A033583 -id:A033583 - cp's OEIS frontend

A064762 a(n) = 21*n^2.

0, 21, 84, 189, 336, 525, 756, 1029, 1344, 1701, 2100, 2541, 3024, 3549, 4116, 4725, 5376, 6069, 6804, 7581, 8400, 9261, 10164, 11109, 12096, 13125, 14196, 15309, 16464, 17661, 18900, 20181, 21504, 22869, 24276, 25725, 27216, 28749

Offset: 0

Roberto E. Martinez II, Oct 18 2001

Number of edges in a complete 7-partite graph of order 7n, K_n,n,n,n,n,n,n.

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

Similar sequences are listed in A244630.
Cf. A000217, A000290, A033428, A033581, A033582, A033583.
Cf. A008603, A195049.

[21*n^2 : n in [0..50]]; // Wesley Ivan Hurt, Jul 04 2014

21 Range[0, 50]^2 (* Wesley Ivan Hurt, Jul 04 2014 *)
LinearRecurrence[{3,-3,1},{0,21,84},40] (* Harvey P. Dale, Jul 29 2019 *)

a(n)=21*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 42*n + a(n-1) - 21 for n > 0, a(0)=0. - Vincenzo Librandi, Aug 07 2010
a(n) = 21*A000290(n) = 7*A033428(n) = 3*A033582(n). - Omar E. Pol, Jul 03 2014
a(n) = t(7*n) - 7*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(7*n) - 7*A000217(n). - Bruno Berselli, Aug 31 2017
From Elmo R. Oliveira, Nov 30 2024: (Start)
G.f.: 21*x*(1 + x)/(1-x)^3.
E.g.f.: 21*x*(1 + x)*exp(x).
a(n) = n*A008603(n) = A195049(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A273367 Numbers k such that 10*k+6 is a perfect square.

Original entry on oeis.org

1, 3, 19, 25, 57, 67, 115, 129, 193, 211, 291, 313, 409, 435, 547, 577, 705, 739, 883, 921, 1081, 1123, 1299, 1345, 1537, 1587, 1795, 1849, 2073, 2131, 2371, 2433, 2689, 2755, 3027, 3097, 3385, 3459, 3763, 3841, 4161, 4243, 4579

Offset: 0

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 20 2016

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

Cf. A132356, A273365, A273366, A273368.

Cf. A033583 (perfect squares ending in 0 in base 10 with final 0 removed).

LinearRecurrence[{1, 2, -2, -1, 1}, {1, 3, 19, 25, 57}, 50] (* G. C. Greubel, May 20 2016 *)

is(n)=issquare(10*n+6) \\ Charles R Greathouse IV, Jan 31 2017

a(2n) = 10*n^2 - 8*n + 1.
a(2n+1) = 10*n^2 + 8*n + 1.
G.f.: (x^4+2x^3+14x^2+2x+1)/((1-x)^3*(1+x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - G. C. Greubel, May 20 2016

A277715 Row 5 of A277710: Positions of 5's in A264977; positions of 10's in A277330.

Original entry on oeis.org

9, 21, 45, 93, 189, 381, 657, 765, 873, 1317, 1533, 1749, 2457, 2637, 3069, 3501, 4329, 4917, 5241, 5277, 5745, 6141, 6345, 7005, 8661, 9561, 9837, 10017, 10485, 10557, 11493, 12285, 12693, 14013, 15129, 17325, 17985, 19125, 19677, 20037, 20973, 21117, 21969, 22989, 24573, 25389, 26793, 28029, 30261, 31545, 34653, 35973

Offset: 1

Antti Karttunen, Oct 29 2016

nonn

Positions in A260443 of terms that are ten times a perfect square (terms in A033583, although not all of them are present in A260443).

It seems that A068156 from 9 onward is a subsequence, which (if true) would also be a sufficient condition for this sequence to be infinite.

Antti Karttunen, Table of n, a(n) for n = 1..444

Row 5 of A277710.

Cf. A033583, A068156, A260443, A264977, A277330, A277716.

A277716(n) = a(n)/3.

A158447 a(n) = 10*n^2 - 1.

Original entry on oeis.org

9, 39, 89, 159, 249, 359, 489, 639, 809, 999, 1209, 1439, 1689, 1959, 2249, 2559, 2889, 3239, 3609, 3999, 4409, 4839, 5289, 5759, 6249, 6759, 7289, 7839, 8409, 8999, 9609, 10239, 10889, 11559, 12249, 12959, 13689, 14439, 15209, 15999, 16809, 17639

Offset: 1

Vincenzo Librandi, Mar 19 2009

Sequence found by reading the line from 9, in the direction 9, 39, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Jul 18 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A033583, A085787, A158446.

I:=[9, 39, 89]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];

A158447:=n->10*n^2-1: seq(A158447(n), n=1..100); # Wesley Ivan Hurt, Apr 26 2017

Table[10n^2-1,{n,50}]
LinearRecurrence[{3,-3,1},{9,39,89},50] (* Harvey P. Dale, Dec 08 2017 *)

PARI
```
a(n) = 10*n^2 - 1.
```

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f: x*(9 + 12*x - x^2)/(1 - x)^3.
a(n) = A033583(n) - 1. - Omar E. Pol, Jul 18 2012
From Amiram Eldar, Feb 04 2021: (Start)
Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(10))*cot(Pi/sqrt(10)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(10))*csc(Pi/sqrt(10)) - 1)/2.
Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(10))*csc(Pi/sqrt(10)).
Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(10))*sin(Pi/sqrt(5))/sqrt(2). (End)
E.g.f.: exp(x)*(10*x^2 + 10*x - 1) + 1. - Stefano Spezia, Aug 25 2022

A249327 Rectangular array T(n,k) = f(n)*k^2, where f = A005117 (squarefree numbers); n, k >= 1; read by antidiagonals.

Original entry on oeis.org

1, 4, 2, 9, 8, 3, 16, 18, 12, 5, 25, 32, 27, 20, 6, 36, 50, 48, 45, 24, 7, 49, 72, 75, 80, 54, 28, 10, 64, 98, 108, 125, 96, 63, 40, 11, 81, 128, 147, 180, 150, 112, 90, 44, 13, 100, 162, 192, 245, 216, 175, 160, 99, 52, 14, 121, 200, 243, 320, 294, 252, 250

Offset: 1

Clark Kimberling, Oct 26 2014

Every positive integer occurs exactly once.

			Northwest corner:
1   4    9    16   25    36    49
2   8    18   32   50    72    98
3   12   27   48   75    108   147
5   20   45   80   125   180   245
6   24   54   96   150   216   294

Cf. A005117, A000037 (is partitioned by the rows of A249327, excluding the first).

z = 20; f = Select[Range[10000], SquareFreeQ[#] &];
u[n_, k_] := f[[n]]*k^2; t = Table[u[n, k], {n, 1, 20}, {k, 1, 20}];
TableForm[t] (* A249327 array *)
Table[u[k, n - k + 1], {n, 1, 15}, {k, 1, n}] // Flatten (* A249327 sequence *)

T(1,k) = A000290(k), T(2,k) = A001105(k), T(3,k) = A033428(k), T(4,k) = A033429(k), T(5,.) through T(10,.) are A033581, A033582, A033583, A033584, A152742 and A144555 without initial 0. - M. F. Hasler, Oct 31 2014

A273365 Numbers k such that 10*k+4 is a perfect square.

Original entry on oeis.org

0, 6, 14, 32, 48, 78, 102, 144, 176, 230, 270, 336, 384, 462, 518, 608, 672, 774, 846, 960, 1040, 1166, 1254, 1392, 1488, 1638, 1742, 1904, 2016, 2190, 2310, 2496, 2624, 2822, 2958, 3168, 3312, 3534, 3686, 3920, 4080, 4326, 4494

Offset: 0

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 20 2016

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

Cf. A132356, A273366, A273367, A273368.

Cf. A033583 (perfect squares ending in 0 in base 10 with final 0 removed).

LinearRecurrence[{1, 2, -2, -1, 1}, {0, 6, 14, 32, 48}, 50] (* G. C. Greubel, May 21 2016 *)
Select[Range[0,5000],IntegerQ[Sqrt[10#+4]]&] (* Harvey P. Dale, Apr 19 2019 *)

is(n)=issquare(10*n+4) \\ Charles R Greathouse IV, Jan 31 2017

a(2n) = 10*n^2 + 4*n, n>=0.
a(2n-1) = 10*n^2 - 4*n, n>=1.
G.f.: 2*x*(3*x^2+4x+3)/((1-x)^3*(1+x)^2).
From G. C. Greubel, May 21 2016: (Start)
E.g.f.: (1/2)*((5*x^2 + 11*x)*cosh(x) + (5*x^2 + 9*x + 1)*sinh(x)).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). (End)

A273368 Numbers k such that 10*k+9 is a perfect square.

Original entry on oeis.org

0, 4, 16, 28, 52, 72, 108, 136, 184, 220, 280, 324, 396, 448, 532, 592, 688, 756, 864, 940, 1060, 1144, 1276, 1368, 1512, 1612, 1768, 1876, 2044, 2160, 2340, 2464, 2656, 2788, 2992, 3132, 3348, 3496, 3724, 3880, 4120, 4284, 4536

Offset: 0

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 20 2016

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

Cf. A132356, A273365, A273366, A273367.

Cf. A033583 (perfect squares ending in 0 in base 10 with final 0 removed).

CoefficientList[Series[4*x*(x^2+3x+1)/((1-x)^3*(1+x)^2), {x,0,50}], x] (* or *) LinearRecurrence[{1, 2, -2, -1, 1}, {0, 4, 16, 28, 52}, 50] (* G. C. Greubel, May 20 2016 *)

is(n)=issquare(10*n+9) \\ Charles R Greathouse IV, Jan 31 2017

a(2n) = 10*n^2 + 6*n, n>=0.
a(2n-1) = 10*n^2 - 6*n, n>=1.
G.f.: 4*x*(x^2+3x+1)/((1-x)^3*(1+x)^2).
From G. C. Greubel, May 21 2016: (Start)
E.g.f.: (1/2)*((5*x^2 + 9*x)*cosh(x) + (5*x^2 + 11*x -1)*sinh(x)).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). (End)
a(n) = 4*A085787(n). - R. J. Mathar, Jun 03 2016

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)

A326122 a(n) = 10 * sigma(n).

Original entry on oeis.org

10, 30, 40, 70, 60, 120, 80, 150, 130, 180, 120, 280, 140, 240, 240, 310, 180, 390, 200, 420, 320, 360, 240, 600, 310, 420, 400, 560, 300, 720, 320, 630, 480, 540, 480, 910, 380, 600, 560, 900, 420, 960, 440, 840, 780, 720, 480, 1240, 570, 930, 720, 980, 540, 1200, 720, 1200, 800, 900, 600, 1680, 620, 960

Offset: 1

Omar E. Pol, Jul 13 2019

10 times the sum of the divisors of n.

a(n) is also the total number of horizontal rhombuses in the terraces of the n-th level of an irregular stepped pyramid (starting from the top) where the structure of every 36-degree-three-dimensional sector arises after the 36-degree-zig-zag folding of every row of the diagram of the isosceles triangle A237593. The top of the pyramid is a 10-pointed star formed by 10 rhombuses (see Links section).

k times sigma(n), k=1..9: A000203, A074400, A272027, A239050, A274535, A274536, A319527, A319528, A325299.

Cf. A008592, A033583, A124080, A124252, A142342, A153780, A237593.

List([1..70],n->10*Sigma(n)); # After Muniru A Asiru

[10*DivisorSigma(1, n): n in [1..70]]; // Vincenzo Librandi, Jul 26 2019

with(numtheory): seq(10*sigma(n), n=1..64);

10*DivisorSigma[1,Range[70]] (* After Harvey P. Dale *)

PARI
```
a(n) = 10 * sigma(n);
```

a(n) = 10*A000203(n) = 5*A074400(n) = 2*A274535(n).
a(n) = A000203(n) + A325299(n) = A074400(n) + A319528(n).
Dirichlet g.f.: 10*zeta(s-1)*zeta(s). - (After Ilya Gutkovskiy)

A302576 Numbers k such that k/10 + 1 is a square.

Original entry on oeis.org

-10, 0, 30, 80, 150, 240, 350, 480, 630, 800, 990, 1200, 1430, 1680, 1950, 2240, 2550, 2880, 3230, 3600, 3990, 4400, 4830, 5280, 5750, 6240, 6750, 7280, 7830, 8400, 8990, 9600, 10230, 10880, 11550, 12240, 12950, 13680, 14430, 15200, 15990, 16800, 17630, 18480, 19350, 20240

Offset: 1

Bruno Berselli, Apr 10 2018

Equivalently, numbers k such that (k + 10)*10 is a square.

The positive terms belong to the fourth column of the array in A185781.

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

Cf. A000290, A008592, A033583, A185781.
After -10, subsequence of A174133 because a(n) = ((n-1)^2-1)*(3^2+1).
Similar lists of k for which k/j + 1 is a square: A067998 (j=1), A054000 (j=2), A067725 (j=3), A134582 (j=4), A067724 (j=5), A067726 (j=6), A067727 (j=7), second bisection of A067728 (j=8), A147651 (j=9), this sequence (j=10), A067705 (j=11), second bisection of A067707 (j=12).

GAP
```
List([1..50], n -> 10*n*(n-2));
```
Julia
```
[10*n*(n-2) for n in 1:50] |> println
```
Magma
```
[10*n*(n-2): n in [1..50]];
```
Mathematica
```
Table[10 n (n - 2), {n, 1, 50}]
```
Maxima
```
makelist(10*n*(n-2), n, 1, 50);
```
PARI
```
vector(50, n, nn; 10*n*(n-2))
```
Python
```
[10*n*(n-2) for n in range(1, 50)]
```
Sage
```
[10*n*(n-2) for n in (1..50)]
```

O.g.f.: -10*x*(1 - 3*x)/(1 - x)^3.
E.g.f.: -10*x*(1 - x)*exp(x).
a(n) = a(2-n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 10*n*(n - 2) = 10*A067998(n).
a(n) = A033583(n-1) - 10. - Altug Alkan, Apr 10 2018

cp's OEIS Frontend

A064762 a(n) = 21*n^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A273367 Numbers k such that 10*k+6 is a perfect square.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A277715 Row 5 of A277710: Positions of 5's in A264977; positions of 10's in A277330.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A158447 a(n) = 10*n^2 - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A249327 Rectangular array T(n,k) = f(n)*k^2, where f = A005117 (squarefree numbers); n, k >= 1; read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A273365 Numbers k such that 10*k+4 is a perfect square.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A273368 Numbers k such that 10*k+9 is a perfect square.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links