A005476 -id:A005476 - cp's OEIS frontend

A057569 Numbers of the form k(5k+1)/2 or k(5k-1)/2.

0, 2, 3, 9, 11, 21, 24, 38, 42, 60, 65, 87, 93, 119, 126, 156, 164, 198, 207, 245, 255, 297, 308, 354, 366, 416, 429, 483, 497, 555, 570, 632, 648, 714, 731, 801, 819, 893, 912, 990, 1010, 1092, 1113, 1199, 1221, 1311, 1334, 1428, 1452, 1550

Offset: 1

N. J. A. Sloane, Oct 04 2000

a(n) is the set of all m such that 40*m+1 is a perfect square. - Gary Detlefs, Feb 22 2010
Integers of the form (n^2 - n) / 10. Numbers of the form  n * (5*n - 1) / 2 where n is an integer. - Michael Somos, Jan 13 2012
Also integers of the form sum_{k=1..n} k/5. - Alonso del Arte, Jan 20 2012
These numbers appear in a theta function identity. See the Hardy-Wright reference, Theorem 356 on p. 284. See the G.f. of A113428. - Wolfdieter Lang, Oct 28 2016

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, p. 284.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A074378, A001318, A057570, A154260. - Vladimir Joseph Stephan Orlovsky, Jan 06 2009

Cf. A001622, A113428.

[(10*(n^2-n)+12*(-1)^n*(n div 2))/16: n in [1..60]]; // Vincenzo Librandi, Oct 29 2016

Select[Table[Plus@@Range[n]/5, {n, 0, 199}], IntegerQ] (* Alonso del Arte, Jan 20 2012 *)
LinearRecurrence[{1,2,-2,-1,1},{0,2,3,9,11},50] (* Harvey P. Dale, Jul 05 2021 *)

{a(n) = (10 * (n^2 - n) + 12 * (-1)^n * (n\2)) / 16}; \\ Michael Somos, Jan 13 2012

Vec(x^2*(2*x^2+x+2) / ((1-x)^3*(1+x)^2) + O(x^60)) \\ Colin Barker, Jun 13 2017

A005475 UNION A005476. G.f.: x^2*(2x^2+x+2)/((1-x)^3*(1+x)^2). a(n) = A132356(n+1)/4. - R. J. Mathar, Apr 07 2008
a(n) = (A090771(n)^2 -1)/40. - Gary Detlefs, Feb 22 2010
|A113428(n)| is the characteristic function of the numbers a(n).
a(n) = a(1 - n) for all n in Z. - Michael Somos, Jan 13 2012
From Colin Barker, Jun 13 2017: (Start)
a(n) = n*(5*n - 2)/8 for n even.
a(n) = (5*n - 3)*(n - 1)/8 for n odd.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5.
(End)
From Amiram Eldar, Mar 17 2022: (Start)
Sum_{n>=2} 1/a(n) = 10 - 2*sqrt(1+2/sqrt(5))*Pi.
Sum_{n>=2} (-1)^n/a(n) = 2*sqrt(5)*log(phi) - 5*(2-log(5)), where phi is the golden ratio (A001622). (End)

A033994 a(n) = n(n+1)(5*n+1)/6.

Original entry on oeis.org

2, 11, 32, 70, 130, 217, 336, 492, 690, 935, 1232, 1586, 2002, 2485, 3040, 3672, 4386, 5187, 6080, 7070, 8162, 9361, 10672, 12100, 13650, 15327, 17136, 19082, 21170, 23405, 25792, 28336, 31042, 33915, 36960, 40182, 43586, 47177, 50960, 54940

Offset: 1

Barry E. Williams, Dec 16 1999

Partial sums of A005476.

a(n) is the dot product of the vectors of the first n positive integers and the next n integers. - Michel Marcus, Sep 02 2020

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005476, A016873, A000330, A132124, A132112, A050409.

Cf. A000292, A001622, A002411, A002412.

a:=List([1..40],n->n*(n+1)*(5*n+1)/6);; Print(a); # Muniru A Asiru, Jan 01 2019

[n*(n+1)*(5*n+1)/6 : n in [1..40]]; // Vincenzo Librandi, Jan 01 2019

[n*(n+1)*(5*n+1)/6$n=1..40]; # Muniru A Asiru, Jan 01 2019

Table[Range[x].Range[x+1,2x],{x,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{2,11,32,70},40] (* Harvey P. Dale, Jun 01 2018 *)

PARI
```
a(n) = n*(n+1)*(5*n+1)/6;
```

G.f.: x*(2+3*x)/(1-x)^4.
a(n) = A132121(n,1). - Reinhard Zumkeller, Aug 12 2007
a(n) = A000292(n) + A002412(n) = A000330(n) + A002411(n). - Omar E. Pol, Jan 11 2013
a(n) = Sum_{i=1..n} Sum_{j=1..n} i+min(i,j). - Enrique Pérez Herrero, Jan 15 2013
a(n) = Sum_{i=1..n} i*(n+i). - Charlie Marion, Apr 10 2013
Sum_{n>=1} 1/a(n) = 36 - 3*Pi*5^(3/4)*phi^(3/2)/4 - 15*sqrt(5)*log(phi)/4 - 75*log(5)/8 = 0.66131826232008423794478..., where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 01 2018
E.g.f.: exp(x)*x*(12 + 21*x + 5*x^2)/6. - Stefano Spezia, Feb 21 2024

More terms from James Sellers, Jan 19 2000

A022288 a(n) = n(31n-1)/2.

Original entry on oeis.org

0, 15, 61, 138, 246, 385, 555, 756, 988, 1251, 1545, 1870, 2226, 2613, 3031, 3480, 3960, 4471, 5013, 5586, 6190, 6825, 7491, 8188, 8916, 9675, 10465, 11286, 12138, 13021, 13935, 14880, 15856, 16863, 17901

Offset: 0

N. J. A. Sloane

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

Cf. A000217, A022289.

Cf. similar sequences of the form n*((2*k+1)*n - 1)/2: A161680 (k=0), A000326 (k=1), A005476 (k=2), A022264 (k=3), A022266 (k=4), A022268 (k=5), A022270 (k=6), A022272 (k=7), A022274 (k=8), A022276 (k=9), A022278 (k=10), A022280 (k=11), A022282 (k=12), A022284 (k=13), A022286 (k=14), this sequence (k=15).

Table[n (31 n - 1)/2, {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 15, 61}, 40] (* Harvey P. Dale, Mar 31 2014 *)

a(n)=n*(31*n-1)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 31*n + a(n-1) - 16 for n>0, a(0)=0. - Vincenzo Librandi, Aug 04 2010
a(0)=0, a(1)=15, a(2)=61; for n>2, a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Mar 31 2014
G.f.: x*(15 + 16*x)/(1 - x)^3. - R. J. Mathar, Sep 02 2016
a(n) = A000217(16*n-1) - A000217(15*n-1). In general, n*((2*k+1)*n - 1)/2 = A000217((k+1)*n-1) - A000217(k*n-1), and the ordinary generating function is x*(k + (k+1)*x)/(1 - x)^3. - Bruno Berselli, Oct 14 2016
E.g.f.: (x/2)*(31*x + 30)*exp(x). - G. C. Greubel, Aug 24 2017

A195014 Vertex number of a square spiral whose edges have length A195013.

Original entry on oeis.org

0, 2, 5, 9, 15, 21, 30, 38, 50, 60, 75, 87, 105, 119, 140, 156, 180, 198, 225, 245, 275, 297, 330, 354, 390, 416, 455, 483, 525, 555, 600, 632, 680, 714, 765, 801, 855, 893, 950, 990, 1050, 1092, 1155, 1199, 1265, 1311, 1380, 1428, 1500, 1550, 1625, 1677

Offset: 0

Omar E. Pol, Sep 09 2011

Zero together with the partial partial sums of A195013.
Second bisection is 2, 9, 21, 38, 60, 87, 119, ...: A005476. - Omar E. Pol, Sep 25 2011
Number of pairs (x,y) with even x in {0,...,n}, odd y in {0,...,3n}, and xClark Kimberling, Jul 02 2012

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

Cf. A005476, A028895, A195013, A195020.

[(10*n^2 + 18*n + 3 + (2*n - 3)*(-1)^n)/16 : n in [0..50]]; // Vincenzo Librandi, Oct 26 2014

LinearRecurrence[{1,2,-2,-1,1},{0,2,5,9,15},60] (* Harvey P. Dale, May 20 2019 *)

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: f(x)/g(x), where f(x) = 2*x + 3*x^2 and g(x) = (1+x)^2 * (1-x)^3. - Clark Kimberling, Jul 02 2012
a(n) = (10*n^2 + 18*n + 3 + (2*n - 3)*(-1)^n)/16. - Luce ETIENNE, Aug 11 2014

A008732 Molien series for 3-dimensional group [2,n] = *22n.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 18, 21, 24, 27, 30, 34, 38, 42, 46, 50, 55, 60, 65, 70, 75, 81, 87, 93, 99, 105, 112, 119, 126, 133, 140, 148, 156, 164, 172, 180, 189, 198, 207, 216, 225, 235, 245, 255, 265

Offset: 0

N. J. A. Sloane

			From _Philippe Deléham_, Apr 05 2013: (Start)
Stored in five columns:
    1   2   3   4   5
    7   9  11  13  15
   18  21  24  27  30
   34  38  42  46  50
   55  60  65  70  75
   81  87  93  99 105
  112 119 126 133 140
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 188
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 8a, lambda=5]
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A130520.

List([0..50], n-> Int((n+3)*(n+4)/10)); # G. C. Greubel, Jul 30 2019

[Floor((n+3)*(n+4)/10): n in [0..50] ]; // Vincenzo Librandi, Aug 21 2011

A092202 := proc(n) op(1+(n mod 5),[0,1,0,-1,0]) ; end proc:
A010891 := proc(n) op(1+(n mod 5),[1,-1,0,0,0]) ; end proc:
A008732 := proc(n) (n+2)*(n+5)/10+(A010891(n-1)+2*A092202(n-1))/5 ; end proc:

LinearRecurrence[{2, -1, 0, 0, 1, -2, 1}, {1, 2, 3, 4, 5, 7, 9}, 50] (* Jean-François Alcover, Jan 18 2018 *)

a(n)=(n+3)*(n+4)\10 \\ Charles R Greathouse IV, Oct 07 2015

[floor((n+3)*(n+4)/10) for n in (0..50)] # G. C. Greubel, Jul 30 2019

a(n) = floor( (n+3)*(n+4)/10 ) = (n+2)*(n+5)/10 + b(n)/5 where b(n) = A010891(n-2) + 2*A092202(n-1) = 0, 1, 1, 0, -2, ... with period length 5.
G.f.: 1/((1-x)^2*(1-x^5)).
a(n) = a(n-5) + n + 1. - Paul Barry, Jul 14 2004
From Mitch Harris, Sep 08 2008: (Start)
a(n) = Sum_{j=0..n+5} floor(j/5).
a(n-5) = (1/2)floor(n/5)*(2*n - 3 - 5*floor(n/5)). (End)
a(n) = A130520(n+5). - Philippe Deléham, Apr 05 2013
a(5n) = A000566(n+1), a(5n+1) = A005476(n+1), a(5n+2) = A005475(n+1), a(5n+3) = A147875(n+2), a(5n+4) = A028895(n+1); these formulas correspond to the 5 columns of the array shown in example. - Philippe Deléham, Apr 05 2013

A195015 Main axis of the square spiral whose edges have length A195013 and whose vertices are the numbers A195014.

Original entry on oeis.org

0, 2, 12, 24, 44, 66, 96, 128, 168, 210, 260, 312, 372, 434, 504, 576, 656, 738, 828, 920, 1020, 1122, 1232, 1344, 1464, 1586, 1716, 1848, 1988, 2130, 2280, 2432, 2592, 2754, 2924, 3096, 3276, 3458, 3648, 3840, 4040, 4242, 4452, 4664, 4884

Offset: 0

Omar E. Pol, Sep 26 2011

Sequence found by reading the line from 0, in the direction 0, 2, ..., and the same line from 0, in the direction 0, 12, ..., in the square spiral mentioned above. Axis perpendicular to A195016 in the same spiral.

Also four times A005475 and positives A152965 interleaved.

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

Cf. A000566, A005475, A005476, A028895, A033571, A152965, A153126, A195013, A195014, A195016.

[(2*n*(5*n+2)+3*(-1)^n-3)/4: n in [0..50]]; // Vincenzo Librandi, Oct 28 2011

LinearRecurrence[{2, 0, -2, 1}, {0, 2, 12, 24}, 50] (* Paolo Xausa, Feb 09 2024 *)

From Bruno Berselli, Oct 14 2011: (Start)
G.f.: 2*x*(1+4*x)/((1+x)*(1-x)^3).
a(n) = (2*n*(5*n+2) + 3*(-1)^n-3)/4.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
a(n) + a(n-1) = A135706(n). (End)

A172043 a(n) = 5*n^2 - n + 1.

Original entry on oeis.org

1, 5, 19, 43, 77, 121, 175, 239, 313, 397, 491, 595, 709, 833, 967, 1111, 1265, 1429, 1603, 1787, 1981, 2185, 2399, 2623, 2857, 3101, 3355, 3619, 3893, 4177, 4471, 4775, 5089, 5413, 5747, 6091, 6445, 6809, 7183, 7567, 7961, 8365, 8779, 9203, 9637, 10081, 10535

Offset: 0

Vincenzo Librandi, Jan 29 2010

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

Cf. A005476.

Magma
```
[ 5*n^2-n+1: n in [0..50] ];
```

CoefficientList[Series[(7*x^2+2*x+1)/(1-x)^3,{x,0,40}],x] (* Vincenzo Librandi, Jul 06 2012 *)
Table[5n^2-n+1,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,5,19},50] (* Harvey P. Dale, Aug 06 2022 *)

a(n)=5*n^2-n+1 \\ Charles R Greathouse IV, Jun 17 2017

From Vincenzo Librandi, Jul 06 2012: (Start)
G.f.: (1+2*x+7*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = 2*A005476(n) + 1. - Bruno Berselli, Jul 06 2012
E.g.f.: exp(x)*(1 + 4*x + 5*x^2). - Elmo R. Oliveira, Oct 31 2024

Replaced definition with formula. - N. J. A. Sloane, Mar 03 2010

A306383 Number of ways to write n as x(2x+1) + y(2y+1) + z*(2z+1), where x,y,z are nonnegative integers with x <= y <= z.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Feb 11 2019

nonn

Conjecture 1: a(n) > 0 for any integer n > 138158.
We have verified this for n up to 2*10^6. Note that n*(2n+1) (n = 0,1,...) are the second hexagonal numbers (A014105).
Conjecture 2: Any integer n > 146858 can be written as the sum of three hexagonal numbers (A000384).
Conjecture 3: Any integer n > 33066 can be written as the sum of three pentagonal numbers (A000326).
Conjecture 4: Any integer n > 24036 can be written as the sum of three second pentagonal numbers (A005449).
Conjecture 5: Let N(1) = 114862, N(-1) = 166897, N(3) = 196987 and N(-3) = 273118. Then, for any r among 1, -1, 3 and -3, each integer n > N(r) can be written as x*(5x+r)/2 + y*(5y+r)/2 + z*(5z+r)/2 with x,y,z nonnegative integers.
We have verified Conjectures 2-5 for n up to 10^6.

			a(223595) = 1 with 223595 = 95*(2*95+1) + 200*(2*200+1) + 250*(2*250+1).
a(290660) = 1 with 290660 = 136*(2*136+1) + 149*(2*149+1) + 323*(2*323+1).

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Universal sums of three quadratic polynomials, Sci. China Math., in press.

Cf. A000217, A000326, A000384, A000566, A005449, A005475, A005476, A008443, A014105, A147875, A306382.

QQ[n_]:=QQ[n]=IntegerQ[Sqrt[8n+1]]&&Mod[Sqrt[8n+1],4]==1;
tab={};Do[r=0;Do[If[QQ[n-x(2x+1)-y(2y+1)],r=r+1],{x,0,(Sqrt[8n/3+1]-1)/4},{y,x,(Sqrt[4(n-x(2x+1))+1]-1)/4}];tab=Append[tab,r],{n,0,100}];Print[tab]

A167469 a(n) = 3n(5*n-1)/2.

Original entry on oeis.org

6, 27, 63, 114, 180, 261, 357, 468, 594, 735, 891, 1062, 1248, 1449, 1665, 1896, 2142, 2403, 2679, 2970, 3276, 3597, 3933, 4284, 4650, 5031, 5427, 5838, 6264, 6705, 7161, 7632, 8118, 8619, 9135, 9666, 10212, 10773, 11349, 11940, 12546, 13167, 13803, 14454

Offset: 1

A.K. Devaraj, Nov 05 2009

This represents the nontrivial imaginary part of the decomposition of the trivariate rational polynomial described in A167467.
Old name was: 3*A005476(n).
Sum of the numbers from n to 4*n-1 for n>=1. - Wesley Ivan Hurt, May 08 2016

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

Cf. A005476, A167467.

Similar sequences are listed in A316466.

List([1..50], n-> 3*n*(5*n-1)/2); # G. C. Greubel, Sep 01 2019

[3*n*(5*n-1)/2 : n in [1..50]]; // Wesley Ivan Hurt, May 08 2016

A167469:=n->3*n*(5*n-1)/2: seq(A167469(n), n=1..50); # Wesley Ivan Hurt, May 08 2016

Table[3n(5n-1)/2, {n, 50}] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)

x='x+O('x^50); Vec(3*x*(2+3*x)/(1-x)^3) \\ Altug Alkan, May 14 2016

[3*n*(5*n-1)/2 for n in (1..50)] # G. C. Greubel, Sep 01 2019

G.f.: 3*x*(2+3*x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
a(n) = Sum_{i=n..4*n-1} i. - Wesley Ivan Hurt, May 08 2016
E.g.f.: 3*x*(4 + 5*x)*exp(x)/2. - Ilya Gutkovskiy, May 14 2016
a(n) = Sum_{i = 2..7} P(i,n), where P(i,m) = m*((i-2)*m-(i-4))/2. - Bruno Berselli, Jul 04 2018

a(1) corrected, definition simplified, sequence extended by R. J. Mathar, Nov 12 2009

Name changed by Wesley Ivan Hurt, May 08 2016

A195016 a(n) = (n(5n+7)-(-1)^n+1)/2.

Original entry on oeis.org

0, 7, 17, 34, 54, 81, 111, 148, 188, 235, 285, 342, 402, 469, 539, 616, 696, 783, 873, 970, 1070, 1177, 1287, 1404, 1524, 1651, 1781, 1918, 2058, 2205, 2355, 2512, 2672, 2839, 3009, 3186, 3366, 3553, 3743, 3940, 4140, 4347, 4557, 4774, 4994

Offset: 0

Omar E. Pol, Sep 26 2011

Sequence found by reading the line from 0, in the direction 0, 7,..., and the same line from 0, in the direction 0, 17,..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. Axis perpendicular to the main axis A195015 in the same spiral.

Also sequence found by reading the line from 0, in the direction 0, 7,..., and the same line from 0, in the direction 0, 17,..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. This line is parallel to A153126 in the same spiral.

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

Cf. A000566, A005476, A028895, A033571, A085787, A152965, A153126, A195013, A195014, A195015.

&cat[[n*t,(n+1)*t] where t is 10*n+7: n in [0..22]]; // Bruno Berselli, Oct 14 2011

LinearRecurrence[{2, 0, -2, 1}, {0, 7, 17, 34}, 50] (* Paolo Xausa, Feb 09 2024 *)

n*(10*n-3), if n >= 1, and (2*n+1)*(5*n+1)-1, if n >= 0, interleaved.

G.f.: x*(7+3*x)/((1+x)*(1-x)^3). - Bruno Berselli, Oct 14 2011

Concise definition by Bruno Berselli, Oct 14 2011

cp's OEIS Frontend

A057569 Numbers of the form k*(5*k+1)/2 or k*(5*k-1)/2.

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A033994 a(n) = n*(n+1)*(5*n+1)/6.

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A022288 a(n) = n*(31*n-1)/2.

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A195014 Vertex number of a square spiral whose edges have length A195013.

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A008732 Molien series for 3-dimensional group [2,n] = *22n.

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A195015 Main axis of the square spiral whose edges have length A195013 and whose vertices are the numbers A195014.

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Magma

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A172043 a(n) = 5*n^2 - n + 1.

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A057569 Numbers of the form k(5k+1)/2 or k(5k-1)/2.

A033994 a(n) = n(n+1)(5*n+1)/6.

A022288 a(n) = n(31n-1)/2.

A306383 Number of ways to write n as x(2x+1) + y(2y+1) + z*(2z+1), where x,y,z are nonnegative integers with x <= y <= z.

A167469 a(n) = 3n(5*n-1)/2.

A195016 a(n) = (n(5n+7)-(-1)^n+1)/2.