 2022

nonn

The terms in this sequence can be considered as a solution to the "near miss" problem which occurs frequently while solving Diophantine equations. It is known that if a number k can be written as the sum of two nonzero distinct squares then so can k^2 and k^2+1. Thus, finding numbers k such that k^2+2 satisfies the same property makes it quite interesting.

			40 is a term since 40 = 2^2 + 6^2 as well as 40^2 + 2 = 1602 = 9^2 + 39^2.
320 is a term since 320 = 8^2 + 16^2 as well as 320^2 + 2 = 102402 = 201^2 + 249^2.

Cf. A000290, A000404, A000415.

is1(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)); \\ A000404
isok(k) = is1(k) && is1(k^2+2); \\ Michel Marcus, Jul 18 2022

from itertools import count, islice
from sympy import factorint
def A355778_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:
                c = False
                for p in (f:=factorint(n**2+2)):
                    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
A355778_list = list(islice(A355778_gen(),30)) # Chai Wah Wu, Sep 14 2022

cp's OEIS Frontend

A254371 Sum of cubes of the first n even numbers (A016743).

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A073412 Lesser of three consecutive nonsquare integers each of which is the sum of two squares.

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A172001 Nonsquare positive integers n such that Pell equation y^2 - n*x^2 = -1 has rational solutions but the norm of fundamental unit of quadratic field Q(sqrt(n)) is 1.

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A263765 Minimum number of squares necessary to write n as a sum or difference of squares.

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A355769 Numbers k such that both k and k+1 can be written as the sum of two nonzero squares.

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PARI

A355778 Numbers k such that both k and k^2 + 2 can be written as the sum of two nonzero squares.

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PARI

Python