A135628 -id:A135628 - cp's OEIS frontend

A008596 Multiples of 14.

0, 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, 182, 196, 210, 224, 238, 252, 266, 280, 294, 308, 322, 336, 350, 364, 378, 392, 406, 420, 434, 448, 462, 476, 490, 504, 518, 532, 546, 560, 574, 588, 602, 616, 630, 644, 658, 672, 686, 700, 714, 728

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 326.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008589, A008594, A008595, A033571, A135628, A158186.

Range[0, 1000, 14] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
CoefficientList[Series[14 x / (x - 1)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 10 2013 *)

a(n)=14*n \\ Charles R Greathouse IV, Sep 28 2015

a(n) = A033571(n) - A158186(n). - Reinhard Zumkeller, Mar 13 2009
From R. J. Mathar, Jun 23 2009: (Start)
a(n) = 14*n.
a(n) = 2*a(n-1) - a(n-2).
G.f.: 14*x/(x-1)^2. (End)
From Elmo R. Oliveira, Apr 08 2025: (Start)
E.g.f.: 14*x*exp(x).
a(n) = 2*A008589(n) = A135628(n)/2. (End)

A135631 Multiples of 31.

Original entry on oeis.org

0, 31, 62, 93, 124, 155, 186, 217, 248, 279, 310, 341, 372, 403, 434, 465, 496, 527, 558, 589, 620, 651, 682, 713, 744, 775, 806, 837, 868, 899, 930, 961, 992, 1023, 1054, 1085, 1116, 1147, 1178, 1209, 1240, 1271, 1302, 1333, 1364, 1395, 1426, 1457, 1488

Offset: 0

Omar E. Pol, Nov 25 2007

a(1) = 31 is the third Mersenne prime. a(8) = 248 is the dimensions of E_8. a(16) = 496 is the third perfect number. - Pol
a(n)^340 = 155 mod 341 unless a(n) is also divisible by 11. - Alonso del Arte, Feb 15 2012
Number of sides on n triacontakaihenagons (31-gons). - Wesley Ivan Hurt, Oct 25 2016

			a(8) = 31 * 8 = 248. a(16) = 31 * 16 = 496.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081, 2014
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A135628.

with(numtheory):a:=proc(n) if n=0 then 0 else mcombine(7*n,3*n,5*n,11*n) fi end: seq(a(n), n=0..45); # Zerinvary Lajos, Apr 11 2008

Range[0, 3000, 31] (* Vladimir Joseph Stephan Orlovsky, Jun 03 2011 *)
LinearRecurrence[{2,-1},{0,31},25] (* G. C. Greubel, Oct 24 2016 *)

a(n) = 31*n.
From G. C. Greubel, Oct 24 2016: (Start)
G.f.: (31*x)/(1 - x)^2.
E.g.f.: 31*x*exp(x).
a(n) = 2*a(n-1) - a(n-2). (End)

A174312 a(n) = 32*n.

Original entry on oeis.org

0, 32, 64, 96, 128, 160, 192, 224, 256, 288, 320, 352, 384, 416, 448, 480, 512, 544, 576, 608, 640, 672, 704, 736, 768, 800, 832, 864, 896, 928, 960, 992, 1024, 1056, 1088, 1120, 1152, 1184, 1216, 1248, 1280, 1312, 1344, 1376, 1408, 1440, 1472, 1504, 1536, 1568, 1600

Offset: 0

Paul Curtz, Nov 27 2010

Subsequence of squares is A017066 (see 2nd formula). - Bernard Schott, Mar 03 2023

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

Cf. A001105, A008598, A017066, A135628, A135631, A152691.

[32*n: n in [0..60]]; // Vincenzo Librandi, Aug 04 2011

Range[0, 3000, 32] (* Vladimir Joseph Stephan Orlovsky, Jun 03 2011 *)

a(n)=n<<5 \\ Charles R Greathouse IV, Dec 21 2011

G.f.: 32*x/(1-x)^2.
a(A001105(n)) = A017066(n). - Bernard Schott, Mar 05 2023
From Elmo R. Oliveira, Apr 07 2025: (Start)
E.g.f.: 32*x*exp(x).
a(n) = 2*A008598(n) = A152691(n)/2.
a(n) = 2*a(n-1) - a(n-2). (End)

A274824 Triangle read by rows: T(n,k) = (n-k+1)*sigma(k), n>=1, 1<=k<=n.

Original entry on oeis.org

1, 2, 3, 3, 6, 4, 4, 9, 8, 7, 5, 12, 12, 14, 6, 6, 15, 16, 21, 12, 12, 7, 18, 20, 28, 18, 24, 8, 8, 21, 24, 35, 24, 36, 16, 15, 9, 24, 28, 42, 30, 48, 24, 30, 13, 10, 27, 32, 49, 36, 60, 32, 45, 26, 18, 11, 30, 36, 56, 42, 72, 40, 60, 39, 36, 12, 12, 33, 40, 63, 48, 84, 48, 75, 52, 54, 24, 28, 13, 36, 44, 70, 54, 96, 56, 90, 65, 72, 36, 56, 14

Offset: 1

Omar E. Pol, Oct 02 2016

Theorem: for any sequence S the partial sums of the partial sums are also the antidiagonal sums of the square array in which the n-th row gives n times the sequence S. Therefore they are also the row sums of the triangular array in which the n-th diagonal gives n times the sequence S.
In this case the sequence S is A000203.
The n-th diagonal of this triangle gives n times A000203.
The row sums give A175254 which gives the partial sums of A024916 which gives the partial sums of A000203.
T(n,k) is also the total number of unit cubes that are exactly below the terraces of the k-th level (starting from the top) up the base of the stepped pyramid with n levels described in A245092. This fact is because the mentioned terraces have the same shape as the symmetric representation of sigma(k). For more information see A237593 and A237270.
In the definition of this sequence the value n-k+1 is also the height of the terraces associated to sigma(k) in the mentioned pyramid with n levels, or in other words, the distance between the mentioned terraces and the base of the pyramid.
The sum of the n-th row of triangle equals the volume (also the number of cubes) of the mentioned pyramid with n levels.
For an illustration of the pyramid, see the Links section.
The sum of the n-th row is also 1/4 of the volume of the stepped pyramid described in A244050 with n levels.
Column k lists the positive multiples of sigma(k).
The k-th term in the n-th diagonal is equal to n*sigma(k).
Note that this is also a square array read by antidiagonals upwards: T(i,j) = i*sigma(j), i>=1, j>=1. The first row of the array is A000203. So consider that the pyramid is upside down. The value of "i" is the distance between the base of the pyramid and the terraces associated to sigma(j). The antidiagonal sums give the partial sums of the partial sums of A000203.

			Triangle begins:
1;
2,  3;
3,  6,  4;
4,  9,  8,  7;
5,  12, 12, 14, 6;
6,  15, 16, 21, 12, 12;
7,  18, 20, 28, 18, 24,  8;
8,  21, 24, 35, 24, 36,  16, 15;
9,  24, 28, 42, 30, 48,  24, 30,  13;
10, 27, 32, 49, 36, 60,  32, 45,  26,  18;
11, 30, 36, 56, 42, 72,  40, 60,  39,  36,  12;
12, 33, 40, 63, 48, 84,  48, 75,  52,  54,  24, 28;
13, 36, 44, 70, 54, 96,  56, 90,  65,  72,  36, 56,  14;
14, 39, 48, 77, 60, 108, 64, 105, 78,  90,  48, 84,  28, 24;
15, 42, 52, 84, 66, 120, 72, 120, 91,  108, 60, 112, 42, 48, 24;
16, 45, 56, 91, 72, 132, 80, 135, 104, 126, 72, 140, 56, 72, 48, 31;
...
For n = 16 and k = 10 the sum of the divisors of 10 is 1 + 2 + 5 + 10 = 18, and 16 - 10 + 1 = 7, and 7*18 = 126, so T(16,10) = 126.
On the other hand, the symmetric representation of sigma(10) has two parts of 9 cells, giving a total of 18 cells. In the stepped pyramid described in A245092, with 16 levels, there are 16 - 10 + 1 = 7 cubes exactly below every cell of the symmetric representation of sigma(10) up the base of pyramid; hence the total numbers of cubes exactly below the terraces of the 10th level (starting from the top) up the base of the pyramid is equal to 7*18 = 126. So T(16,10) = 126.
The sum of the 16th row of the triangle is 16 + 45 + 56 + 91 + 72 + 132 + 80 + 135 + 104 + 126 + 72 + 140 + 56 + 72 + 48 + 31 = A175254(16) = 1276, equaling the volume (also the number of cubes) of the stepped pyramid with 16 levels described in A245092 (see Links section).

Indranil Ghosh, Rows 1..100 of triangle, flattened
Omar E. Pol, Illustration of the stepped pyramid with 16 levels

Row sums of triangle give A175254.
Diagonals 1-6: A000203, A074400, A272027, A239050, A274535, A274536.
Column 1 is A000027.
Initial zeros should be omitted in the following sequences related to the columns of triangle:
Columns 2-5: A008585, A008586, A008589, A008588.
Columns 6 and 11: A008594.
Columns 7-9: A008590, A008597, A008595.
Columns 10 and 17: A008600.
Columns 12-13: A135628, A008596.
Columns 14, 15 and 23: A008606.
Columns 16 and 25: A135631.
(There are many other OEIS sequences that are also columns of this triangle.)
Cf. A004736, A024916, A054973, A196020, A236104, A235791, A237591, A237593, A237270, A237271, A244050, A245092, A245093, A262612.

T(n,k) = (n-k+1) * A000203(k).

T(n,k) = A004736(n,k) * A245093(n,k).

A064763 a(n) = 28*n^2.

Original entry on oeis.org

0, 28, 112, 252, 448, 700, 1008, 1372, 1792, 2268, 2800, 3388, 4032, 4732, 5488, 6300, 7168, 8092, 9072, 10108, 11200, 12348, 13552, 14812, 16128, 17500, 18928, 20412, 21952, 23548, 25200, 26908, 28672, 30492, 32368, 34300, 36288, 38332

Offset: 0

Roberto E. Martinez II, Oct 18 2001

Number of edges in a complete 8-partite graph of order 8n, K_n,n,n,n,n,n,n,n.

Sequence found by reading the line from 0, in the direction 0, 28, ..., in the square spiral whose vertices are the generalized 16-gonal numbers. - Omar E. Pol, Jul 03 2014

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Similar sequences are listed in A244630.
Cf. A000217, A000290, A001105, A016742, A033428, A033581, A033582, A033583.
Cf. A144555, A135628.

[28*n^2: n in [0..40]]; // Vincenzo Librandi, Mar 30 2015

I:=[0,28,112]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Mar 30 2015

CoefficientList[Series[28 x (1 + x) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 30 2015 *)

a(n)=28*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 56*n + a(n-1) - 28 (with a(0)=0). - Vincenzo Librandi, Aug 07 2010
a(n) = 28*A000290(n) = 14*A001105(n) = 7*A016742(n) = 4*A033582(n) = 2*A144555(n). - Omar E. Pol, Jul 03 2014
From Vincenzo Librandi, Mar 30 2015: (Start)
G.f.: 28*x*(1+x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = t(8*n) - 8*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(8*n) - 8*A000217(n). - Bruno Berselli, Aug 31 2017
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 28*x*(1 + x)*exp(x).
a(n) = n*A135628(n). (End)

A319073 Square array read by antidiagonals upwards: T(n,k) = k*sigma(n), n >= 1, k >= 1.

Original entry on oeis.org

1, 3, 2, 4, 6, 3, 7, 8, 9, 4, 6, 14, 12, 12, 5, 12, 12, 21, 16, 15, 6, 8, 24, 18, 28, 20, 18, 7, 15, 16, 36, 24, 35, 24, 21, 8, 13, 30, 24, 48, 30, 42, 28, 24, 9, 18, 26, 45, 32, 60, 36, 49, 32, 27, 10, 12, 36, 39, 60, 40, 72, 42, 56, 36, 30, 11, 28, 24, 54, 52, 75, 48, 84, 48, 63, 40, 33, 12

Offset: 1

Omar E. Pol, Sep 22 2018

			The corner of the square array begins:
         A000203 A074400 A272027 A239050 A274535 A274536 A319527 A319528
A000027:       1,      2,      3,      4,      5,      6,      7,      8, ...
A008585:       3,      6,      9,     12,     15,     18,     21,     24, ...
A008586:       4,      8,     12,     16,     20,     24,     28,     32, ...
A008589:       7,     14,     21,     28,     35,     42,     49,     56, ...
A008588:       6,     12,     18,     24,     30,     36,     42,     48, ...
A008594:      12,     24,     36,     48,     60,     72,     84,     96, ...
A008590:       8,     16,     24,     32,     40,     48,     56,     64, ...
A008597:      15,     30,     45,     60,     75,     90,    105,    120, ...
A008595:      13,     26,     39,     52,     65,     78,     91,    104, ...
A008600:      18,     36,     54,     72,     90,    108,    126,    144, ...
...

Index entries for sequences related to sigma(n)

Another version of A274824.
Antidiagonal sums give A175254.
Main diagonal gives A064987.
Row n lists the multiples of A000203(n).
Row 1 is A000027.
Initial zeros should be omitted in the following sequences related to the rows of the array:
Row 2-5: A008585, A008586, A008589, A008588.
Rows 6 and 11: A008594.
Rows 7-9: A008590, A008597, A008595.
Rows 10 and 17: A008600.
Rows 12-13: A135628, A008596.
Rows 14, 15 and 23: A008606.
Rows 16 and 25: A135631.
(Note that in the OEIS there are many other sequences that are also rows of this square array.)
Cf. A000203, A237593, A319526.

T:=Flat(List([1..12],n->List([1..n],k->k*Sigma(n-k+1))));; Print(T); # Muniru A Asiru, Jan 01 2019

with(numtheory): T:=(n,k)->k*sigma(n-k+1): seq(seq(T(n,k),k=1..n),n=1..12); # Muniru A Asiru, Jan 01 2019

Table[k DivisorSigma[1, #] &[m - k + 1], {m, 12}, {k, m}] // Flatten (* Michael De Vlieger, Dec 31 2018 *)

A139608 a(n) = 28*n + 8.

Original entry on oeis.org

8, 36, 64, 92, 120, 148, 176, 204, 232, 260, 288, 316, 344, 372, 400, 428, 456, 484, 512, 540, 568, 596, 624, 652, 680, 708, 736, 764, 792, 820, 848, 876, 904, 932, 960, 988, 1016, 1044, 1072, 1100, 1128, 1156, 1184, 1212, 1240, 1268, 1296, 1324, 1352, 1380

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 8th column of positive numbers in the square array of nonnegative and polygonal numbers A139600.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189. - From N. J. A. Sloane, Dec 01 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017005, A057145, A135628, A139600, A316466.

[4*(7*n + 2): n in [0..50]]; // Vincenzo Librandi, Jul 13 2011

Range[8, 7000, 28] (* Vladimir Joseph Stephan Orlovsky, Jul 13 2011 *)
LinearRecurrence[{2,-1},{8,36},50] (* or *) NestList[28+#&,8,50] (* Harvey P. Dale, Dec 14 2012 *)

a(n)=28*n+8 \\ Charles R Greathouse IV, Oct 05 2011

a(n) = A057145(n+2,8).
a(n) = 2*a(n-1) - a(n-2); a(0)=8, a(1)=36. - Harvey P. Dale, Dec 14 2012
G.f.: 4*(2+5*x)/(x-1)^2. - R. J. Mathar, Jul 28 2016
From Elmo R. Oliveira, Apr 16 2024: (Start)
E.g.f.: 4*exp(x)*(2 + 7*x).
a(n) = 4*A017005(n) = A135628(n) + 8 = A316466(n+1) - A316466(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)

A169825 Multiples of 420.

Original entry on oeis.org

0, 420, 840, 1260, 1680, 2100, 2520, 2940, 3360, 3780, 4200, 4620, 5040, 5460, 5880, 6300, 6720, 7140, 7560, 7980, 8400, 8820, 9240, 9660, 10080, 10500, 10920, 11340, 11760, 12180, 12600, 13020, 13440, 13860, 14280, 14700, 15120, 15540, 15960, 16380, 16800

Offset: 0

N. J. A. Sloane, May 30 2010

Numbers that are divisible by all of 1,2,3,4,5,6,7.

Erich Friedman, What's Special About This Number? (See the entry for "420")
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001477, A008589, A008596, A008597, A008602, A008603.

Cf. A135628, A169823, A169827, A249674.

Range[0, 100]*420 (* Paolo Xausa, Apr 17 2024 *)

a(n)=420*n \\ Charles R Greathouse IV, Apr 16 2024

a(n) = 420*n. - Wesley Ivan Hurt, Apr 11 2021
From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 420*x/(x-1)^2.
E.g.f.: 420*x*exp(x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 7*A169823(n) = 14*A249674(n) = 15*A135628(n) = 20*A008603(n) = 21*A008602(n) = 28*A008597(n) = 30*A008596(n) = 60*A008589(n) = 420*A001477(n) = A169827(n)/2. (End)

A169827 Multiples of 840.

Original entry on oeis.org

0, 840, 1680, 2520, 3360, 4200, 5040, 5880, 6720, 7560, 8400, 9240, 10080, 10920, 11760, 12600, 13440, 14280, 15120, 15960, 16800, 17640, 18480, 19320, 20160, 21000, 21840, 22680, 23520, 24360, 25200, 26040, 26880, 27720, 28560, 29400, 30240, 31080, 31920

Offset: 0

N. J. A. Sloane, May 30 2010

Numbers that are divisible by all of 1,2,3,4,5,6,7,8.

Erich Friedman, What's Special About This Number?
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001477, A005843, A008596, A008603, A135628, A169823, A169825.

Cf. A249674, A317095.

840*Range[0,40] (* Harvey P. Dale, Nov 03 2015 *)

a(n)=840*n \\ Charles R Greathouse IV, Apr 16 2024

From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 840*x/(x-1)^2.
E.g.f.: 840*x*exp(x).
a(n) = 840*n = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 2*A169825(n) = 14*A169823(n) = 21*A317095(n) = 28*A249674(n) = 30*A135628(n) = 40*A008603(n) = 60*A008596(n) = 420*A005843(n) = 840*A001477(n). (End)

cp's OEIS Frontend

A008596 Multiples of 14.

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A135631 Multiples of 31.

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A174312 a(n) = 32*n.

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A274824 Triangle read by rows: T(n,k) = (n-k+1)*sigma(k), n>=1, 1<=k<=n.

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A064763 a(n) = 28*n^2.

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A319073 Square array read by antidiagonals upwards: T(n,k) = k*sigma(n), n >= 1, k >= 1.

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A139608 a(n) = 28*n + 8.

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