A000217 -id:A000217 - cp's OEIS frontend

A278310 Numbers m such that T(m) + 3*T(m+1) is a square, where T = A000217.

3, 143, 4899, 166463, 5654883, 192099599, 6525731523, 221682772223, 7530688524099, 255821727047183, 8690408031080163, 295218051329678399, 10028723337177985443, 340681375412721826703, 11573138040695364122499, 393146012008229658338303, 13355391270239113019379843

Offset: 1

Bruno Berselli, Nov 17 2016

Equivalently, both m+1 and 2*m+3 are squares for nonnegative m.
Corresponding triangular numbers T(m): 6, 10296, 12002550, 13855048416, 15988853699286, 18451128064030200, 21292585958400815526, ...
Square roots of T(m) + 3*T(m+1) are listed by A082405 (after 0).
Negative values of m for which T(m) + 3*T(m+1) is a square: -1, -2, -26, -842, -28562, -970226, -32959082, ...

			3 is in the sequence because T(3) + 3*T(4) = 6 + 3*10 = 6^2.
For n=5 is a(5) = 5654883, therefore floor(sqrt(5654883)) = 2377 = A182189(5) - 2 = 2379 - 2.

Colin Barker, Table of n, a(n) for n = 1..650
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Subsequence of A000466.
Cf. A000217, A001109, A029546, A046176, A082405, A156164, A182189.
Cf. A278438: numbers m such that T(m) + 2*T(m+1) is a square.
Cf. A078522: numbers m such that 3*T(m) + T(m+1) is a square.
Cf. similar sequences with closed form ((1 + sqrt(2))^(4*r) + (1 - sqrt(2))^(4*r))/8 + k/4: A084703 (k=-1), A076218 (k=3), this sequence (k=-5).

Iv:=[3,143]; [n le 2 select Iv[n] else 34*Self(n-1)-Self(n-2)+40: n in [1..20]];

P:=proc(q) local n; for n from 3 to q do if type(sqrt(2*n^2+5*n+3),integer) then print(n); fi; od; end: P(10^9); # Paolo P. Lava, Nov 18 2016

Table[((1 + Sqrt[2])^(4 n) + (1 - Sqrt[2])^(4 n))/8 - 5/4, {n, 1, 20}]
RecurrenceTable[{a[1] == 3, a[2] == 143, a[n] == 34 a[n - 1] - a[n - 2] + 40}, a, {n, 1, 20}]
LinearRecurrence[{35, -35, 1}, {3, 143, 4899}, 50] (* G. C. Greubel, Nov 20 2016 *)

Vec(x*(3 + 38*x - x^2)/((1 - x)*(1 - 34*x + x^2)) + O(x^50)) \\ G. C. Greubel, Nov 20 2016

def A278310():
    a, b = 3, 143
    yield a
    while True:
        yield b
        a, b = b, 34*b - a + 40
a = A278310(); print([next(a) for  in range(18)]) # _Peter Luschny, Nov 18 2016

O.g.f.: x*(3 + 38*x - x^2)/((1 - x)*(1 - 34*x + x^2)).
E.g.f.: (exp((1-sqrt(2))^4*x) + exp((1+sqrt(2))^4*x) - 10*exp(x))/8 + 1.
a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3) for n>3.
a(n) = 34*a(n-1) - a(n-2) + 40 for n>2.
a(n) = a(-n) = ((1 + sqrt(2))^(4*n) + (1 - sqrt(2))^(4*n))/8 - 5/4.
a(n) = 4*A001109(n)^2 - 1.
a(n) = -A029546(n) + 38*A029546(n-1) + 3*A029546(n-2) for n>1.
Lim_{n -> infinity} a(n)/a(n-1) = A156164.
Floor(sqrt(a(n))) = A182189(n) - 2.
a(n) - a(n-1) = 4*A046176(n) for n>1.

A285955 Numbers a(n) = T(b(n))*sqrt(T(b(n))+1), where T(b(n)) is the triangular number of b(n)= A000217(b(n)) and b(n)=A006451(n). Also a(n) = y solutions of the Bachet Mordell equation y^2=x^3+K, where x= T(b(n)) = A006454(n) and K = (T(b(n)))^2= A285985(n).

Original entry on oeis.org

0, 6, 60, 1320, 12144, 262080, 2405970, 51894744, 476378760, 10274921850, 94320640056, 2034382775040, 18675010652760, 402797515372356, 3697557790357470, 79751873665825680, 732097767490332144, 15790468188346521390, 144951660405354891060, 3126432949419110989944

Offset: 0

Vladimir Pletser, Apr 29 2017

Numbers a(n) which are the products of the triangular number T(b(n)) and the square root of this triangular number plus one, sqrt(T(b(n))+1), where  b(n) is the sequence A006451(n) of numbers n such that T(n)+1 is a square.
This sequence a(n) gives also the y solutions of the 3rd degree Diophantine Bachet-Mordell equation y^2=x^3+K, with x= T(b(n)) = A006454(n) and K = (T(b(n)))^2 = A285985(n), where T(b(n)) is the triangular number of b(n)= A006451(n).
Also: A000217(A006451(n)) * sqrt(A000217(A006451(n))+1).

			For n=2, b(n)=5, a(n)=60.
For n=5, b(n)=90, a(n)= 262080.
For n = 3, A006451(n) = 15. Therefore, A000217(A006451(n)) = A000217(15) = 120. This gives A000217(A006451(n)) * sqrt(A000217(A006451(n)) + 1) = 120 * sqrt(120 + 1) = 1320. - _David A. Corneth_, Apr 29 2017

V. Pletser, On some solutions of the Bachet-Mordell equation for large parameter values, to be submitted, April 2017.

Vladimir Pletser, Table of n, a(n) for n = 0..1000
M.A. Bennett and A. Ghadermarzi, Data on Mordell's curve.
Michael A. Bennett, Amir Ghadermarzi, Mordell's equation : a classical approach, arXiv:1311.7077 [math.NT], 2013.
Eric Weisstein's World of Mathematics, Mordell Curve

Cf. A285985, A006454, A000217, A006451, A081119, A054504.

restart: bm2:=-1: bm1:=0: bp1:=2: bp2:=5: print (‘0,0’,’1,6’,’2,60’); for n from 3 to 1000 do b:= 8*sqrt((bp1^2+bp1)/2+1)+bm2; a:=(b*(b+1)/2)* sqrt((b*(b+1)/2)+1); print(n,a); bm2:=bm1; bm1:=bp1; bp1:=bp2; bp2:=b; end do:

Since b(n) = 8*sqrt(T(b(n-2))+1)+ b(n-4) = 8*sqrt((b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4), with b(-1)=-1, b(0)=0, b(1)=2, b(2)=5 (see A006451) and a(n) = T(b(n))*sqrt(T(b(n))+1) (this sequence), one has :
a(n) = ([8*sqrt((b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4)]*[ 8*sqrt((b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4)+1]/2)* sqrt(([8*sqrt((b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4)]*[ 8*sqrt((b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4)+1]/2)+1).
Empirical g.f.: 6*x*(1 - x)*(1 + 11*x + 27*x^2 + 11*x^3 + x^4) / ((1 + 14*x - x^2)*(1 + 2*x - x^2)*(1 - 2*x - x^2)*(1 - 14*x - x^2)). - Colin Barker, Apr 30 2017

A309507 Number of ways the n-th triangular number T(n) = A000217(n) can be written as the difference of two positive triangular numbers.

Original entry on oeis.org

0, 1, 1, 1, 3, 3, 1, 2, 5, 3, 3, 3, 3, 7, 3, 1, 5, 5, 3, 7, 7, 3, 3, 5, 5, 7, 7, 3, 7, 7, 1, 3, 7, 7, 11, 5, 3, 7, 7, 3, 7, 7, 3, 11, 11, 3, 3, 5, 8, 11, 7, 3, 7, 15, 7, 7, 7, 3, 7, 7, 3, 11, 5, 3, 15, 7, 3, 7, 15, 7, 5, 5, 3, 11, 11, 7, 15, 7, 3, 9, 9, 3, 7

Offset: 1

Alois P. Heinz, Aug 05 2019

nonn

Equivalently, a(n) is the number of triples [n,k,m] with k>0 satisfying the Diophantine equation n*(n+1) + k*(k+1) - m*(m+1) = 0. Any such triple satisfies a triangle inequality, n+k > m. The n for which there is a triple [n,n,m] are listed in A053141. - Bradley Klee, Mar 01 2020; edited by N. J. A. Sloane, Mar 31 2020

			a(5) = 3: T(5) = T(6)-T(3) = T(8)-T(6) = T(15)-T(14).
a(7) = 1: T(7) = T(28)-T(27).
a(8) = 2: T(8) = T(13)-T(10) = T(36)-T(35).
a(9) = 5: T(9) = T(10)-T(4) = T(11)-T(6) = T(16)-T(13) = T(23)-T(21) = T(45)-T(44).
a(49) = 8: T(49) = T(52)-T(17) = T(61)-T(36) = T(94)-T(80) = T(127)-T(117) = T(178)-T(171) = T(247)-T(242) = T(613)-T(611) = T(1225)-T(1224).
The triples with n <= 16 are:
2, 2, 3
3, 5, 6
4, 9, 10
5, 3, 6
5, 6, 8
5, 14, 15
6, 5, 8
6, 9, 11
6, 20, 21
7, 27, 28
8, 10, 13
8, 35, 36
9, 4, 10
9, 6, 11
9, 13, 16
9, 21, 23
9, 44, 45
10, 8, 13
10, 26, 28
10, 54, 55
11, 14, 18
11, 20, 23
11, 65, 66
12, 17, 21
12, 24, 27
12, 77, 78
13, 9, 16
13, 44, 46
13, 90, 91
14, 5, 15
14, 11, 18
14, 14, 20
14, 18, 23
14, 33, 36
14, 51, 53
14, 104, 105
15, 21, 26
15, 38, 41
15, 119, 120
16, 135, 136. - _N. J. A. Sloane_, Mar 31 2020

Alois P. Heinz, Table of n, a(n) for n = 1..20000
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.
M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.

Cf. A000217, A001108, A046079 (the same for squares), A068194, A100821 (the same for primes for n>1), A309332.
See also A053141. The monotonic triples [n,k,m] with n <= k <= m are counted in A333529.
Cf. A092517, A063440.

with(numtheory): seq(tau(n*(n+1))-tau(n*(n+1)/2)-1, n=1..80); # Ridouane Oudra, Dec 08 2023

TriTriples[TNn_] := Sort[Select[{TNn, (TNn + TNn^2 - # - #^2)/(2 #),
      (TNn + TNn^2 - # + #^2)/(2 #)} & /@
    Complement[Divisors[TNn (TNn + 1)], {TNn}],
   And[And @@ (IntegerQ /@ #), And @@ (# > 0 & /@ #)] &]]
Length[TriTriples[#]] & /@ Range[100]
(* Bradley Klee, Mar 01 2020 *)

a(n) = 1 <=> n in { A068194 } \ { 1 }.
a(n) is even <=> n in { A001108 } \ { 0 }.
a(n) = number of odd divisors of n*(n+1) (or, equally, of T(n)) that are greater than 1. - N. J. A. Sloane, Apr 03 2020
a(n) = A092517(n) - A063440(n) - 1. - Ridouane Oudra, Dec 08 2023

A373330 a(n) is the difference between T = A000217(n^2) and the greatest square not exceeding T.

Original entry on oeis.org

0, 0, 1, 9, 15, 1, 41, 0, 55, 72, 9, 156, 36, 204, 262, 144, 135, 289, 209, 316, 111, 117, 406, 309, 527, 261, 342, 860, 804, 36, 954, 1200, 624, 605, 1257, 969, 1400, 741, 849, 1856, 1639, 0, 1721, 2076, 855, 701, 1770, 1101, 1719, 397, 426, 1980, 1416, 2449, 1142

Offset: 0

Hugo Pfoertner, Jun 02 2024

Michael De Vlieger, Table of n, a(n) for n = 0..9998
Hugo Pfoertner, Logarithmic plot of a(n) vs n, n <= 10^5, lower envelope of terms > 0 shown in red, zoom for details.

Cf. A000217, A000290, A002315, A037270, A061288, A373329, A373333.

A373331 and A373332 are the coordinates of the observed lower envelope of this sequence.

Array[PolygonalNumber[#^2] - Floor[Sqrt[(#^4 + #^2)/2]]^2 &, 55, 0] (* Michael De Vlieger, Jun 02 2024 *)

a(n) = my(T=(n^4+n^2)/2); T-sqrtint(T)^2

from sympy import integer_nthroot
def A373330(n): return (T:=(n**4 + n**2) // 2)-(integer_nthroot(T,2)[0])**2
# Karl-Heinz Hofmann, Jul 01 2024

a(n) = A000217(n^2) - A373329(n)^2.

a(A002315(n)) = 0.

A025555 Least common multiple (or LCM) of first n positive triangular numbers (A000217).

Original entry on oeis.org

1, 3, 6, 30, 30, 210, 420, 1260, 1260, 13860, 13860, 180180, 180180, 180180, 360360, 6126120, 6126120, 116396280, 116396280, 116396280, 116396280, 2677114440, 2677114440, 13385572200, 13385572200, 40156716600, 40156716600

Offset: 1

Clark Kimberling and Asher Auel

			a(5) = lcm{1, 3, 6, 10, 15} = 30.

T. D. Noe, Table of n, a(n) for n = 1..200
Peter Luschny and Stefan Wehmeier, The lcm(1,2,...,n) as a product of sine values sampled over the points in Farey sequences, arXiv:0909.1838 [math.CA], 2009.

Cf. A051543, A051538.

a025555 n = a025555_list !! (n-1)
a025555_list = scanl1 lcm $ tail a000217_list
-- Reinhard Zumkeller, Nov 22 2013

HalfFarey := proc (n) local a,b,c,d,k,s; if n<2 then RETURN([1]) fi; a:=0; b:=1; c:=1; d:=n; s:=NULL; do k := iquo(n+b,d); a,b,c,d := c, d, k*c-a, k*d-b; if b < 2*a then break fi; s := s, a/b od; [s] end:
A025555 := proc(n) local r; HalfFarey(n+1); subsop(nops(%) = NULL,%); mul(2*sin(Pi*r),r = %)^2 end: seq(round(evalf(A025555(i))),i=1..27); # Peter Luschny, Jun 09 2011

nn=30;With[{trnos=Accumulate[Range[nn]]},Table[LCM@@Take[trnos,n], {n,nn}]] (* Harvey P. Dale, Oct 21 2011 *)
f[x_] := x + 1; a[1] = f[1]; a[n_] := LCM[f[n], a[n - 1]]; Array[a, 30]/2 (* Robert G. Wilson v, Jan 04 2013 *)

S=1;for(n=1,20,S=lcm(S,n*(n+1)/2);print1(S,",")) \\ Edward Jiang, Sep 08 2014

a(n) = A003418(n+1)/2. - Matthew Vandermast, Jun 04 2012

Corrected by James Sellers

Definition rendered more precisely by Reinhard Zumkeller, Nov 22 2013

A075113 a(n) = A000217(n) - A048702(n).

Original entry on oeis.org

0, 0, 0, 1, -1, 0, 4, 7, -7, -6, 0, 3, 13, 18, 28, 35, -35, -34, -24, -21, -5, 0, 14, 21, 43, 52, 70, 81, 105, 118, 140, 155, -155, -154, -136, -133, -105, -100, -78, -71, -35, -26, 0, 11, 47, 60, 90, 105, 151, 168, 202, 221, 265, 286, 324, 347, 399, 424, 466, 493, 545, 574, 620, 651, -651, -650, -616, -613, -561

Offset: 0

Antti Karttunen, Sep 02 2002

sign

The positions of the zeros seem to be given by A000975.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000217, A048701, A048702.

Cf. A000975, A000225, A006095.

A048702 := Join[{0}, Reap[For[k = 1, k < 1500, k += 2, bb = IntegerDigits[k, 2]; If[bb == Reverse[bb], If[EvenQ[Length[bb]], Sow[k/3]]]]][[2, 1]]]; Table[n*(n + 1)/2 - A048702[[n + 1]], {n, 0, 50}] (* G. C. Greubel, Sep 26 2017 *)

a01(n) = my(f); f = length(binary(n)) - 1; 2^(f+1)*n + sum(i=0, f, bittest(n, i) * 2^(f-i)); \\ A048701
a(n) = n*(n+1)/2 - a01(n)/3; \\ A006095

def A075113(n: int) -> int:
    s = bin(n)[2:]
    return n * (n + 1) // 2 - int(s + s[::-1], 2) // 3
print([A075113(n) for n in range(69)]) # Peter Luschny, Dec 14 2022

a(A000225(n)) = ((2^n)-1)*(2^(n-1)) - (2^(2n) - 1)/3 = A006095(n).

Definition corrected by Georg Fischer, Dec 13 2022

A086737 a(n) = A000217(A000041(n)).

Original entry on oeis.org

1, 1, 3, 6, 15, 28, 66, 120, 253, 465, 903, 1596, 3003, 5151, 9180, 15576, 26796, 44253, 74305, 120295, 196878, 314028, 502503, 788140, 1241100, 1917861, 2968266, 4531555, 6913621, 10421895, 15705210, 23409903, 34857075, 51445296, 75774205, 110759286

Offset: 0

Jon Perry, Jul 29 2003

nonn

a(n) is the number of partitions of 2n that are sum-symmetric. That is, a(n) is the number of partitions of 2n that can be divided into two subsequences (no central summand) that each total to n. Example: Of the 11 partitions of 6, there are 6 that are sum-symmetric (partition subsequences bracketed [] and listed in descending order for clarity:) [3][3], [3][2,1], [3][1,1,1], [2,1][2,1], [2,1][1,1,1], [1,1,1][1,1,1]. As this example suggests, a(n) = p(n)*(p(n)+1)/2. - Gregory L. Simay, Oct 26 2015

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..1000 from Robert Israel)
Tara Kalsi, Alessandro Romito, and Henning Schomerus, Hierarchical analytical approach to universal spectral correlations in Brownian Quantum Chaos, arXiv:2410.15872 [cond-mat.mes-hall], 2024. See p. 21.

Cf. A000041, A000217, A108796.

f:= proc(n) local p;
  p:= combinat:-numbpart(n);
  p*(p+1)/2
end proc:
map(f, [$1..100]); # Robert Israel, Oct 26 2015

pp = Array[PartitionsP, 40, 0]; pp (pp + 1)/2 (* Jean-François Alcover, Mar 19 2019 *)

a(n) = apply(x->x*(x+1)/2, numbpart(n)); \\ Michel Marcus, Oct 26 2015

a(0)=1 prepended by Alois P. Heinz, Mar 25 2017

A092858 "Sum" of the sequences of primes and the triangular numbers (A000217).

Original entry on oeis.org

5, 6, 7, 10, 11, 13, 15, 17, 19, 21, 23, 28, 29, 31, 36, 37, 41, 43, 45, 47, 53, 55, 59, 61, 66, 67, 71, 73, 78, 79, 83, 89, 91, 97, 101, 103, 105, 107, 109, 113, 120, 127, 131, 136, 137, 139, 149, 151, 153, 157, 163, 167, 171, 173, 179, 181, 190, 191, 193, 197, 199

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu)

If two monotonic sequences are mapped into the real codomain of (0,1) as it is defined in A051006, then the fractional part of the sum of the two reals can be mapped back into a sequence as defined in A092855, yielding the "sum" of the two sequences.

Ferenc Adorjan, Binary mapping of monotonic sequences and the Aronson function

Cf. A092855, A051006, A092857, A092859, A092860, A092861, A092862, A092863, A092874.

{ssum(a,b)= /*Returns the "sum" monotonic sequences a and b */ return(mtinv(mt(a)+mt(b))) /* the functions mt(a) and mtinv(r) are defined in A051006 and A092855, respectively */ }

A092859 "Difference" of the sequences of triangular numbers (A000217) and the primes (cf. A092858).

Original entry on oeis.org

3, 4, 5, 7, 12, 13, 16, 18, 19, 22, 23, 30, 31, 38, 39, 40, 42, 43, 46, 48, 49, 50, 51, 52, 53, 56, 57, 58, 60, 61, 68, 69, 70, 72, 73, 80, 81, 82, 84, 85, 86, 87, 88, 89, 92, 93, 94, 95, 96, 98, 99, 100, 102, 103, 106, 108, 110, 111, 112, 113, 121, 122, 123, 124, 125, 126

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu)

Here the complement of the sequence of primes (1 and the composites) is "added" to the sequence of triangulars, according to the definition outlined in A092858.

Ferenc Adorjan, Binary mapping of monotonic sequences and the Aronson function

Cf. A092855, A051006, A092857, A092858, A092860, A092861, A092862, A092863, A092874.

{sdif(a,b)= /*Returns the "difference" of monotonic sequences a and b */ return(mtinv(mt(a)+mt(compl(b)))) /* the functions mt(a) and mtinv(r) are defined in A051006 and A092855, respectively */ } {compl(v)=/* Returns the complement of v monotonic positive sequence */ local(n,p=0,vv=[]);n=matsize(v)[2];for(i=1,n, for(j=p+1,v[i]-1,vv=concat (vv,j));p=v[i]);return(vv)}

A096032 Take pairs (a, b), sorted on a, such that T(a)+T(b)=concatenation of a and b, where T(k) is the k-th triangular number A000217(k). Sequence gives values of b.

Original entry on oeis.org

1, 415, 1545, 1726, 2196, 910, 3676, 3846, 910, 5226, 415, 6970, 7171, 8526, 9231, 9300, 9756, 9850, 9880, 44835, 9880, 9850, 9756, 9300, 9231, 52830, 8526, 7171, 6970, 5226, 3846, 3676, 2196, 1726, 1545, 84906, 89386, 99580, 99580, 89386, 84906

Offset: 1

Lekraj Beedassy, Jun 16 2004

For values of a see A096031.

It is easier to generate the pairs sorted by b. A d-digit number b is a member iff 4*(10^(2*d)-10^d-b^2+b)+1 is a square. All such b occur twice, except for 1, which occurs once. There are no members with 2, 6, 7, or 8 digits. There are six distinct nine-digit members. - David Wasserman, May 15 2007

			1726 of the sequence forms a pair with 150 and we indeed have T(150)+T(1726)=11325+1490401=1501726.

J. S. Madachy, Madachy's Mathematical Recreations, pp. 166 Dover NY 1979.

Chai Wah Wu, Table of n, a(n) for n = 1..294

f[n_] := Block[{k = n + 1, t1 = n(n + 1)/2, td = IntegerDigits[n]}, While[k < 15*n && t1 + k(k + 1)/2 != FromDigits[ Join[ td, IntegerDigits[k]]], k++ ]; If[k != 15*n, k, 0]]; Do[ k = f[n]; If[k != 0, Print[n, " & ", k]], {n, 10^6}] (* Robert G. Wilson v, Jun 21 2004 *)

Two more terms from Robert G. Wilson v, Jun 21 2004

Terms from a(19) onwards from David Wasserman, May 15 2007

cp's OEIS Frontend

A278310 Numbers m such that T(m) + 3*T(m+1) is a square, where T = A000217.

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A285955 Numbers a(n) = T(b(n))*sqrt(T(b(n))+1), where T(b(n)) is the triangular number of b(n)= A000217(b(n)) and b(n)=A006451(n). Also a(n) = y solutions of the Bachet Mordell equation y^2=x^3+K, where x= T(b(n)) = A006454(n) and K = (T(b(n)))^2= A285985(n).

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A309507 Number of ways the n-th triangular number T(n) = A000217(n) can be written as the difference of two positive triangular numbers.

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A373330 a(n) is the difference between T = A000217(n^2) and the greatest square not exceeding T.

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A025555 Least common multiple (or LCM) of first n positive triangular numbers (A000217).

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Haskell

Maple

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A075113 a(n) = A000217(n) - A048702(n).

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A086737 a(n) = A000217(A000041(n)).

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