A252994 -id:A252994 - cp's OEIS frontend

A008595 Multiples of 13.

0, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 169, 182, 195, 208, 221, 234, 247, 260, 273, 286, 299, 312, 325, 338, 351, 364, 377, 390, 403, 416, 429, 442, 455, 468, 481, 494, 507, 520, 533, 546, 559, 572, 585, 598, 611, 624, 637, 650, 663, 676

Offset: 0

N. J. A. Sloane

Complement of A113763. - Reinhard Zumkeller, Apr 26 2011

Ivan Panchenko, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 325.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008594, A017533, A017545, A076310, A113763, A252994.

A008595:=n->13*n; seq(A008595(n), n=0..100); # Wesley Ivan Hurt, Jan 30 2014

Range[0, 1000, 13] (* Vladimir Joseph Stephan Orlovsky, May 29 2011 *)

a(n)=13*n \\ Charles R Greathouse IV, Jul 10 2016

(floor(a(n)/10) + 4*(a(n) mod 10)) == 0 modulo 13, see A076310. - Reinhard Zumkeller, Oct 06 2002
From Vincenzo Librandi, Dec 24 2010: (Start)
a(n) = 13*n.
a(n) = 2*a(n-1) - a(n-2).
G.f.: 13*x/(x-1)^2. (End)
From Elmo R. Oliveira, Apr 08 2025: (Start)
E.g.f.: 13*x*exp(x).
a(n) = A252994(n)/2. (End)

A244633 a(n) = 26*n^2.

Original entry on oeis.org

0, 26, 104, 234, 416, 650, 936, 1274, 1664, 2106, 2600, 3146, 3744, 4394, 5096, 5850, 6656, 7514, 8424, 9386, 10400, 11466, 12584, 13754, 14976, 16250, 17576, 18954, 20384, 21866, 23400, 24986, 26624, 28314, 30056, 31850, 33696, 35594, 37544, 39546, 41600, 43706

Offset: 0

Vincenzo Librandi, Jul 03 2014

Sequence found by reading the line from 0, in the direction 0, 26, ..., in the square spiral whose vertices are the generalized 15-gonal numbers. - Omar E. Pol, Jul 03 2014

Norms of purely imaginary numbers in Z[sqrt(-26)]. - Alonso del Arte, Dec 25 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. similar sequences listed in A244630.

Cf. A000290, A001105, A005843, A008595, A152742, A252994, A277082.

Magma
```
[26*n^2: n in [0..40]];
```

A244633:=n->26*n^2: seq(A244633(n), n=0..50); # Wesley Ivan Hurt, Jul 04 2014

Table[26 n^2, {n, 0, 40}]
26 Range[0, 50]^2 (* Wesley Ivan Hurt, Jul 04 2014 *)

a(n)=26*n^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 26*x*(1 + x)/(1 - x)^3. [corrected by Bruno Berselli, Jul 03 2014]
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = 26*A000290(n) = 13*A001105(n) = 2*A152742(n). - Omar E. Pol, Jul 03 2014
From Elmo R. Oliveira, Dec 02 2024: (Start)
E.g.f.: 26*x*(1 + x)*exp(x).
a(n) = n*A252994(n) = A005843(n)*A008595(n). (End)

A305548 a(n) = 27*n.

Original entry on oeis.org

0, 27, 54, 81, 108, 135, 162, 189, 216, 243, 270, 297, 324, 351, 378, 405, 432, 459, 486, 513, 540, 567, 594, 621, 648, 675, 702, 729, 756, 783, 810, 837, 864, 891, 918, 945, 972, 999, 1026, 1053, 1080, 1107, 1134, 1161, 1188, 1215, 1242, 1269, 1296, 1323, 1350, 1377, 1404, 1431, 1458, 1485, 1512

Offset: 0

Eric Chen, Jun 05 2018

Eric Chen, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

For a(n) = k*n: A001489 (k=-1), A000004 (k=0), A001477 (k=1), A005843 (k=2), A008585 (k=3), A008591 (k=9), A008607 (k=25), A252994 (k=26), this sequence (k=27), A135628 (k=28), A195819 (k=29), A249674 (k=30), A135631 (k=31), A174312 (k=32), A044102 (k=36), A085959 (k=37), A169823 (k=60), A152691 (k=64).

Mathematica
```
Range[0,2000,27]
```
PARI
```
a(n)=27*n
```

a(n) = 27*n.
a(n) = A008585(A008591(n)) = A008591(A008585(n)).
G.f.: 27*x/(x-1)^2.
From Elmo R. Oliveira, Apr 10 2025: (Start)
E.g.f.: 27*x*exp(x).
a(n) = 2*a(n-1) - a(n-2). (End)

A317326 Multiples of 26 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 26, 3, 52, 5, 78, 7, 104, 9, 130, 11, 156, 13, 182, 15, 208, 17, 234, 19, 260, 21, 286, 23, 312, 25, 338, 27, 364, 29, 390, 31, 416, 33, 442, 35, 468, 37, 494, 39, 520, 41, 546, 43, 572, 45, 598, 47, 624, 49, 650, 51, 676, 53, 702, 55, 728, 57, 754, 59, 780, 61, 806, 63, 832, 65, 858, 67, 884, 69

Offset: 0

Omar E. Pol, Jul 25 2018

a(n) is the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 30-gonal numbers (A316729).
Partial sums give the generalized 30-gonal numbers.
More generally, the partial sums of the sequence formed by the multiples of m and the odd numbers interleaved, give the generalized k-gonal numbers, with m >= 1 and k = m + 4.
From Bruno Berselli, Jul 27 2018: (Start)
Also, this type of sequence is characterized by:
O.g.f.: x*(1 + m*x + x^2)/(1 - x^2)^2;
E.g.f.: x*(2 - m + (2 + m)*exp(2*x))*exp(-x)/4;
a(n) = -a(-n) = (2 + m - (2 - m)*(-1)^n)*n/4;
a(n) = (m/2)^((1 + (-1)^n)/2)*n;
a(n) = 2*a(n-2) - a(n-4), with signature (0,2,0,-1). (End)

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A252994 and A005408 interleaved.
Column 26 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14), A317311 (k=15), A317312 (k=16), A317313 (k=17), A317314 (k=18), A317315 (k=19), A317316 (k=20), A317317 (k=21), A317318 (k=22), A317319 (k=23), A317320 (k=24), A317321 (k=25), A317322 (k=26), A317323 (k=27), A317324 (k=28), A317325 (k=29), this sequence (k=30).
Cf. A316729.

[13^div(1+(-1)^n,2)*n for n in 0:70] |> println # Bruno Berselli, Jul 28 2018

Table[(7 + 6 (-1)^n) n, {n, 0, 70}] (* Bruno Berselli, Jul 27 2018 *)

a(2*n) = 26*n, a(2*n+1) = 2*n + 1.
From Bruno Berselli, Jul 27 2018: (Start)
O.g.f.: x*(1 + 26*x + x^2)/(1 - x^2)^2.
E.g.f.: x*(-6 + 7*exp(2*x))*exp(-x).
a(n) = -a(-n) = (7 + 6*(-1)^n)*n.
a(n) = 13^((1 + (-1)^n)/2)*n.
a(n) = 2*a(n-2) - a(n-4). (End)
Multiplicative with a(2^e) = 13*2^e, and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 3*2^(3-s)). - Amiram Eldar, Oct 26 2023

A099943 Number of 5 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01,1) and (11;0).

Original entry on oeis.org

72, 98, 124, 150, 176, 202, 228, 254, 280, 306, 332, 358, 384, 410, 436, 462, 488, 514, 540, 566, 592, 618, 644, 670, 696, 722, 748, 774, 800, 826, 852, 878, 904, 930, 956, 982, 1008, 1034, 1060, 1086, 1112, 1138, 1164, 1190, 1216, 1242, 1268, 1294, 1320

Offset: 2

Sergey Kitaev, Nov 12 2004

An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i11 and n>1.

Tanya Khovanova, Recursive Sequences.
S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016957 (m=2), A008592 (m=3), A063130 (m=4).

Cf. A252994.

Range[72, 7000, 26] (* Vladimir Joseph Stephan Orlovsky, Jul 13 2011 *)

a(n) = 26*n + 20.
From Elmo R. Oliveira, Jul 01 2025: (Start)
G.f.: 2*x^2*(36-23*x)/(1-x)^2.
E.g.f.: 2*(exp(x)*(10 + 13*x) - (10 + 23*x)).
a(n) = 2*a(n-1) - a(n-2) for n > 3.
a(n) = A252994(n) + 20. (End)

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A008595 Multiples of 13.

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A244633 a(n) = 26*n^2.

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A305548 a(n) = 27*n.

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A317326 Multiples of 26 and odd numbers interleaved.

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Julia

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A099943 Number of 5 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01,1) and (11;0).

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