A269044 a(n) = 13*n + 7.

7, 20, 33, 46, 59, 72, 85, 98, 111, 124, 137, 150, 163, 176, 189, 202, 215, 228, 241, 254, 267, 280, 293, 306, 319, 332, 345, 358, 371, 384, 397, 410, 423, 436, 449, 462, 475, 488, 501, 514, 527, 540, 553, 566, 579, 592, 605, 618, 631, 644, 657, 670, 683, 696, 709, 722, 735

Offset: 0

Author

Bruno Berselli, Feb 18 2016

Keywords

• nonn
• easy

Comments

After 7 (which corresponds to n=0), all terms belong to A090767 because a(n) = 3*n*2*1 + 2*(n*2+2*1+n*1) + (n+2+1).

This sequence is related to A152741 by the recurrence A152741(n+1) = (n+1)*a(n+1) - Sum_{k = 0..n} a(k).

Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 (A010376) but not 7, and for this reason there are no squares in the sequence. Likewise, any cube mod 13 is one of 0, 1, 5, 8 or 12, therefore no a(k) is a cube.

The sum of the squares of any two terms of the sequence is also a term of the sequence, that is: a(h)^2 + a(k)^2 = a(h*(13*h+14) + k*(13*k+14) + 7). Therefore: a(h)^2 + a(k)^2 > a(a( h*(h+1) + k*(k+1) )) for h+k > 0.

The primes of the sequence are listed in A140371.

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Bruno Berselli, A description of the recursive method shown in the third comment: website Matem@ticamente (in Italian), 2008.
• Tanya Khovanova, Recursive Sequences.
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A010376, A022271 (partial sums), A088227, A090767, A140371, A152741.

Similar sequences with closed form (2*k-1)*n+k: A001489 (k=0), A000027 (k=1), A016789 (k=2), A016885 (k=3), A017029 (k=4), A017221 (k=5), A017461 (k=6), this sequence (k=7), A164284 (k=8).

Sequences of the form 13*n+q: A008595 (q=0), A190991 (q=1), A153080 (q=2), A127547 (q=4), A154609 (q=5), A186113 (q=6), this sequence (q=7), A269100 (q=11).

Programs

Magma

[13*n+7: n in [0..60]];

Mathematica

13 Range[0, 60] + 7 (* or *) Range[7, 800, 13] (* or *) Table[13 n + 7, {n, 0, 60}]
LinearRecurrence[{2, -1}, {7, 20}, 60] (* Vincenzo Librandi, Feb 19 2016 *)

Maxima

makelist(13*n+7, n, 0, 60);

PARI

vector(60, n, n--; 13*n+7)

Sage

[13*n+7 for n in (0..60)]

Formula

G.f.: (7 + 6*x)/(1 - x)^2.

a(n) = A088227(4*n+3).

a(n) = -A186113(-n-1).

Sum_{i=h..h+13*k} a(i) = a(h*(13*k + 1) + k*(169*k + 27)/2).

Sum_{i>=0} 1/a(i)^2 = 0.0257568950542502716970... = polygamma(1, 7/13)/13^2.

E.g.f.: exp(x)*(7 + 13*x). - Stefano Spezia, Aug 02 2021