A174312 -id:A174312 - cp's OEIS frontend

A008598 Multiples of 16.

0, 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240, 256, 272, 288, 304, 320, 336, 352, 368, 384, 400, 416, 432, 448, 464, 480, 496, 512, 528, 544, 560, 576, 592, 608, 624, 640, 656, 672, 688, 704, 720, 736, 752, 768, 784, 800, 816, 832

Offset: 0

N. J. A. Sloane

If X is an n-set and Y_i (i=1,2,3,4) mutually disjoint 2-subsets of X then a(n-6) is equal to the number of 5-subsets of X intersecting each Y_i (i=1,2,3,4). - Milan Janjic, Aug 26 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 328.
Milan Janjic, Two Enumerative Functions.
Tanya Khovanova, Recursive Sequences.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Leo Tavares, Illustration: Square Block Star Frames
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008590, A008596, A008597, A014641, A174312, A185212.

A008598:=n->16*n; seq(A008598(n), n=0..100); # Wesley Ivan Hurt, Nov 13 2013

Range[0, 1000, 16] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
CoefficientList[Series[16 x / (x - 1)^2, {x, 0, 60}], x] (* Vincenzo Librandi Jun 10 2013 *)

a(n)=16*n \\ Charles R Greathouse IV, Sep 24 2015

a(n) = Sum_{k=1..8n} (i^k+1)*(i^(8n-k)+1), where i=sqrt(-1). - Bruno Berselli, Mar 19 2012
G.f.: 16*x/(x-1)^2. - Vincenzo Librandi, Jun 10 2013
a(n) = A014641(n) - A185212(n). - Leo Tavares, May 24 2022
From Elmo R. Oliveira, Apr 07 2025: (Start)
E.g.f.: 16*x*exp(x).
a(n) = 16*n = 2*A008590(n) = A174312(n)/2.
a(n) = 2*a(n-1) - a(n-2). (End)

A044102 Multiples of 36.

Original entry on oeis.org

0, 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, 1728

Offset: 0

Clark Kimberling

Also, k such that Fibonacci(k) mod 27 = 0. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 18 2004
A033183(a(n)) = n+1. - Reinhard Zumkeller, Nov 07 2009
A122841(a(n)) > 1 for n > 0. - Reinhard Zumkeller, Nov 10 2013
Sum of the numbers from 4*(n-1) to 4*(n+1). - Bruno Berselli, Oct 25 2018

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

Cf. A005843, A008600, A033183, A122841, A135631, A167632, A174312.

Subsequence of A008588.

a:=[0,36];; for n in [3..50] do a[n]:=2*a[n-1]-a[n-2]; od; a; # Muniru A Asiru, Oct 25 2018

a044102 = (* 36)
a044102_list = [0, 36 ..]  -- Reinhard Zumkeller, Nov 10 2013

[36*n: n in [0..50]]; // Vincenzo Librandi, May 20 2014

seq(coeff(series(36*x/(1-x)^2,x,n+1), x, n), n = 0 .. 50); # Muniru A Asiru, Oct 25 2018

Range[0, 2000, 36] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
CoefficientList[Series[36 x/(1 - x)^2, {x, 0, 100}], x] (* Vincenzo Librandi, May 20 2014 *)

a(n)=36*n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 36*x/(1 - x)^2.
a(n) = A167632(n+1). - Reinhard Zumkeller, Nov 07 2009
a(n) = 36*n. - Vincenzo Librandi, Jan 26 2011
From Elmo R. Oliveira, Apr 10 2025: (Start)
E.g.f.: 36*x*exp(x).
a(n) = 18*A005843(n) = 2*A008600(n).
a(n) = 2*a(n-1) - a(n-2). (End)

A152691 Multiples of 64.

Original entry on oeis.org

0, 64, 128, 192, 256, 320, 384, 448, 512, 576, 640, 704, 768, 832, 896, 960, 1024, 1088, 1152, 1216, 1280, 1344, 1408, 1472, 1536, 1600, 1664, 1728, 1792, 1856, 1920, 1984, 2048, 2112, 2176, 2240, 2304, 2368, 2432, 2496, 2560, 2624, 2688, 2752, 2816, 2880

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 10 2008

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

Cf. A005843, A048169, A044187, A174312.

[64*n: n in [0..50]]; // Vincenzo Librandi, Feb 11 2012

Mathematica
```
64*Range[0, 47]
```

vector(50, n, n--; 64*n) \\ G. C. Greubel, Sep 02 2018

a(n) = 64*n.
G.f.: 64*x/(1-x)^2. - Bruno Berselli, Feb 11 2012
E.g.f: 64*x*exp(x). - G. C. Greubel, Sep 02 2018
From Elmo R. Oliveira, Apr 07 2025: (Start)
a(n) = 2*a(n-1) - a(n-2).
a(n) = 2*A174312(n) = 32*A005843(n). (End)

Definition corrected by R. J. Mathar, Dec 12 2008

A244082 a(n) = 32*n^2.

Original entry on oeis.org

0, 32, 128, 288, 512, 800, 1152, 1568, 2048, 2592, 3200, 3872, 4608, 5408, 6272, 7200, 8192, 9248, 10368, 11552, 12800, 14112, 15488, 16928, 18432, 20000, 21632, 23328, 25088, 26912, 28800, 30752, 32768, 34848, 36992, 39200, 41472, 43808, 46208, 48672, 51200

Offset: 0

Wesley Ivan Hurt, Jun 19 2014

Geometric connections of a(n) to the area and perimeter of a square.
Area:
. half the area of a square with side 8n (cf. A008590);
. area of a square with diagonal 8n (cf. A008590);
. twice the area of a square with side 4n (cf. A008586);
. four times the area of a square with diagonal 4n (cf. A008586);
. eight times the area of a square with side 2n (cf. A005843);
. sixteen times the area of a square with diagonal 2n (cf. A005843);
. thirty two times the area of a square with side n (cf. A001477);
. sixty four times the area of a square with diagonal n (cf. A001477).
Perimeter:
. perimeter of a square with side 8n^2 (cf. A139098);
. twice the perimeter of a square with side 4n^2 (cf. A016742);
. four times the perimeter of a square with side 2n^2 (cf. A001105);
. eight times the perimeter of a square with side n^2 (cf. A000290).
Sequence found by reading the line from 0, in the direction 0, 32, ..., in the square spiral whose vertices are the generalized 18-gonal numbers. - Omar E. Pol, May 10 2018

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

Pentasection of A077221, A181900.
Cf. A000290, A001105, A010021, A016742, A016802, A139098.
Cf. A001477, A005843, A008586, A008590, A139098, A174312.

Magma
```
[32*n^2 : n in [0..50]];
```

A244082:=n->32*n^2; seq(A244082(n), n=0..50);

32 Range[0, 50]^2 (* or *)
Table[32 n^2, {n, 0, 50}] (* or *)
CoefficientList[Series[32 x (1 + x)/(1 - x)^3, {x, 0, 30}], x]

a(n)=32*n^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 32*x*(1+x)/(1-x)^3.
a(n) =  2 * A016802(n).
a(n) =  4 * A139098(n).
a(n) =  8 * A016742(n).
a(n) = 16 * A001105(n).
a(n) = 32 * A000290(n).
a(n) = A010021(n) - 2 for n > 0. - Bruno Berselli, Jun 24 2014
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Nov 19 2021
From Elmo R. Oliveira, Dec 02 2024: (Start)
E.g.f.: 32*x*(1 + x)*exp(x).
a(n) = n*A174312(n) = A139098(2*n). (End)

A254312 Rectangular array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = (2^a(n)(6k - (3 - (-1)^a(n))(1 - (-1)^n)/2) - 2^n + 4)/6, n,k >= 1, where {a(n)} is the Beatty sequence A117630 defined by a(n) = floor(nlog(3)/log(3/2)).

Original entry on oeis.org

3, 32, 7, 170, 64, 11, 1022, 426, 96, 15, 2726, 2046, 682, 128, 19, 65526, 10918, 3070, 938, 160, 23, 174742, 131062, 19110, 4094, 1194, 192, 27, 2097110, 436886, 196598, 27302, 5118, 1450, 224, 31, 11184726, 4194262, 699030, 262134, 35494, 6142, 1706, 256, 35

Offset: 1

L. Edson Jeffery, May 03 2015

Conjecture: The array A contains without duplication all natural numbers m such that m < S(m), where the function S is as defined in A257480; i.e., the sequence is a permutation of A254311.

			Array A begins:
.         3       7      11      15       19       23       27       31
.        32      64      96     128      160      192      224      256
.       170     426     682     938     1194     1450     1706     1962
.      1022    2046    3070    4094     5118     6142     7166     8190
.      2726   10918   19110   27302    35494    43686    51878    60070
.     65526  131062  196598  262134   327670   393206   458742   524278
.    174742  436886  699030  961174  1223318  1485462  1747606  2009750
.   2097110 4194262 6291414 8388566 10485718 12582870 14680022 16777174

Cf. A117630, A254311, A257480.

Cf. A004767, A174312 (rows 1 and 2).

(* Array antidiagonals flattened: *)
a[n_] := Floor[n*Log[3/2, 3]]; A254312[n_, k_] := (2^a[n]*(6*k - (3 - (-1)^a[n])*(1 - (-1)^n)/2) - 2^n + 4)/6; Flatten[Table[A254312[n - k + 1, k], {n, 9}, {k, n}]]

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)

A362841 Numbers with at least one 5 in their prime signature.

Original entry on oeis.org

32, 96, 160, 224, 243, 288, 352, 416, 480, 486, 544, 608, 672, 736, 800, 864, 928, 972, 992, 1056, 1120, 1184, 1215, 1248, 1312, 1376, 1440, 1504, 1568, 1632, 1696, 1701, 1760, 1824, 1888, 1944, 1952, 2016, 2080, 2144, 2208, 2272, 2336, 2400, 2430, 2464, 2528, 2592, 2656, 2673, 2720, 2784, 2848, 2912, 2976

Offset: 1

R. J. Mathar, May 05 2023

nonn

Contains all odd multiples of 2^5: Each second term of A174312 is in this sequence.

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/p^5 + 1/p^6) = 0.01863624892... . - Amiram Eldar, May 05 2023

			Contains 2^5, 2^5*3, 2^5*5, 2^5*7, 3^5, 2^5*3^2, 2^5*11, 2^5*13, 2^5*3*5, 2*3^5, etc.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A038109 (at least one 2), A176297 (at least one 3), A050997 (subsequence), A178740 (subsequence), A179646 (subsequence), A179667 (subsequence), A179671 (subsequence), A174312.

Select[Range[3000], MemberQ[FactorInteger[#][[;;, 2]], 5] &] (* Amiram Eldar, May 05 2023 *)

cp's OEIS Frontend

A008598 Multiples of 16.

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A044102 Multiples of 36.

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A152691 Multiples of 64.

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A244082 a(n) = 32*n^2.

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A254312 Rectangular array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = (2^a(n)*(6*k - (3 - (-1)^a(n))*(1 - (-1)^n)/2) - 2^n + 4)/6, n,k >= 1, where {a(n)} is the Beatty sequence A117630 defined by a(n) = floor(n*log(3)/log(3/2)).

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Mathematica

A305548 a(n) = 27*n.

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Mathematica

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Formula

A362841 Numbers with at least one 5 in their prime signature.

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A254312 Rectangular array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = (2^a(n)(6k - (3 - (-1)^a(n))(1 - (-1)^n)/2) - 2^n + 4)/6, n,k >= 1, where {a(n)} is the Beatty sequence A117630 defined by a(n) = floor(nlog(3)/log(3/2)).