A089982 -id:A089982 - cp's OEIS frontend

A214838 Triangular numbers of the form k^2 + 2.

3, 6, 66, 171, 2211, 5778, 75078, 196251, 2550411, 6666726, 86638866, 226472403, 2943171003, 7693394946, 99981175206, 261348955731, 3396416785971, 8878171099878, 115378189547778, 301596468440091, 3919462027838451, 10245401755863186, 133146330756959526, 348042063230908203

Offset: 1

Alex Ratushnyak, Mar 07 2013

Corresponding k values are in A077241.

Except 3, all terms are in A089982: in fact, a(2) = 3+3 and a(n) = (k-2)*(k-1)/2+(k+1)*(k+2)/2, where k = sqrt(a(n)-2) > 2 for n > 2. [Bruno Berselli, Mar 08 2013]

			2211 is in the sequence because 2211 = 47^2 + 2.

Index entries for linear recurrences with constant coefficients, signature (1,34,-34,-1,1).

Cf. A000217, A001110, A001219, A006454, A029549, A069017, A077241, A089982, A164055, A165892.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(-3*(x^4+x^3-14*x^2+x+1)/((x-1)*(x^2-6*x+1)*(x^2+6*x+1)))); // Bruno Berselli, Mar 08 2013

LinearRecurrence[{1, 34, -34, -1, 1}, {3, 6, 66, 171, 2211}, 25] (* Bruno Berselli, Mar 08 2013 *)

t[n]:=((5-2*sqrt(2))*(1+(-1)^n*sqrt(2))^(2*floor(n/2))+(5+2*sqrt(2))*(1-(-1)^n*sqrt(2))^(2*floor(n/2))-2)/4$
makelist(expand(t[n]*(t[n]+1)/2), n, 1, 25); /* Bruno Berselli, Mar 08 2013 */

for(n=1, 10^9, t=n*(n+1)/2; if(issquare(t-2), print1(t,", "))); \\ Joerg Arndt, Mar 08 2013

import math
for i in range(2, 1<<32):
      t = i*(i+1)//2 - 2
      sr = int(math.sqrt(t))
      if sr*sr == t:
          print(f'{sr:10} {i:10} {t+2}')

G.f.: -3*x*(x^4+x^3-14*x^2+x+1)/((x-1)*(x^2-6*x+1)*(x^2+6*x+1)). - Joerg Arndt, Mar 08 2013

a(n) = A000217(t), where t = ((5-2*sqrt(2))*(1+(-1)^n*sqrt(2))^(2*floor(n/2))+(5+2*sqrt(2))*(1-(-1)^n*sqrt(2))^(2*floor(n/2))-2)/4. - Bruno Berselli, Mar 08 2013

A112352 Triangular numbers that are the sum of two distinct positive triangular numbers.

Original entry on oeis.org

21, 36, 55, 66, 91, 120, 136, 171, 231, 276, 351, 378, 406, 496, 561, 666, 703, 741, 820, 861, 946, 990, 1035, 1081, 1176, 1225, 1326, 1378, 1431, 1485, 1540, 1596, 1653, 1711, 1770, 1891, 1953, 2016, 2080, 2211, 2278, 2346, 2556, 2701, 2775, 2850, 2926

Offset: 1

Rick L. Shepherd, Sep 05 2005

nonn

Subsequence of A089982: it doesn't require the two positive triangular numbers to be distinct.
Subsequence of squares: 36, 1225, 41616, 1413721,... is also in A001110. - Zak Seidov, May 07 2015
First term with 2 representations is 231: 21+210=78+153, first term with 3 representations is 276: 45+211=66+120=105+171; apparently the number of representations is unbounded. - Zak Seidov, May 11 2015

			36 is a term because 36 = 15 + 21 and these three numbers are distinct triangular numbers (A000217(8) = A000217(5) + A000217(6)).

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A000217 (triangular numbers), A112353 (triangular numbers that are the sum of three distinct positive triangular numbers), A089982.

Cf. A001110. - Zak Seidov, May 07 2015

N:= 10^5: # to get all terms <= N
S:= {}:
for a from 1 to floor(sqrt(1+8*N)/2) do
  for b from 1 to a-1 do
    y:= a*(a+1)/2 + b*(b+1)/2;
      if y > N then break fi;
      if issqr(8*y+1) then S:= S union {y} fi
  od
od:
sort(convert(S,list)); # Robert Israel, May 13 2015

Select[Union[Total/@Subsets[Accumulate[Range[100]],{2}]],OddQ[ Sqrt[ 1+8#]]&] (* Harvey P. Dale, Feb 28 2016 *)

Offset corrected by Arkadiusz Wesolowski, Aug 06 2012

A112355 Triangular numbers that are the sum of three positive triangular numbers.

Original entry on oeis.org

3, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326

Offset: 1

Rick L. Shepherd, Sep 06 2005

nonn

A112353 is a subsequence: it requires the three positive triangular numbers to be distinct. The positive terms of A076140 are a subsequence.

			21 is a term because 21 = 3 + 3 + 15 and these four numbers are positive triangular numbers (A000217(6) = A000217(2) + A000217(2) + A000217(5)). {Also 21 = 1 + 10 + 10 = A000217(1) + A000217(4) + A000217(4).}.

Cf. A000217 (triangular numbers), A089982 (triangular numbers that are the sum of two positive triangular numbers), A112353, A076140 (the three triangular numbers summed are identical).

A136346 Octagonal numbers which are the sums of exactly two positive octagonal numbers.

Original entry on oeis.org

560, 736, 1541, 3201, 5461, 6816, 7400, 9976, 11041, 11408, 13333, 14981, 15408, 15841, 19521, 21000, 21505, 25761, 28616, 30401, 41536, 45141, 50440, 51221, 52008, 54405, 56856, 61920, 63656, 65416, 69008, 75525, 76480, 81345, 82336, 85345, 87381, 89441

Offset: 1

Jonathan Vos Post, Dec 25 2007

For sums of two positive octagonal numbers, see A136345. This is to octagonal numbers A000567 as A089982 is to triangular numbers A000217, as A009000 is to squares A000290, as A136117 is to pentagonal numbers A000326, as A133215 is to hexagonal numbers A000384, and as A117104 is to heptagonal numbers A000566. If Oc(a) + Oc(b) = Oc(c) then a(3a-2) + b(3b+2) = c(3c+2), so solving the quadratic equations for c we have (when an integer): c = (2 + sqrt(4 + 36a^2 + 36b^2 - 24a - 24b))/6.

			Where Oc(n) = A000567(n) = n-th octagonal number:
a(1) = 560 = Oc(14) = 280 + 280 = Oc(10) + Oc(10).
a(2) = 736 = Oc(16) = 560 + 176 = Oc(14) + Oc(8).
a(3) = 1541 = Oc(23) = 1408 + 133 = Oc(22) + Oc(7).
a(4) = 3201 = Oc(33) = 2465 + 736 = Oc(29) + Oc(16).
a(5) = 5461 = Oc(43) = 2821 + 2640 = Oc(31) + Oc(30).

B. D. Swan, Table of n, a(n) for n = 1..1800
Eric Weisstein's World of Mathematics, Octagonal Number

Cf. A000567, A009000, A089982, A136117, A136345.

Module[{nn=300,ono},ono=PolygonalNumber[8,Range[nn]];Union[Select[ Total/@ Tuples[ono,2],MemberQ[ono,#]&]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 26 2019 *)

A000567 INTERSECTION {A000567(i) + A000567(j), i, j > 0}. {i*(3*i-2)} INTERSECTION {i*(3*i-2) + j(3*j-2), i > 0}.

Corrected and edited by B. D. Swan (bdswan(AT)gmail.com), Dec 20 2008

A343426 Triangular numbers T(i) that can be expressed as the sum of 2 positive triangular numbers, T(j)+T(k), and for which i+j+k is a triangular number, where T is A000217.

Original entry on oeis.org

276, 741, 17766, 30876, 42778, 43071, 44850, 54946, 73920, 99681, 163306, 184528, 254541, 310866, 446040, 524800, 963966, 1006071, 1046181, 1160526, 1258491, 1873080, 1929630, 2793066, 3034416, 3108771, 3121251, 3454506, 3635556, 4305645, 4317391, 4435731, 4831386, 4859403

Offset: 1

Michel Marcus, Apr 15 2021

nonn

			276 = T(23) = 105 + 171 = T(14) + T(18) and 23+14+18 = 55 = T(10), so 276 is a term.

K. R. S. Sastry, Problem 518, The College Mathematics Journal, p. 64, Vol. 25, No. 1, Jan., 1994; Solution, Pythagorean Triples of Triangular Numbers, p. 69, Vol. 26, No. 1, Jan., 1995.

Cf. A000217. Subsequence of A089982.

lista(nn) = {for (n=1, nn, my(t = n*(n+1)/2, kk); for (k=1, n-1, if (ispolygonal(t - k*(k+1)/2, 3, &kk), if (ispolygonal(n+k+kk, 3), print1(t, ", "); break;););););}

A350367 Triangular numbers that are the sum of two distinct nonzero triangular numbers in more than one way.

Original entry on oeis.org

231, 276, 406, 666, 861, 1081, 1225, 1431, 1711, 1891, 2211, 2556, 3081, 3741, 3916, 4186, 4371, 4560, 4656, 5151, 5356, 5671, 5886, 6786, 7021, 7381, 7875, 8001, 8128, 8256, 8778, 9316, 10731, 11781, 12246, 12561, 12720, 13366, 13861, 14196, 14706, 15576

Offset: 1

Shyam Sunder Gupta, Dec 27 2021

nonn

			231 = 21 + 210 = 78 + 153.
276 = 45 + 231 = 66 + 210 = 105 + 171.

Intersection of A000217 and A262749.

Cf. A089982, A112352.

(P=PolygonalNumber)[3,Select[Range@176,Length@Select[Subsets[P[3,Range[s=#]],{2}],Total@#==P[3,s]&]>1&]] (* Giorgos Kalogeropoulos, Dec 31 2021 *)

from collections import Counter
from itertools import count, takewhile, combinations as combs
def aupto(limit):
    tris = takewhile(lambda x: x <= limit, (k*(k+1)//2 for k in count(1)))
    trilst = list(tris); triset = set(trilst)
    tri2ct = Counter(sum(c) for c in combs(trilst, 2) if sum(c) in triset)
    return sorted(t for t in tri2ct if t <= limit and tri2ct[t] > 1)
print(aupto(16000)) # Michael S. Branicky, Dec 27 2021

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A214838 Triangular numbers of the form k^2 + 2.

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A112352 Triangular numbers that are the sum of two distinct positive triangular numbers.

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A112355 Triangular numbers that are the sum of three positive triangular numbers.

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A136346 Octagonal numbers which are the sums of exactly two positive octagonal numbers.

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A343426 Triangular numbers T(i) that can be expressed as the sum of 2 positive triangular numbers, T(j)+T(k), and for which i+j+k is a triangular number, where T is A000217.

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A350367 Triangular numbers that are the sum of two distinct nonzero triangular numbers in more than one way.

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