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

0,1

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).

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

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 *)

Prog

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

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).

Sequence in context: A333858 A319966 A200773 * A063235 A063151 A297882

Adjacent sequences: A269041 A269042 A269043 * A269045 A269046 A269047

Keyword

nonn,easy

Author

Bruno Berselli, Feb 18 2016

Status

approved