cp's OEIS Frontend

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

%I A154293 #98 Jan 26 2025 21:06:56
%S A154293 0,1,6,11,13,20,35,46,50,63,88,105,111,130,165,188,196,221,266,295,
%T A154293 305,336,391,426,438,475,540,581,595,638,713,760,776,825,910,963,981,
%U A154293 1036,1131,1190,1210,1271,1376,1441,1463,1530,1645,1716,1740,1813,1938,2015
%N A154293 Integers of the form t/6, where t is a triangular number (A000217).
%C A154293 Old definition was "Integers of the form: 1/6+2/6+3/6+4/6+5/6+...".
%C A154293 1/6 + 2/6 + 3/6 = 1, 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6 + 7/6 + 8/6 = 6, ...
%C A154293 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
%C A154293 Conjecture: A193828 divided by 2. - _Omar E. Pol_, Aug 19 2011
%C A154293 The above conjecture is correct. - _Charles R Greathouse IV_, Jan 02 2012
%C A154293 Quasipolynomial of order 4. - _Charles R Greathouse IV_, Jan 02 2012
%C A154293 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
%C A154293 From _Peter Bala_, Dec 12 2024 (Start)
%C A154293 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.
%C A154293 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)
%C A154293 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
%H A154293 G. C. Greubel, <a href="/A154293/b154293.txt">Table of n, a(n) for n = 1..1000</a>
%H A154293 B. C. Berndt, B. Kim, and A. J. Yee, <a href="http://dx.doi.org/10.1016/j.jcta.2009.07.005">Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions</a>, J. Comb. Thy. Ser. A, 117 (2010), 957-973.
%H A154293 Mircea Merca, <a href="http://dx.doi.org/10.1016/j.jnt.2015.05.008">The bisectional pentagonal number theorem</a>, Journal of Number Theory, Volume 157 (December 2015), Pages 223-232.
%H A154293 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-5,7,-7,5,-3,1).
%F A154293 From _R. J. Mathar_, Jan 07 2009: (Start)
%F A154293 a(n) = A000217(A108752(n))/6.
%F A154293 G.f.: x^2*(x^2-x+1)*(x^2+4*x+1)/((1+x^2)^2*(1-x)^3) (conjectured). (End)
%F A154293 The conjectured g.f. is correct. - _Charles R Greathouse IV_, Jan 02 2012
%F A154293 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
%F A154293 a(n) = a(1-n) for all n in Z. - _Michael Somos_, Oct 27 2012
%F A154293 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
%F A154293 Sum_{n>=2} 1/a(n) = 12 - (1+4/sqrt(3))*Pi. - _Amiram Eldar_, Mar 18 2022
%F A154293 a(n) = A069497(n)/6. - _Hugo Pfoertner_, Nov 19 2024
%F A154293 From _Peter Bala_, Jan 21 2025: (Start)
%F A154293 a(4*n) = 12*n^2 - n; a(4*n+1) = 12*n^2 + n;
%F A154293 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).
%F A154293 Let T(n) = n*(n + 1)/2 denote the n-th triangular number. Then
%F A154293 a(4*n) = (1/6) * T(12*n-1); a(4*n+1) = (1/6) * T(12*n);
%F A154293 a(4*n+2) = (1/6) * T(12*n+3); a(4*n+3) = (1/6) * T(12*n+8). (End)
%e A154293 G.f. = x^2 + 6*x^3 + 11*x^4 + 13*x^5 + 20*x^6 + 35*x^7 + 46*x^8 + ...
%p A154293 f:=n-> 8*(n-floor(n/4))+(-1)^n:seq((f(n)^2-1)/48,n=0..51); # _Gary Detlefs_, Mar 01 2010
%t A154293 lst={}; s=0; Do[s+=n/6; If[Floor[s]==s, AppendTo[lst, s]], {n, 0, 7!}]; lst (* Orlovsky *)
%t A154293 Join[{0}, Select[Table[Plus@@Range[n]/6, {n, 200}], IntegerQ]] (* _Alonso del Arte_, Jan 20 2012 *)
%t A154293 LinearRecurrence[{3,-5,7,-7,5,-3,1},{0,1,6,11,13,20,35},60] (* _Charles R Greathouse IV_, Jan 20 2012 *)
%t A154293 a[ n_] := (3 n^2 + If[ OddQ[ Quotient[ n + 1, 2]], -5 n + 2, -n]) / 4; (* _Michael Somos_, Feb 10 2015 *)
%t A154293 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 *)
%o A154293 (PARI) a(n)=n--;(8*(n-n\4)+(-1)^n)^2\48 \\ _Charles R Greathouse IV_, Jan 02 2012
%o A154293 (PARI) {a(n) = (3*n^2 + if( (n+1)\2%2, -5*n+2,-n)) / 4}; /* _Michael Somos_, Feb 10 2015 */
%o A154293 (PARI) {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 */
%o A154293 (Magma) /* By definition: */ [t/6: n in [0..160] | IsIntegral(t/6) where t is n*(n+1)/2]; // _Bruno Berselli_, Mar 07 2016
%Y A154293 Cf. A000217, A001318, A069497, A074378, A057569, A057570, A154292, A193828, A218171.
%K A154293 nonn,easy
%O A154293 1,3
%A A154293 _Vladimir Joseph Stephan Orlovsky_, Jan 06 2009
%E A154293 Definition rewritten by _M. F. Hasler_, Dec 31 2012