A060544 -id:A060544 - cp's OEIS frontend

A182427 Triangular numbers that can be represented as a sum of a nonzero square number and a nonzero triangular number.

10, 15, 28, 45, 55, 91, 136, 190, 210, 231, 253, 325, 378, 406, 435, 496, 561, 595, 666, 703, 741, 820, 861, 903, 946, 990, 1081, 1128, 1176, 1225, 1378, 1431, 1540, 1596, 1711, 1770, 1830, 1891, 2080, 2145, 2211, 2278, 2346, 2415, 2485, 2556, 2701, 2926, 3160, 3321

Offset: 1

Ivan N. Ianakiev, Apr 28 2012

nonn

Theorem (by Ivan N. Ianakiev): There are infinitely many such numbers. Proof: Any triangular number of the form A000217(n^2) for n>1 is such a number, as A000217(n^2) = A000217(n^2-1) + A000290(n), for n>=1. Observation: Other numbers not of the form A000217(n^2), for example 15 and 28, are also in A182427. - Ivan N. Ianakiev, May 30 2012

For any integer k>1, all triangular numbers with indices of the form 3*k-2 (A060544) are terms as (3*k-2)*(3*k-1)/2 = (2*k-1)^2 + (k-1)*k/2. - Ivan N. Ianakiev, Nov 25 2015

			10, 15, 28 are in the sequence because 10 = 2^2 + 3*4/2 = 3^2 + 1*2/2, 15 = 3^2 + 3*4/2, 28 = 5^2 + 2*3/2.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000217, A000290, A037270.

isok(t) = {for (k=1, sqrtint(t), my(tt = t - k^2); if ((tt) && ispolygonal(tt, 3), return (1)););}
lista(nn) = {for (n=1, nn, my(t = n*(n+1)/2); if (isok(t), print1(t, ", ")););} \\ Michel Marcus, Nov 25 2015

A188621 Smallest number k>1 such that k*(n-th triangular number) is also a triangular number.

Original entry on oeis.org

3, 2, 6, 12, 3, 5, 42, 56, 14, 18, 8, 10, 33, 2, 27, 240, 60, 68, 15, 3, 13, 105, 61, 67, 138, 150, 47, 51, 24, 26, 930, 117, 21, 6, 40, 66, 315, 41, 7, 231, 35, 37, 118, 5, 83, 495, 220, 230, 564, 55, 28, 147, 663, 98, 10, 50, 92, 798, 221, 229, 885, 12, 741, 615

Offset: 1

Zak Seidov, Apr 06 2011

nonn

There is a sequence of triangular numbers >3 which are not divisible by any smaller triangular number > 1, primitive triangular numbers in that sense: 3, 10, 28, 55, 91, 136, 253.... whose indices are in A137281.

(This is apparently a subsequence of A060544. - R. J. Mathar, Apr 06 2011)

			a(1)=3 because A000217(1)=1, 3*1 is triangular and k*1 for 1<k<3 is not triangular.
a(2)=2 because A000217(2)=3, 2*3 is triangular and k*3 for 1<k<2 (empty condition) is not triangular.
a(3)=6 because A000217(3)=6, 6*6 is triangular and k*6 for 1<k<6 is not triangular.
a(1000)=153 because A000217(1000)=500500, 153*500500=76576500 is triangular and k*500500 for 1<k<153 is not triangular.

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

Cf. A000217, A068084, A069752, A137281.

TriangularQ[n_] := IntegerQ[Sqrt[1 + 8 n]]; Table[t = (n + 1)*n/2; k = 2; While[! TriangularQ[k*t], k++]; k, {n, 100}] (* T. D. Noe, Apr 06 2011 *)
snk[n_]:=Module[{k=2},While[!OddQ[Sqrt[8k*n+1]],k++];k]; snk/@Accumulate[ Range[ 70]] (* Harvey P. Dale, Apr 29 2018 *)

a(n) = A068084(n)/A000217(n).

A249348 a(n) = (A001147(n+1)^2-1)/8, where A001147(n+1) = 35...*(2n+1).

Original entry on oeis.org

0, 1, 28, 1378, 111628, 13507003, 2282683528, 513603793828, 148431496416328, 53583770206294453, 23630442660975853828, 12500504167656226675078, 7812815104785141671923828, 5695542211388368278832470703, 4789950999777617722498107861328

Offset: 0

M. F. Hasler, Oct 26 2014

nonn

These are the numerators of the partial sums S(n) = Sum_{k=1..n} A000217(k)/A001147(k+1)^2 before simplification, i.e., a(n) = S(n)*A001147(n+1)^2, where A000217(n) = n(n+1)/2. The series S(n) has sum 1/8, actually S(n) = 1/8 - 1/(8*A001147(n+1)^2). (Similarly, Sum_{n=1..oo} A249354(n)/A007559(n+1)^3 = 1/9, where A249354(n) = 3n^3+3n^2+n.)

This is a subsequence of the centered 9-gonal numbers A060544, which are a subsequence of the triangular numbers A000217.

S. Klein, A neat infinite sum ..., "Number Theory" group on LinkedIn, Oct. 2014.

A249348 := proc(n)
    (doublefactorial(2*n+1)^2-1)/8 ;
end proc:
seq(A249348(n),n=0..20) ;

PARI
```
a(n)=(prod(k=1,n,2*k+1)^2-1)/8
```

(-n+1)*a(n) +2*n*(2*n^2-1)*a(n-1) -(n+1)*(-1+2*n)^2*a(n-2)=0. - R. J. Mathar, Oct 28 2014

a(11)/a(12) corrected by Georg Fischer, Mar 12 2020

A302537 a(n) = (n^2 + 13*n + 2)/2.

Original entry on oeis.org

1, 8, 16, 25, 35, 46, 58, 71, 85, 100, 116, 133, 151, 170, 190, 211, 233, 256, 280, 305, 331, 358, 386, 415, 445, 476, 508, 541, 575, 610, 646, 683, 721, 760, 800, 841, 883, 926, 970, 1015, 1061, 1108, 1156, 1205, 1255, 1306, 1358, 1411, 1465, 1520, 1576

Offset: 0

Franck Maminirina Ramaharo, Apr 09 2018

Binomial transform of [1, 7, 1, 0, 0, 0, ...].

Numbers m > 0 such that 8*m + 161 is a square.

			Illustration of initial terms (by the formula a(n) = A052905(n) + 3*n):
.                                                                    o
.                                                                  o o
.                                                    o           o o o
.                                                  o o         o o o o
.                                      o         o o o       o o o o o
.                                    o o       o o o o     o o o o o o
.                          o       o o o     o o o o o   o . . . . . o
.                        o o     o o o o   o . . . . o   o . . . . . o
.                o     o o o   o . . . o   o . . . . o   o . . . . . o
.              o o   o . . o   o . . . o   o . . . . o   o . . . . . o
.        o   o . o   o . . o   o . . . o   o . . . . o   o . . . . . o
.      o o   o . o   o . . o   o . . . o   o . . . . o   o . . . . . o
.  o   o o   o o o   o o o o   o o o o o   o o o o o o   o o o o o o o
.        o     o o     o o o     o o o o     o o o o o     o o o o o o
.        o     o o     o o o     o o o o     o o o o o     o o o o o o
.        o     o o     o o o     o o o o     o o o o o     o o o o o o
----------------------------------------------------------------------
.  1     8      16        25          35            46              58

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Sequences whose n-th terms are of the form binomial(n, 2) + n*k + 1:
A152947 (k = 0); A000124 (k = 1); A000217 (k = 2); A034856 (k = 3);
A052905 (k = 4); A051936 (k = 5); A246172 (k = 6).
Cf. A000578, A002939, A004120, A008589, A056119, A060544, A162261.

A302537:= func< n | ((n+1)^2 +12*n +1)/2 >;
[A302537(n): n in [0..50]]; // G. C. Greubel, Jan 21 2025

a := n -> (n^2 + 13*n + 2)/2;
seq(a(n), n = 0 .. 100);

Table[(n^2 + 13 n + 2)/2, {n, 0, 100}]
CoefficientList[ Series[(5x^2 - 5x - 1)/(x - 1)^3, {x, 0, 50}], x] (* or *)
LinearRecurrence[{3, -3, 1}, {1, 8, 16}, 51] (* Robert G. Wilson v, May 19 2018 *)

makelist((n^2 + 13*n + 2)/2, n, 0, 100);

a(n) = (n^2 + 13*n + 2)/2; \\ Altug Alkan, Apr 12 2018

def A302537(n): return (n**2 + 13*n + 2)//2
print([A302537(n) for n in range(51)]) # G. C. Greubel, Jan 21 2025

a(n) = binomial(n + 1, 2) + 6*n + 1 = binomial(n, 2) + 7*n + 1.
a(n) = a(n-1) + n + 6.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3, where a(0) = 1, a(1) = 8 and a(2) = 16.
a(n) = 2*a(n-1) - a(n-2) + 1.
a(n) = A004120(n+1) for n > 1.
a(n) = A056119(n) + 1.
a(n) = A152947(n+1) + A008589(n).
a(n) = A060544(n+1) - A002939(n).
a(n) = A000578(n+1) - A162261(n) for n > 0.
G.f.: (1 + 5*x - 5*x^2)/(1 - x)^3.
E.g.f.: (1/2)*(2 + 14*x + x^2)*exp(x).
Sum_{n>=0} 1/a(n) = 24097/45220 + 2*Pi*tan(sqrt(161)*Pi/2) / sqrt(161) = 1.4630922534498496... - Vaclav Kotesovec, Apr 11 2018

A061793 a(n) = 25n(n + 1)/2 + 3.

Original entry on oeis.org

3, 28, 78, 153, 253, 378, 528, 703, 903, 1128, 1378, 1653, 1953, 2278, 2628, 3003, 3403, 3828, 4278, 4753, 5253, 5778, 6328, 6903, 7503, 8128, 8778, 9453, 10153, 10878, 11628, 12403, 13203, 14028, 14878, 15753, 16653, 17578, 18528, 19503, 20503, 21528, 22578, 23653

Offset: 0

Jason Earls, Jun 22 2001

"If m is a triangular number, then so are 9*m+1, 25*m+3 and 49*m+6. (Euler, 1775)", see Burton in References. Note that A060544 is the same as 9*m+1 when m is triangular and that 9*(m*(m+1)/2)+1 is another formula for it.
9*m+1, 25*m+3 and 49*m+6 are special cases of the identity A000290(2*r + 1)*A000217(s) + A000217(r) = A000217((2*r + 1)*s + r). - Bruno Berselli, Mar 01 2018
Complementing the previous comment, with T(n) = A000217(n), 4*T(s)+1+s = T(2*s+1), 16*T(s)+3+2s = T(4*s+2) and 36*T(s)+6+3s = T(6*s+3) are special cases of the identity A000290(2*r)*T(s) + T(r) + r*s = T(2*r*s + r). - Charlie Marion, Mar 28 2018

D. M. Burton, Elementary Number Theory, Allyn and Bacon, Inc. Boston, MA, 1976, p. 17.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000290, A060544, A123296.

List([0..40],n->25*(n*(n+1)/2)+3); # Muniru A Asiru, Mar 30 2018

[seq(25*(n*(n+1)/2)+3,n=0..40)]; # Muniru A Asiru, Mar 30 2018

25*Accumulate[Range[0,40]]+3 (* Harvey P. Dale, Aug 26 2013 *)

PARI
```
a(n) = 25*n*(n + 1)/2 + 3
```

a(n) = 25*A000217(n) + 3 = A123296(n) + 3.
From Elmo R. Oliveira, Oct 23 2024: (Start)
G.f.: (3 + 19*x + 3*x^2)/(1 - x)^3.
E.g.f.: (3 + 25*x + 25*x^2/2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

Corrected by T. D. Noe, Oct 25 2006

A081275 Shallow diagonal of triangular spiral in A051682.

Original entry on oeis.org

1, 31, 97, 199, 337, 511, 721, 967, 1249, 1567, 1921, 2311, 2737, 3199, 3697, 4231, 4801, 5407, 6049, 6727, 7441, 8191, 8977, 9799, 10657, 11551, 12481, 13447, 14449, 15487, 16561, 17671, 18817, 19999, 21217, 22471, 23761, 25087, 26449, 27847, 29281, 30751, 32257

Offset: 0

Paul Barry, Mar 15 2003

Reflection of A060544 in the horizontal A051682.

Binomial transform of (1, 30, 36, 0, 0, 0, ...).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051682, A060544, A092296.

Table[30Binomial[n,1]+36Binomial[n,2]+1,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{1,31,97},40] (* Harvey P. Dale, Jun 30 2011 *)
CoefficientList[Series[(1 + 28 x + 7 x^2) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 07 2013 *)

a(n)=18*n^2+12*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = C(n,0) + 30*C(n,1) + 36*C(n,2).
a(n) = 18*n^2 + 12*n + 1.
G.f.: (1 + 28*x + 7*x^2)/(1-x)^3.
a(0)=1, a(1)=31, a(2)=97, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jun 30 2011
E.g.f.: exp(x)*(1 + 30*x + 18*x^2). - Elmo R. Oliveira, Nov 13 2024

A248579 a(n) = the smallest numbers k such that nT(k)-1 and nT(k)+1 are twin primes or 0 if no solution exists for n where T(k) = A000217(k) = k-th triangular number.

Original entry on oeis.org

3, 2, 3, 1, 3, 1, 3, 0, 0, 2, 12, 1, 11, 2, 4, 5, 3, 1, 12, 2, 24, 6, 3, 2, 3, 12, 4, 5, 20, 1, 27, 3, 3, 2, 11, 2, 56, 3, 7, 3, 32, 1, 44, 5, 3, 2, 3, 11, 12, 2, 7, 3, 15, 5, 20, 14, 4, 3, 32, 1, 27, 6, 8, 2, 8, 2, 11, 5, 7, 3, 167, 1, 20, 9, 12, 2, 3, 18

Offset: 1

Jaroslav Krizek, Oct 25 2014

nonn

For n = 8 and 9 there are no triangular numbers T(k) such that n*T(k) +/- 1 are twin primes.
a(8) = 0 because 8*T(k) + 1 = A016754(k) = composite number for k >= 1.
a(9) = 0 because 9*T(k) + 1 = A060544(k+1) = composite number for k >= 1.
Are there numbers n > 9 such that a(n) = 0? If a(n) = 0 for n > 9, n must be bigger than 4000.
a(n) > 0 for 10 <= n <= 100000. - Robert Israel, Aug 10 2023

			a(5) = 3 because 3 is the smallest number k with this property: 5*T(3) -/+ 1 = 5*6 -+ 1 = 29 and 31 (twin primes).

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

Cf. A000217, A016754, A060544, A248580.

a(n) = 1: A014574.

A248579:=func; [A248579(n): n in[1..100]]

f:= proc(n) local k;
   for k from 1 do if isprime(n*k*(k+1)/2+1) and isprime(n*k*(k+1)/2-1) then return k fi od:
end proc;
f(8):= 8: f(9):= 0:
map(f, [$1..100]); # Robert Israel, Aug 10 2023

a(n) = {if ((n==8) || (n==9), return (0)); k = 1; while (!isprime(n*k*(k+1)/2-1) || !isprime(n*k*(k+1)/2+1), k++); k;} \\ Michel Marcus, Nov 05 2014

A248580 a(n) = the smallest triangular number T(k) such that nT(k)-1 and nT(k)+1 are twin primes or 0 if no solution exists for n; T(k) = A000217(k) = k-th triangular number.

Original entry on oeis.org

6, 3, 6, 1, 6, 1, 6, 0, 0, 3, 78, 1, 66, 3, 10, 15, 6, 1, 78, 3, 300, 21, 6, 3, 6, 78, 10, 15, 210, 1, 378, 6, 6, 3, 66, 3, 1596, 6, 28, 6, 528, 1, 990, 15, 6, 3, 6, 66, 78, 3, 28, 6, 120, 15, 210, 105, 10, 6, 528, 1, 378, 21, 36, 3, 36, 3, 66, 15, 28, 6

Offset: 1

Jaroslav Krizek, Oct 25 2014

nonn

For n = 8 and 9 there are no triangular numbers T(k) such that n*T(k)-+1 are twin primes.
a(8) = 0 because 8*T(k)+1 = A016754(k) = composite number for k >= 1.
a(9) = 0 because 9*T(k)+1 = A060544(k+1) = composite number for k >= 1.
Are there numbers n > 9 such that a(n) = 0? If a(n) = 0 for n > 9, n must be bigger than 4000.

			a(5) = 6 because 6 is the smallest smallest triangular number with this property: 5*6 -+ 1 = 29 and 31 (twin primes).

Cf. A000217, A016754, A060544, A248579.

A248580:=func; [A248580(n): n in[1..100]]

a248580[n_Integer] := Catch@Module[{T, k}, T[i_] := i (i + 1)/2; Do[If[And[PrimeQ[n*T[k] + 1], PrimeQ[n*T[k] - 1]], Throw[T[k]], 0], {k, 1, 10^4}] /. Null -> 0]; a248580 /@ Range[70] (* Michael De Vlieger, Nov 12 2014 *)

a(n) = {if ((n==8) || (n==9), return (0)); k = 1; while (!isprime(n*k*(k+1)/2-1) || !isprime(n*k*(k+1)/2+1), k++); k*(k+1)/2; } \\ Michel Marcus, Nov 12 2014

a(n) = A000217(A248579(n)).

A249349 (A001147(n+1)-1)/2, equals the index of A249348(n) within the triangular numbers A000217.

Original entry on oeis.org

0, 1, 7, 52, 472, 5197, 67567, 1013512, 17229712, 327364537, 6874655287, 158117071612, 3952926790312, 106729023338437, 3095141676814687, 95949391981255312, 3166329935381425312, 110821547738349885937, 4100397266318945779687, 159915493386438885407812

Offset: 0

M. F. Hasler, Oct 26 2014

nonn

Also a(n) = floor(sqrt(A249348(n)*2)).

The positive terms are of the form 3k-2; this k (= 1, 3, 18, 157, ...) is the index of A249348(n) within the centered 9-gonal numbers A060544.

PARI
```
a(n)=A001147(n+1)\2
```

vector(10,n,A001147(n)\2) \\ To get the initial term a(0) for n=1.

a(n) +(-2*n-3)*a(n-1) +(4*n-1)*a(n-2) +(-2*n+3)*a(n-3)=0. - R. J. Mathar, Oct 28 2014

A318255 Associated Omega numbers of order 3, triangle T(n,k) read by rows for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 10, -9, 1, 28, -504, 477, 1, 55, -4158, 78705, -74601, 1, 91, -18018, 1432431, -27154764, 25740261, 1, 136, -55692, 11595870, -923261976, 17503377480, -16591655817, 1, 190, -139536, 60087690, -12529983960, 997692516360, -18914487631380, 17929265150637

Offset: 0

Peter Luschny, Aug 26 2018

See the comments in A318254.

			Triangle starts:
[0] 1
[1] 1,   1
[2] 1,  10,     -9
[3] 1,  28,   -504,      477
[4] 1,  55,  -4158,    78705,     -74601
[5] 1,  91, -18018,  1432431,  -27154764,    25740261
[6] 1, 136, -55692, 11595870, -923261976, 17503377480, -16591655817

T(n, 0) = A060544, T(n, n) = A293951(n+1) (up to signs), row sums are A040000.

Cf. A318146, A318253, A318254 (m=2).

# The function TNum is defined in A318253.
T := (m, n, k) -> `if`(k=0, 1, binomial(m*n-1, m*(n-k))*TNum(m, k)):
for n from 0 to 6 do seq(T(3, n, k), k=0..n) od;

# uses[AssociatedOmegaNumberTriangle from A318254]
A318255Triangle = lambda dim: AssociatedOmegaNumberTriangle(3, dim)
print(A318255Triangle(8))

T(m, n, k) = binomial(m*n-1, m*(n-k))*A318253(m, k) for k>0 and 1 for k=0. We consider here the case m=3.

cp's OEIS Frontend

A182427 Triangular numbers that can be represented as a sum of a nonzero square number and a nonzero triangular number.

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A188621 Smallest number k>1 such that k*(n-th triangular number) is also a triangular number.

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A249348 a(n) = (A001147(n+1)^2-1)/8, where A001147(n+1) = 3*5*...*(2n+1).

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A302537 a(n) = (n^2 + 13*n + 2)/2.

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A061793 a(n) = 25*n*(n + 1)/2 + 3.

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A081275 Shallow diagonal of triangular spiral in A051682.

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A248579 a(n) = the smallest numbers k such that n*T(k)-1 and n*T(k)+1 are twin primes or 0 if no solution exists for n where T(k) = A000217(k) = k-th triangular number.

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A249348 a(n) = (A001147(n+1)^2-1)/8, where A001147(n+1) = 35...*(2n+1).

A061793 a(n) = 25n(n + 1)/2 + 3.

A248579 a(n) = the smallest numbers k such that nT(k)-1 and nT(k)+1 are twin primes or 0 if no solution exists for n where T(k) = A000217(k) = k-th triangular number.

A248580 a(n) = the smallest triangular number T(k) such that nT(k)-1 and nT(k)+1 are twin primes or 0 if no solution exists for n; T(k) = A000217(k) = k-th triangular number.