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A128880 Triangular numbers congruent to 1 or 5 mod 6. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 30 05:55 EST 2023. Contains 359939 sequences. (Running on oeis4.)