A131229 Numbers congruent to {1,7} mod 10.

1, 7, 11, 17, 21, 27, 31, 37, 41, 47, 51, 57, 61, 67, 71, 77, 81, 87, 91, 97, 101, 107, 111, 117, 121, 127, 131, 137, 141, 147, 151, 157, 161, 167, 171, 177, 181, 187, 191, 197, 201, 207, 211, 217, 221, 227, 231, 237, 241, 247, 251, 257, 261, 267, 271, 277, 281

Offset

1,2

Comments

Row sums of triangle A131228.

Take the sum of the squares of the first n triangular numbers and divide it by the sum of these n triangular numbers. The sum evenly divides the sum of the squares for the n in this sequence. - J. M. Bergot, May 09 2012

a(n) = the difference between the sum of the terms in antidiagonal(n) and antidiagonal(n-1) in A204008. - J. M. Bergot, Jul 15 2013

Links

David Lovler, Table of n, a(n) for n = 1..1000

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

Formula

From R. J. Mathar, Apr 17 2008: (Start)

a(n) = a(n-2) + 10 = 5*n + ((-1)^n - 7)/2.

G.f.: x*(1 + 6*x + 3*x^2)/((-1+x)^2*(1+x)). (End)

a(n) = 10*n - a(n-1) - 12 with n > 1, a(1)=1. - Vincenzo Librandi, Nov 23 2010

a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3; a(1)=1, a(2)=7, a(3)=11. - Harvey P. Dale, May 20 2012

a(n) = A063097(n) for n >= 2. - Georg Fischer, Nov 02 2018

E.g.f.: 3 + ((10*x - 7)*exp(x) + exp(-x))/2. - David Lovler, Sep 07 2022

Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt((5+sqrt(5))/2)*Pi/10 + 3*log(phi)/(2*sqrt(5)), where phi is the golden ratio (A001622). - Amiram Eldar, Apr 15 2023

Example

11 = a(3) = 1 mod 10.
11 = a(3) = row 4 sums, triangle A131228: (1 + 3 + 7).

Mathematica

Select[Range[300], MemberQ[{1, 7}, Mod[#, 10]]&] (* or *) LinearRecurrence[ {1, 1, -1}, {1, 7, 11}, 60] (* Harvey P. Dale, May 20 2012 *)

Prog

(PARI) is(n)=n%10==1 || n%10==7 \\ Charles R Greathouse IV, Jul 16 2013

Crossrefs

Cf. A001622, A128174, A051340, A063097, A131228.

Sequence in context: A352630 A360396 A299978 * A092737 A072669 A108101

Adjacent sequences: A131226 A131227 A131228 * A131230 A131231 A131232

Keyword

nonn,easy

Author

Gary W. Adamson, Jun 20 2007

Extensions

More terms from R. J. Mathar, Apr 17 2008

Status

approved