A128880 Triangular numbers congruent to 1 or 5 mod 6.

1, 55, 91, 253, 325, 595, 703, 1081, 1225, 1711, 1891, 2485, 2701, 3403, 3655, 4465, 4753, 5671, 5995, 7021, 7381, 8515, 8911, 10153, 10585, 11935, 12403, 13861, 14365, 15931, 16471, 18145, 18721, 20503, 21115, 23005, 23653, 25651, 26335, 28441

Offset

1,2

Comments

Or, except for the first term, triangular numbers the least prime factor of which is >=5.

There are no triangular numbers that are congruent to 5 mod 6. - Amiram Eldar, Aug 18 2022

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

a(1)=Tr(1), a(2)=Tr(10), where Tr(k)=k(k+1)/2 is triangular number; for n>=3 a(n)=Tr(k(n)), where k(n)=k(n-2)+12 with k(1)=1, k(2)=10.

G.f.: -x*(1+54*x+34*x^2+54*x^3+x^4) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Jul 07 2015

From Colin Barker, Jan 26 2016: (Start)

a(n) = (36*n^2+18*(-1)^n*n-36*n-9*(-1)^n+11)/2.

a(n) = 18*n^2-9*n+1 for n even.

a(n) = 18*n^2-27*n+10 for n odd.

(End)

Sum_{n>=1} 1/a(n) = Pi/3. - Amiram Eldar, Aug 18 2022

Mathematica

c=0; Do[tr=n(n+1)/2; If[Abs[Mod[tr, 6]]==1, c++; a[c]=tr], {n, 300}]; Table[a[i], {i, c}]
Select[Accumulate[Range[500]], MemberQ[{1, 5}, Mod[#, 6]]&] (* Harvey P. Dale, Sep 28 2013 *)

Prog

(PARI) Vec(-x*(1+54*x+34*x^2+54*x^3+x^4)/((1+x)^2*(x-1)^3) + O(x^100)) \\ Colin Barker, Jan 26 2016

Crossrefs

Intersection of A000217 and A007310.

Sequence in context: A111192 A063873 A063131 * A039596 A013543 A115377

Adjacent sequences: A128877 A128878 A128879 * A128881 A128882 A128883

Keyword

nonn,easy

Author

Zak Seidov, Apr 18 2007, Apr 25 2007

Status

approved