A044102 -id:A044102 - cp's OEIS frontend

A008588 Nonnegative multiples of 6.

0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 246, 252, 258, 264, 270, 276, 282, 288, 294, 300, 306, 312, 318, 324, 330, 336, 342, 348

Offset: 0

N. J. A. Sloane

For n > 3, the number of squares on the infinite 3-column half-strip chessboard at <= n knight moves from any fixed point on the short edge.
Second differences of A000578. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004
A008615(a(n)) = n. - Reinhard Zumkeller, Feb 27 2008
A157176(a(n)) = A001018(n). - Reinhard Zumkeller, Feb 24 2009
These numbers can be written as the sum of four cubes (i.e., 6*n = (n+1)^3 + (n-1)^3 + (-n)^3 + (-n)^3). - Arkadiusz Wesolowski, Aug 09 2013
A122841(a(n)) > 0 for n > 0. - Reinhard Zumkeller, Nov 10 2013
Surface area of a cube with side sqrt(n). - Wesley Ivan Hurt, Aug 24 2014
a(n) is representable as a sum of three but not two consecutive nonnegative integers, e.g., 6 = 1 + 2 + 3, 12 = 3 + 4 + 5, 18 = 5 + 6 + 7, etc. (see A138591). - Martin Renner, Mar 14 2016 (Corrected by David A. Corneth, Aug 12 2016)
Numbers with three consecutive divisors: for some k, each of k, k+1, and k+2 divide n. - Charles R Greathouse IV, May 16 2016
Numbers k for which {phi(k),phi(2k),phi(3k)} is an arithmetic progression. - Ivan Neretin, Aug 12 2016

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 81.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 318
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Essentially the same as A008458.
Cf. A016921, A016933, A016945, A016957, A016969, A138591.
Cf. A044102 (subsequence).

a008588 = (* 6)
a008588_list = [0, 6 ..]  -- Reinhard Zumkeller, Nov 10 2013

[6*n: n in [0..60] ]; // Vincenzo Librandi, Jul 16 2011

Maple
```
[ seq(6*n,n=0..45) ];
```

Range[0, 500, 6] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
NestList[6+#&,0,60] (* Harvey P. Dale, Nov 17 2024 *)

makelist(6*n,n,0,30); /* Martin Ettl, Nov 12 2012 */

a(n)=6*n \\ Charles R Greathouse IV, Feb 08 2012

[6*n for n in range(59)] # Stefano Spezia, Mar 09 2025

From Vincenzo Librandi, Dec 24 2010: (Start)
a(n) = 6*n = 2*a(n-1) - a(n-2).
G.f.: 6*x/(1-x)^2. (End)
a(n) = Sum_{k>=0} A030308(n,k)*6*2^k. - Philippe Deléham, Oct 24 2011
a(n) = Sum_{k=2n-1..2n+1} k. - Wesley Ivan Hurt, Nov 22 2015
From Ilya Gutkovskiy, Aug 12 2016: (Start)
E.g.f.: 6*x*exp(x).
Convolution of A010722 and A057427.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/6 = A002162*A020793. (End)
a(n) = 6 * A001477(n). - David A. Corneth, Aug 12 2016

A122841 Greatest k such that 6^k divides n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Sep 13 2006

nonn

See A054895 for the partial sums. - Hieronymus Fischer, Jun 08 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A122840, A007814, A007949, A054895, A234959.

a122841 = f 0 where
   f y x = if r > 0 then y else f (y + 1) x'
           where (x', r) = divMod x 6
-- Reinhard Zumkeller, Nov 10 2013

Table[IntegerExponent[n, 6], {n, 1, 100}] (* Amiram Eldar, Sep 14 2020 *)

a(n) = valuation(n, 6); \\ Michel Marcus, Jan 17 2022

From Hieronymus Fischer, Jun 03 2012: (Start)
With m = floor(log_6(n)), frac(x) = x-floor(x):
a(n) = Sum_{j=1..m} (1 - ceiling(frac(n/6^j))).
a(n) = m + Sum_{j=1..m} (floor(-frac(n/6^j))).
a(n) = A054895(n) - A054895(n-1).
G.f.: Sum_{j>0} x^6^j/(1-x^6^j). (End)
a(A047253(n)) = 0; a(A008588(n)) > 0; a(A044102(n)) > 1. - Reinhard Zumkeller, Nov 10 2013
6^a(n) = A234959(n), n >= 1. - Wolfdieter Lang, Jun 30 2014
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/5. - Amiram Eldar, Jan 17 2022
a(n) = min(A007814(n), A007949(n)). - Jianing Song, Jul 23 2022

A008600 Multiples of 18.

Original entry on oeis.org

0, 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, 216, 234, 252, 270, 288, 306, 324, 342, 360, 378, 396, 414, 432, 450, 468, 486, 504, 522, 540, 558, 576, 594, 612, 630, 648, 666, 684, 702, 720, 738, 756, 774, 792, 810, 828, 846, 864, 882, 900, 918, 936

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 330.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008591, A008598, A044102, A154612.

Range[0, 1000, 18] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)
CoefficientList[Series[18 x / (x - 1)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 10 2013 *)
18*Range[0,60] (* Harvey P. Dale, Jan 15 2023 *)

a(n)=18*n \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 18*x/(x-1)^2. - Vincenzo Librandi, Jun 10 2013
From Elmo R. Oliveira, Apr 10 2025: (Start)
E.g.f.: 18*x*exp(x).
a(n) = 18*n = 2*A008591(n) = A044102(n)/2.
a(n) = 2*a(n-1) - a(n-2). (End)

A208379 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 1 1 horizontally and 0 0 1 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 8, 16, 100, 72, 64, 10, 26, 256, 240, 108, 100, 12, 42, 676, 704, 420, 144, 144, 14, 68, 1764, 2080, 1344, 640, 180, 196, 16, 110, 4624, 6216, 4212, 2176, 900, 216, 256, 18, 178, 12100, 18496, 13860, 7072, 3200, 1200, 252, 324, 20, 288

Offset: 1

R. H. Hardin Feb 25 2012

Table starts
..2...4...6...10...16....26....42.....68.....110.....178......288......466
..4..16..36..100..256...676..1764...4624...12100...31684....82944...217156
..6..36..72..240..704..2080..6216..18496...55000..163760...487296..1450192
..8..64.108..420.1344..4212.13860..44880..144540..468852..1517184..4906980
.10.100.144..640.2176..7072.25200..87040..296560.1028128..3545856.12198016
.12.144.180..900.3200.10660.40740.148240..526900.1931300..7015680.25336420
.14.196.216.1200.4416.14976.60984.231744..851400.3276624.12438144.46737936
.16.256.252.1540.5824.20020.86436.340816.1285900.5170900.20393856.79220932

			Some solutions for n=4 k=3
..0..1..0....0..1..1....1..1..0....0..0..1....0..0..1....1..0..1....1..0..0
..1..1..0....0..1..1....1..1..0....0..1..1....1..0..1....1..1..0....1..0..0
..1..1..0....0..0..1....1..1..0....0..1..0....0..0..1....0..1..0....1..0..0
..1..0..0....0..0..1....1..1..0....0..1..0....0..0..1....0..1..0....1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..1463

Column 1 is A004275(n+1)
Column 2 is A016742
Column 3 is A044102(n-1) for n>1
Row 1 is A006355(n+2)
Row 2 is A206981
Row 3 is A207840

Empirical for column k:
k=1: a(n) = 2*n
k=2: a(n) = 4*n^2
k=3: a(n) = 36*n - 36 for n>1
k=4: a(n) = 20*n^2 + 40*n - 60 for n>1
k=5: a(n) = 96*n^2 - 32*n - 64 for n>1
k=6: a(n) = 364*n^2 - 416*n + 52 for n>1
k=7: a(n) = 84*n^3 + 840*n^2 - 1344*n + 420 for n>1
Empirical for row n:
n=1: a(k)=a(k-1)+a(k-2)
n=2: a(k)=2*a(k-1)+2*a(k-2)-a(k-3)
n=3: a(k)=a(k-1)+4*a(k-2)+5*a(k-3)+2*a(k-4)-a(k-5)+a(k-6) for k>8
n=4: a(k)=a(k-1)+4*a(k-2)+9*a(k-3)+5*a(k-4)-2*a(k-5)+4*a(k-6) for k>8
n=5: a(k)=a(k-1)+4*a(k-2)+13*a(k-3)+8*a(k-4)-3*a(k-5)+9*a(k-6) for k>8
n=6: a(k)=a(k-1)+4*a(k-2)+17*a(k-3)+11*a(k-4)-4*a(k-5)+16*a(k-6) for k>8
n=7: a(k)=a(k-1)+4*a(k-2)+21*a(k-3)+14*a(k-4)-5*a(k-5)+25*a(k-6) for k>8

A249726 Numbers n such that there is a multiple of 36 on row n of Pascal's triangle with property that it is also the least multiple of 4 and the least multiple of 9 on the same row.

Original entry on oeis.org

36, 72, 73, 108, 110, 144, 145, 147, 180, 216, 217, 218, 221, 252, 288, 289, 291, 295, 324, 326, 360, 361, 396, 432, 433, 434, 435, 437, 443, 468, 504, 505, 540, 542, 576, 577, 579, 583, 612, 648, 649, 650, 653, 684, 720, 721, 723, 756, 758, 792, 793, 828, 864, 865, 866, 867, 869, 871, 875, 887, 900, 936, 937, 972, 974, 1008, 1009, 1011, 1044, 1080

Offset: 1

Antti Karttunen, Nov 04 2014

nonn

All n such that both on row n of A034931 (Pascal's triangle reduced modulo 4) and on row n of A095143 (Pascal's triangle reduced modulo 9) there is at least one zero and the distance from the edge to the nearest zero is same on both rows.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Subsequence of A249724.
A044102 is a subsequence (after zero).
Natural numbers (A000027) is a disjoint union of the sequences A048278, A249722, A249723 and A249726.
Cf. A007318, A034931, A095143, A048645, A051382.

A249726list(upto_n) = { my(i=0, n=0); while(i

A139609 a(n) = 36*n + 9.

Original entry on oeis.org

9, 45, 81, 117, 153, 189, 225, 261, 297, 333, 369, 405, 441, 477, 513, 549, 585, 621, 657, 693, 729, 765, 801, 837, 873, 909, 945, 981, 1017, 1053, 1089, 1125, 1161, 1197, 1233, 1269, 1305, 1341, 1377, 1413, 1449, 1485, 1521, 1557, 1593, 1629, 1665, 1701

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 9th column of positive numbers in the square array of nonnegative and polygonal numbers A139600.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016813, A044102, A057145, A139600, A152994.

[9*(4*n + 1): n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

Range[9, 7000, 36] (* Vladimir Joseph Stephan Orlovsky, Jul 23 2011 *)
36*Range[0,50]+9 (* or *) LinearRecurrence[{2,-1},{9,45},50] (* Harvey P. Dale, Jan 04 2018 *)

a(n)=36*n+9 \\ Charles R Greathouse IV, Oct 05 2011

a(n) = A057145(n+2,9).
G.f.: 9*(1+3*x)/(x-1)^2. - R. J. Mathar, Jul 28 2016
From Elmo R. Oliveira, Apr 16 2024: (Start)
E.g.f.: 9*exp(x)*(1 + 4*x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 9*A016813(n) = A044102(n) + 9 = A152994(n+1) - A152994(n). (End)

A085959 Multiples of 37.

Original entry on oeis.org

0, 37, 74, 111, 148, 185, 222, 259, 296, 333, 370, 407, 444, 481, 518, 555, 592, 629, 666, 703, 740, 777, 814, 851, 888, 925, 962, 999, 1036, 1073, 1110, 1147, 1184, 1221, 1258, 1295, 1332, 1369, 1406, 1443, 1480, 1517, 1554, 1591, 1628, 1665, 1702, 1739

Offset: 0

Reinhard Zumkeller, Aug 17 2003

If a(k) = 100*u + 10*v + w with 0 <= u, v, w < 10, then 100*v + 10*w + u is also a term.

Numbers for which the sum of "digits" base 1000 is divisible by 37, since 999 = 3^3 * 37. For instance, 10089601558 gives 10 + 089 + 601 + 558 = 1258, 1 + 258 = 259 = 7 * 37. - Daniel Forgues, Feb 22 2016

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1986.

Tanya Khovanova, Recursive Sequences.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A044102.

[37*n: n in [0..50]]; // Vincenzo Librandi, Feb 23 2016

Range[0, 2000, 37] (* Vladimir Joseph Stephan Orlovsky, Jun 03 2011 *)
CoefficientList[Series[37 x / (1 - x)^2, {x, 0, 33}], x] (* Vincenzo Librandi, Feb 23 2016 *)

a(n) = a(n-1) + 37; a(0)=0.
G.f.: 37*x/(1-x)^2. - Vincenzo Librandi, Feb 23 2016
From Elmo R. Oliveira, Apr 10 2025: (Start)
E.g.f.: 37*x*exp(x).
a(n) = 37*n.
a(n) = 2*a(n-1) - a(n-2). (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)

A167632 Smallest m such that A033183(m) = n.

Original entry on oeis.org

0, 4, 36, 72, 108, 144, 180, 216, 252, 288, 324, 360, 396, 432, 468, 504, 540, 576, 612, 648, 684, 720, 756, 792, 828, 864, 900, 936, 972, 1008, 1044, 1080, 1116, 1152, 1188, 1224, 1260, 1296, 1332, 1368, 1404, 1440, 1476, 1512, 1548, 1584, 1620, 1656, 1692

Offset: 0

Reinhard Zumkeller, Nov 07 2009

nonn

A033183(a(n)) = n and A033183(m) < n for m < a(n);

for n>1: a(n) = A044102(n-1).

A351381 Table read by downward antidiagonals: T(n,k) = n*(k+1)^2.

Original entry on oeis.org

4, 9, 8, 16, 18, 12, 25, 32, 27, 16, 36, 50, 48, 36, 20, 49, 72, 75, 64, 45, 24, 64, 98, 108, 100, 80, 54, 28, 81, 128, 147, 144, 125, 96, 63, 32, 100, 162, 192, 196, 180, 150, 112, 72, 36, 121, 200, 243, 256, 245, 216, 175, 128, 81, 40, 144, 242, 300, 324, 320, 294, 252, 200, 144, 90, 44

Offset: 1

Bernard Schott, Mar 28 2022

When m and k are both positive integers and k | m, with m/k = n, then T(n,k) = S(m,k) = (m+k) + (m-k) + (m*k) + (m/k) = S(n*k,k) = n*(k+1)^2, problem proposed by Yakov Perelman.

All terms are nonsquarefree (A013929).

			Table begins:
  n \ k |   1      2      3      4      5      6      7      8      9     10
  ----------------------------------------------------------------------------
     1  |   4      9     16      25    36     49     64     81    100    121
     2  |   8     18     32      50    72     98    128    162    200    242
     3  |  12     27     48      75   108    147    192    243    300    363
     4  |  16     36     64     100   144    196    256    324    400    484
     5  |  20     45     80     125   180    245    320    405    500    605
     6  |  24     54     96     150   216    294    384    486    600    726
     7  |  28     63    112     175   252    343    448    567    700    847
     8  |  32     72    128     200   288    392    512    648    800    968
     9  |  36     81    144     225   324    441    576    729    900   1089
    10  |  40     90    160     250   360    490    640    810   1000   1210
  ............................................................................
T(3,4) = 75 = 3*(4+1)^2 corresponds to S(3*4,4) = S(12,4) = (12+4) + (12-4) + (12*4) + 12/4 = 75.
S(10,5) = (10+5) + (10-5) + (10*5) + (10/5) = T(10/5,5) = T(2,5) = 72.

I. Perelman, L'Algèbre Récréative, Chapitre IV, Les équations de Diophante, Deux nombres et quatre opérations, Editions en langues étrangères, Moscou, 1959, pp. 101-102.
Ya. I. Perelman, Algebra Can Be Fun, Chapter IV, Diophantine Equations, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.

Ya. I. Perelman, Algebra Can Be Fun, Chapter IV, Diophantine Equations, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.
Wikipedia, Yakov Perelman.

Cf. A013929.
Cf. A000290 \ {0,1} (row 1), A001105 \ {0,2} (row 2), A033428 \ {0,3} (row 3), A016742 \ {0,4} (row 4), A033429 \ {0,5} (row 5), A033581 \ {0,6} (row 6).
Cf. A008586 \ {0} (column 1), A008591 \ {0} (column 2), A008598 \ {0} (column 3), A008607 \ {0} (column 4), A044102 \ {0} (column 5).
Cf. A045991 \ {0} (diagonal).

T[n_, k_] := n*(k + 1)^2; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Mar 29 2022 *)

T(n,k) = n*(k+1)^2.
T(n,n) = (n+1)^3 - (n+1)^2 = A045991(n+1) for n >= 1.
G.f.: x*(1 + y)/((1 - x)^2*(1 - y)^3). - Stefano Spezia, Mar 31 2022

cp's OEIS Frontend

A008588 Nonnegative multiples of 6.

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A122841 Greatest k such that 6^k divides n.

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A008600 Multiples of 18.

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A208379 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 1 1 horizontally and 0 0 1 and 1 0 1 vertically.

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A249726 Numbers n such that there is a multiple of 36 on row n of Pascal's triangle with property that it is also the least multiple of 4 and the least multiple of 9 on the same row.

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A139609 a(n) = 36*n + 9.

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A085959 Multiples of 37.

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