A108752 -id:A108752 - cp's OEIS frontend

A154293 Integers of the form t/6, where t is a triangular number (A000217).

0, 1, 6, 11, 13, 20, 35, 46, 50, 63, 88, 105, 111, 130, 165, 188, 196, 221, 266, 295, 305, 336, 391, 426, 438, 475, 540, 581, 595, 638, 713, 760, 776, 825, 910, 963, 981, 1036, 1131, 1190, 1210, 1271, 1376, 1441, 1463, 1530, 1645, 1716, 1740, 1813, 1938, 2015

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 06 2009

Old definition was "Integers of the form: 1/6+2/6+3/6+4/6+5/6+...".
1/6 + 2/6 + 3/6 = 1, 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6 + 7/6 + 8/6 = 6, ...
a(n) is the set of all integers k such that 48k+1 is a perfect square. The square roots of 48*a(n) + 1 = 1, 7, 17, 23, 25, ... = 8*(n-floor(n/4)) + (-1)^n. - Gary Detlefs, Mar 01 2010
Conjecture: A193828 divided by 2. - Omar E. Pol, Aug 19 2011
The above conjecture is correct. - Charles R Greathouse IV, Jan 02 2012
Quasipolynomial of order 4. - Charles R Greathouse IV, Jan 02 2012
It appears that the sequence terms occur as exponents in the expansion Sum_{n >= 0} x^n/Product_{k = 1..2*n} (1 + x^k) = 1 + x - x^6 - x^11 + x^13 + x^20 - x^35 - x^46  + + - - .... Cf. A218171. [added Jan 21 2025: this is correct  - see Berndt et al., Theorem 3.2.] - Peter Bala, Feb 04 2021
From Peter Bala, Dec 12 2024 (Start)
The sequence terms occur as exponents in the expansion of F(x)*Product_{n >= 1} (1 - x^n) = Product_{n >= 1} (1 - x^n)*(1 + x^(4*n))^2*(1 + x^(4*n-2))*(1 + x^(8*n-3))*(1 + x^(8*n-5)) =  1 - x - x^6 + x^11 + x^13 - x^20 - x^35 + x^46 + x^50 - - + + ..., where F(x) is the g.f. of A069910.
It appears that the sequence terms occur as exponents in the expansion 1/(1 - x) * ( - x^2 + Sum_{n >= 1} x^floor((3*n+1)/2) * 1/Product_{k = 1..n} (1 + x^k) ) = x^6 + x^11 - x^13 - x^20 + x^35 + x^46 - - + + .... (End)
It appears that the sequence terms occur as exponents in the expansion Sum_{n >= 0} x^(n+1)/Product_{k = 1..2*n+2} (1 + x^k) =  x - x^6 - x^11 + x^13 + x^20 - x^35 - x^46  + + - - .... - Peter Bala, Jan 21 2025

			G.f. = x^2 + 6*x^3 + 11*x^4 + 13*x^5 + 20*x^6 + 35*x^7 + 46*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
B. C. Berndt, B. Kim, and A. J. Yee, Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions, J. Comb. Thy. Ser. A, 117 (2010), 957-973.
Mircea Merca, The bisectional pentagonal number theorem, Journal of Number Theory, Volume 157 (December 2015), Pages 223-232.
Index entries for linear recurrences with constant coefficients, signature (3,-5,7,-7,5,-3,1).

Cf. A000217, A001318, A069497, A074378, A057569, A057570, A154292, A193828, A218171.

/* By definition: */ [t/6: n in [0..160] | IsIntegral(t/6) where t is n*(n+1)/2]; // Bruno Berselli, Mar 07 2016

f:=n-> 8*(n-floor(n/4))+(-1)^n:seq((f(n)^2-1)/48,n=0..51); # Gary Detlefs, Mar 01 2010

lst={}; s=0; Do[s+=n/6; If[Floor[s]==s, AppendTo[lst, s]], {n, 0, 7!}]; lst (* Orlovsky *)
Join[{0}, Select[Table[Plus@@Range[n]/6, {n, 200}], IntegerQ]] (* Alonso del Arte, Jan 20 2012 *)
LinearRecurrence[{3,-5,7,-7,5,-3,1},{0,1,6,11,13,20,35},60] (* Charles R Greathouse IV, Jan 20 2012 *)
a[ n_] := (3 n^2 + If[ OddQ[ Quotient[ n + 1, 2]], -5 n + 2, -n]) / 4; (* Michael Somos, Feb 10 2015 *)
a[ n_] := Module[{m = n}, If[ n < 1, m = 1 - n]; SeriesCoefficient[ x^2 (1 + 4 x + x^2) (1 - x^2) (1 - x^6) / ((1 - x)^2 (1 - x^3) (1 - x^4)^2), {x, 0, m}]]; (* Michael Somos, Feb 10 2015 *)

a(n)=n--;(8*(n-n\4)+(-1)^n)^2\48 \\ Charles R Greathouse IV, Jan 02 2012

{a(n) = (3*n^2 + if( (n+1)\2%2, -5*n+2,-n)) / 4}; /* Michael Somos, Feb 10 2015 */

{a(n) = if( n<1, n = 1-n); polcoeff( x^2 * (1 + 4*x + x^2) * (1 - x^2) * (1 - x^6) / ((1 - x)^2 * (1 - x^3) * (1 - x^4)^2) + x * O(x^n), n)}; /* Michael Somos, Feb 10 2015 */

From R. J. Mathar, Jan 07 2009: (Start)
a(n) = A000217(A108752(n))/6.
G.f.: x^2*(x^2-x+1)*(x^2+4*x+1)/((1+x^2)^2*(1-x)^3) (conjectured). (End)
The conjectured g.f. is correct. - Charles R Greathouse IV, Jan 02 2012
a(n) = (f(n)^2-1)/48 where f(n) = 8*(n-floor(n/4))+(-1)^n, with offset 0, a(0)=0. - Gary Detlefs, Mar 01 2010
a(n) = a(1-n) for all n in Z. - Michael Somos, Oct 27 2012
G.f.: x^2 * (1 + 4*x + x^2) * (1 - x^2) * (1 - x^6) / ((1 - x)^2 * (1 - x^3) * (1 - x^4)^2). - Michael Somos, Feb 10 2015
Sum_{n>=2} 1/a(n) = 12 - (1+4/sqrt(3))*Pi. - Amiram Eldar, Mar 18 2022
a(n) = A069497(n)/6. - Hugo Pfoertner, Nov 19 2024
From Peter Bala, Jan 21 2025: (Start)
a(4*n) = 12*n^2 - n; a(4*n+1) = 12*n^2 + n;
a(4*n+2) = (3*n + 1)*(4*n + 1) = A033577(n); a(4*n+3) = (3*n + 2)*(4*n + 3) = A033578(n+1).
Let T(n) = n*(n + 1)/2 denote the n-th triangular number. Then
a(4*n) = (1/6) * T(12*n-1); a(4*n+1) = (1/6) * T(12*n);
a(4*n+2) = (1/6) * T(12*n+3); a(4*n+3) = (1/6) * T(12*n+8). (End)

Definition rewritten by M. F. Hasler, Dec 31 2012

A069497 Triangular numbers of the form 6*k.

Original entry on oeis.org

0, 6, 36, 66, 78, 120, 210, 276, 300, 378, 528, 630, 666, 780, 990, 1128, 1176, 1326, 1596, 1770, 1830, 2016, 2346, 2556, 2628, 2850, 3240, 3486, 3570, 3828, 4278, 4560, 4656, 4950, 5460, 5778, 5886, 6216, 6786, 7140, 7260, 7626, 8256, 8646, 8778, 9180

Offset: 1

Amarnath Murthy, Mar 30 2002

Index entries for linear recurrences with constant coefficients, signature (3,-5,7,-7,5,-3,1).

Cf. A000217, A108752, A112652, A154293.

a[0] := 0:a[1] := 3:a[2] := 8:a[3] := 11:seq((12*(floor(i/4))+a[i mod 4])*(12*(floor(i/4))+a[i mod 4]+1)/2,i=0..100);

CoefficientList[ Series[ 6x (x^2 -x +1) (x^2 +4x +1)/((x^2 +1)^2*(1 -x)^3), {x, 0, 45}], x] (* or *)
LinearRecurrence[{3, -5, 7, -7, 5, -3, 1}, {0, 6, 36, 66, 78, 120, 210}, 46] (* Robert G. Wilson v, May 31 2017 *)
Select[Accumulate[Range[0, 89]], Divisible[#, 6] &] (* Alonso del Arte, May 31 2017 *)

a(n) = 6 * A154293(n). - Joerg Arndt, Aug 18 2022
a(n) = A000217(A112652(n+1)-1). - R. J. Mathar, Aug 21 2007
From R. J. Mathar, Nov 18 2009: (Start)
a(n) = 3*a(n-1) - 5*a(n-2) + 7*a(n-3) - 7*a(n-4) + 5*a(n-5) - 3*a(n-6) + a(n-7).
G.f.: 6*x*(x^2-x+1)*(x^2+4*x+1)/((1+x^2)^2*(1-x)^3) (6*A154293). (End)
From Amiram Eldar, Aug 18 2022: (Start)
a(n) = A000217(A108752(n)).
Sum_{n>=2} 1/a(n) = 2 - (3+4*sqrt(3))*Pi/18. (End)

More terms from Sascha Kurz, Apr 01 2002
More terms from R. J. Mathar, Aug 21 2007
Offset corrected to 1, Joerg Arndt, Aug 18 2022

A072833 Numbers that are congruent to 0, 5, 8, 9 mod 12.

Original entry on oeis.org

0, 5, 8, 9, 12, 17, 20, 21, 24, 29, 32, 33, 36, 41, 44, 45, 48, 53, 56, 57, 60, 65, 68, 69, 72, 77, 80, 81, 84, 89, 92, 93, 96, 101, 104, 105, 108, 113, 116, 117, 120, 125, 128, 129, 132, 137, 140, 141, 144, 149, 152, 153, 156, 161, 164, 165, 168, 173, 176, 177, 180, 185, 188, 189

Offset: 0

N. J. A. Sloane, Jul 25 2002

The exponents occurring in the expansion of F_6(q^2) (see Ahlgren) or, equivalently, the norms of the vectors in the A*5 lattice. - _Andrey Zabolotskiy, Oct 26 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000
Scott Ahlgren, The sixth, eighth, ninth and tenth powers of Ramanujan's theta function, Proc. Amer. Math. Soc. 128 (2000), 1333-1338.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A023917, A108752, A072835.

f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; F[6, q_]:= ( -3*f[q, q]^5 + 5*f[q, q]^3*f[q^3, q^3]^2 + 15*f[q, q]*f[q^3, q^3]^4 + 15*f[q^3, q^3]^6/f[q, q]  )/32; cfs = CoefficientList[Series[F[6, q], {q, 0, 500}], q]; Take[Pick[Range[Length[cfs]] - 1, Sign[Abs[cfs]], 1], 50] (* G. C. Greubel, Apr 16 2018 *)
Flatten[#+{0,5,8,9}&/@(12*Range[0,20])] (* Harvey P. Dale, Apr 10 2022 *)

G.f.: x*(3*x^2-2*x+5) / ((x-1)^2*(x^2+1)). - Colin Barker, Jul 31 2013
Sum_{n>=1} 1/a(n) = Pi*(3-2*sqrt(3))/72 + log(2)/2 - arccoth(sqrt(3))/(2*sqrt(3)). - Amiram Eldar, Jul 26 2024
E.g.f.: exp(x)*(1 + 3*x) - cos(x) + sin(x). - Stefano Spezia, Oct 27 2024

Terms a(33) onward added by G. C. Greubel, Apr 16 2018

Edited by Andrey Zabolotskiy, Aug 14 2020

A193828 Even generalized pentagonal numbers.

Original entry on oeis.org

0, 2, 12, 22, 26, 40, 70, 92, 100, 126, 176, 210, 222, 260, 330, 376, 392, 442, 532, 590, 610, 672, 782, 852, 876, 950, 1080, 1162, 1190, 1276, 1426, 1520, 1552, 1650, 1820, 1926, 1962, 2072, 2262, 2380, 2420, 2542, 2752, 2882, 2926, 3060, 3290, 3432, 3480

Offset: 0

Omar E. Pol, Aug 19 2011

Even numbers in A001318.

Exponents in the expansion of Sum_{n >= 0} q^(2*n)/(Product_{k = 1..2*n} 1 + q^(2*k)) = 1 + q^2 - q^12 - q^22 + q^26 + q^40 - - + + ... (follows from Berndt et al., Theorem 3.3). Cf. A067589. - Peter Bala, Jan 21 2025

G. C. Greubel, Table of n, a(n) for n = 0..1000
B. C. Berndt, B. Kim, and A. J. Yee, Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions, J. Comb. Thy. Ser. A, 117 (2010), 957-973.
Mircea Merca, The bisectional pentagonal number theorem, Journal of Number Theory, Volume 157 (December 2015), Pages 223-232.
Index entries for linear recurrences with constant coefficients, signature (3,-5,7,-7,5,-3,1).

Cf. A001318, A067589, A154293.

CoefficientList[Series[-2*x*(x^2 - x + 1)*(x^2 + 4*x + 1)/((x - 1)^3*(x^2 + 1)^2), {x, 0, 50}], x] (* G. C. Greubel, Jun 06 2017 *)
LinearRecurrence[{3,-5,7,-7,5,-3,1},{0,2,12,22,26,40,70},50] (* Harvey P. Dale, Apr 09 2019 *)

my(x='x+O('x^50)); concat([0], Vec(-2*x*(x^2-x+1)*(x^2+4*x+1)/((x-1)^3*(x^2+1)^2))) \\ G. C. Greubel, Jun 06 2017

a(n) = A000217(A108752(n+1))/3 = 2*A154293(n+1).
G.f.: -2*x*(x^2-x+1)*(x^2+4*x+1)/((x-1)^3*(x^2+1)^2). - Colin Barker, Sep 12 2012
Sum_{n>=1} 1/a(n) = 6 - (1+4/sqrt(3))*Pi/2. - Amiram Eldar, Mar 18 2022

A287765 Period 4: repeat [1, 3, 5, 3].

Original entry on oeis.org

1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1

Offset: 1

Robert G. Wilson v, May 31 2017

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

Inspired by the first difference of A108752.

PadRight[{}, 105, {1, 3, 5, 3}]
CoefficientList[Series[(3 x^2 + 2 x + 1)/(-x^3 + x^2 - x + 1), {x, 0, 104}], x]
LinearRecurrence[{1, -1, 1}, {1, 3, 5}, 105]
RecurrenceTable[{a[n] == a[n - 1] - a[n - 2] + a[n - 3], a[1] == 1, a[2] == 3, a[3] == 5}, a, {n, 105}]
Table[{1, 3, 5, 3}, 10] // Flatten (* Eric W. Weisstein, Feb 07 2025 *)
Table[3 - 2 Sin[n Pi/2], {n, 20}] (* Eric W. Weisstein, Feb 07 2025 *)
3 - 2 Sin[Range[20] Pi/2] (* Eric W. Weisstein, Feb 07 2025 *)

G.f.: x * (3*x^2+2*x+1) / (1-x+x^2-x^3). [Corrected by Georg Fischer, May 19 2019]
a(n) = a(n-1) - a(n-2) + a(n-3) with a(1)=1, a(2)=3 and a(3)=5.
a(2n) = 3, a(4*n+1) = 1 and a(4*n+3) = 5.
a(n) = ((n+3) mod 4) + ((n+4) mod 4). - Aaron J Grech, Aug 30 2024

A108760 Irregular array: n-th row consists of nonnegative integers i less than n such that n divides i(i+1).

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 0, 4, 0, 2, 3, 5, 0, 6, 0, 7, 0, 8, 0, 4, 5, 9, 0, 10, 0, 3, 8, 11, 0, 12, 0, 6, 7, 13, 0, 5, 9, 14, 0, 15, 0, 16, 0, 8, 9, 17, 0, 18, 0, 4, 15, 19, 0, 6, 14, 20, 0, 10, 11, 21, 0, 22, 0, 8, 15, 23, 0, 24, 0, 12, 13, 25, 0, 26, 0, 7, 20, 27, 0, 28, 0, 5, 9, 14, 15, 20, 24, 29

Offset: 2

Robert Phillips (bobp(AT)usca.edu), Jun 24 2005

Row n starts with 0 and ends with n-1.
Row n of this irregular array can be viewed as the first row of an infinite matrix with elements a_{j,i} = T(n,i)+n*j. That matrix consists of all nonnegative integers i such that n divides i(i+1).
I use these matrices to generate subsequences of A012132, as you may see on page 9 of my referenced work.

			Row 12 is 0,3,8,11 which is the first row of the matrix:
   0  3  8 11
  12 15 20 23
  24 27 32 35
  ...
giving all nonnegative integers i such that 12 divides i(i+1) (cf. A108752).
Array begins:
  0, 1;
  0, 2;
  0, 3;
  0, 4;
  0, 2, 3, 5;
  0, 6;
  0, 7;
  0, 8;
  0, 4, 5, 9;
  ...

Robert Phillips, Triangular Numbers which are sums of two Triangular Numbers

Cf. A012132, A108752.

[i for n in range(2, 30) for i in range(0, n) if i*(i+1)%n==0] # Andrey Zabolotskiy, Mar 19 2022

Edited by Andrey Zabolotskiy, Mar 19 2022

cp's OEIS Frontend

A154293 Integers of the form t/6, where t is a triangular number (A000217).

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A069497 Triangular numbers of the form 6*k.

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A072833 Numbers that are congruent to 0, 5, 8, 9 mod 12.

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A193828 Even generalized pentagonal numbers.

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A287765 Period 4: repeat [1, 3, 5, 3].

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A108760 Irregular array: n-th row consists of nonnegative integers i less than n such that n divides i(i+1).

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