A132356 -id:A132356 - cp's OEIS frontend

A090771 Numbers that are congruent to {1, 9} mod 10.

1, 9, 11, 19, 21, 29, 31, 39, 41, 49, 51, 59, 61, 69, 71, 79, 81, 89, 91, 99, 101, 109, 111, 119, 121, 129, 131, 139, 141, 149, 151, 159, 161, 169, 171, 179, 181, 189, 191, 199, 201, 209, 211, 219, 221, 229, 231, 239, 241, 249, 251, 259, 261, 269, 271, 279, 281

Offset: 1

Giovanni Teofilatto, Feb 07 2004

Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n + (h-4)*(-1)^n - h)/4 (h, n natural numbers), therefore ((2*h*n + (h-4)*(-1)^n - h)/4)^2-1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 10). - Bruno Berselli, Nov 17 2010

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000217, A000796, A090298, A005408, A047209, A007310, A019970, A047336, A047522, A091998, A094888, A113801, A175886, A175887, A188593.
Cf. A056020 (n = 1 or 8 mod 9), A175885 (n = 1 or 10 mod 11).
Cf. A045468 (primes), A195142 (partial sums).

a090771 n = a090771_list !! (n-1)
a090771_list = 1 : 9 : map (+ 10) a090771_list
-- Reinhard Zumkeller, Jan 07 2012

Flatten[Table[10n - {9, 1}, {n, 30}]] (* Alonso del Arte, Sep 02 2014 *)
LinearRecurrence[{1,1,-1},{1,9,11},60] (* Harvey P. Dale, Jul 05 2020 *)

a(n)=n\2*10-(-1)^n \\ Charles R Greathouse IV, Sep 24 2015

a(n) = sqrt(40*A057569(n) + 1). - Gary Detlefs, Feb 22 2010
From Bruno Berselli, Sep 16 2010 - Nov 17 2010: (Start)
  G.f.: x*(1 + 8*x + x^2)/((1 + x)*(1 - x)^2).
  a(n) = (10*n + 3*(-1)^n - 5)/2.
  a(n) = -a(-n + 1) = a(n-1) + a(n-2) - a(n-3) = a(n-2) + 10.
  a(n) = 10*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i) for n > 1. (End)
a(n) = 10*n - a(n-1) - 10 (with a(1) = 1). - Vincenzo Librandi, Nov 16 2010
a(n) = sqrt(10*A132356(n-1) + 1). - Ivan N. Ianakiev, Nov 09 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/10)*cot(Pi/10) = A000796 * A019970 / 10 = sqrt(5 + 2*sqrt(5))*Pi/10. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((10*x - 5)*exp(x) + 3*exp(-x))/2. - David Lovler, Sep 03 2022
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(phi+2) (A188593).
Product_{n>=2} (1 + (-1)^n/a(n)) = Pi*phi/5 = A094888/10. (End)

Edited and extended by Ray Chandler, Feb 10 2004

A219257 Numbers k such that 11*k+1 is a square.

Original entry on oeis.org

0, 9, 13, 40, 48, 93, 105, 168, 184, 265, 285, 384, 408, 525, 553, 688, 720, 873, 909, 1080, 1120, 1309, 1353, 1560, 1608, 1833, 1885, 2128, 2184, 2445, 2505, 2784, 2848, 3145, 3213, 3528, 3600, 3933, 4009, 4360, 4440, 4809, 4893, 5280, 5368, 5773, 5865

Offset: 1

Bruno Berselli, Nov 16 2012

Equivalently, numbers of the form m*(11*m+2), where m = 0,-1,1,-2,2,-3,3,...

Also, integer values of h*(h+2)/11.

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

Cf. numbers k such that h*k+1 is a square: A005563 (h=1), A046092 (h=2), A001082 (h=3), A002378 (h=4), A036666 (h=5), A062717 (h=6), A132354 (h=7), A000217 (h=8), A132355 (h=9), A132356 (h=10), A152749 (h=12), A219389 (h=13), A219390 (h=14), A204221 (h=15), A074378 (h=16), A219394 (h=17), A219395 (h=18), A219396 (h=19), A219190 (h=20), A219391 (h=21), A219392 (h=22), A219393 (h=23), A001318 (h=24), A219259 (h=25), A217441 (h=26), A219258 (h=27), A219191 (h=28).

Cf. A175885 (square roots of 11*a(n)+1).

[n: n in [0..7000] | IsSquare(11*n+1)];

I:=[0,9,13,40,48]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

Select[Range[0, 7000], IntegerQ[Sqrt[11 # + 1]] &]
CoefficientList[Series[x (9 + 4 x + 9 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)

G.f.: x^2*(9+4*x+9*x^2)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = (22*n*(n-1)+7*(-1)^n*(2*n-1)-1)/8 + 1 = (1/176)*(22*n+7*(-1)^n-15)*(22*n+7*(-1)^n-7).
Sum_{n>=2} 1/a(n) = 11/4 - cot(2*Pi/11)*Pi/2. - Amiram Eldar, Mar 15 2022

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

Original entry on oeis.org

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)

A273366 a(n) = 10n^2 + 10n + 2.

Original entry on oeis.org

2, 22, 62, 122, 202, 302, 422, 562, 722, 902, 1102, 1322, 1562, 1822, 2102, 2402, 2722, 3062, 3422, 3802, 4202, 4622, 5062, 5522, 6002, 6502, 7022, 7562, 8122, 8702, 9302, 9922, 10562, 11222, 11902, 12602, 13322, 14062, 14822, 15602

Offset: 0

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

These are the numbers k such that 10*k+5 is a perfect square.

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

Cf. A062786, A132356, A273365, A273367, A273368, A350760.

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

LinearRecurrence[{3,-3,1}, {2, 22, 62}, 50] (* G. C. Greubel, May 20 2016 *)
Table[10n^2+10n+2,{n,0,40}] (* Harvey P. Dale, May 21 2024 *)

a(n)=10*n^2+10*n+2 \\ Charles R Greathouse IV, Jan 31 2017

G.f.: 2*(x^2+8x+1)/(1-x)^3.
From G. C. Greubel, May 20 2016: (Start)
E.g.f.: 2*(1 + 10*x + 5*x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = 2*A062786(n+1). - R. J. Mathar, Jun 03 2016
Sum_{n>=0} 1/a(n) = Pi/(2*sqrt(5)) * tan(Pi/(2*sqrt(5))) (A350760). - Amiram Eldar, Jan 20 2022

A071176 Smallest k such that the concatenation of n and k is a square (decimal notation).

Original entry on oeis.org

6, 5, 6, 9, 29, 4, 29, 1, 61, 0, 56, 1, 69, 4, 21, 9, 64, 49, 6, 25, 16, 5, 104, 336, 6, 244, 225, 9, 16, 25, 36, 4, 64, 81, 344, 1, 21, 44, 69, 0, 209, 25, 56, 1, 369, 24, 61, 4, 284, 41, 84, 9, 29, 76, 225, 25, 6, 564, 29, 84, 504, 5, 504

Offset: 1

Reinhard Zumkeller, May 15 2002

a(n) = 1 correspond to n = A132356(m), m > 0. - Bill McEachen, Aug 31 2023

			a(5) = 29 as 529 = 23^2 and 5'i is nonsquare for i<29, A071177(5)=23.

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

Cf. A000196, A071177, A245631, A132356.

import Data.List (findIndex)
import Data.Maybe (fromJust)
a071176 n = fromJust $ findIndex (== 1) $
            map (a010052 . read . (show n ++) . show) [0..]
-- Reinhard Zumkeller, Aug 09 2011

nksq[n_]:=Module[{idn=IntegerDigits[n],k=0},While[!IntegerQ[Sqrt[ FromDigits[Join[ idn,IntegerDigits[k]]]]],k++];k]; Array[nksq,70] (* Harvey P. Dale, Sep 28 2012 *)

a(n)={if(issquare(10*n), 0, my(m=n, b=1); while(1, m*=10; my(r=(sqrtint(m+b-1)+1)^2-m); b*=10; if(rAndrew Howroyd, Jan 13 2023

from math import isqrt
from sympy.ntheory.primetest import is_square
def A071176(n):
    m = 10*n
    if is_square(m): return 0
    a = 1
    while (k:=(isqrt(a*(m+1)-1)+1)**2-m*a)>=10*a:
        a *= 10
    return k # Chai Wah Wu, Feb 15 2023

A000196(n . a(n)) = A071177(n) where "." stands for concatenation.

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

A219190 Numbers of the form k(5k+1), where k = 0,-1,1,-2,2,-3,3,...

Original entry on oeis.org

0, 4, 6, 18, 22, 42, 48, 76, 84, 120, 130, 174, 186, 238, 252, 312, 328, 396, 414, 490, 510, 594, 616, 708, 732, 832, 858, 966, 994, 1110, 1140, 1264, 1296, 1428, 1462, 1602, 1638, 1786, 1824, 1980, 2020, 2184, 2226, 2398, 2442, 2622, 2668, 2856, 2904, 3100

Offset: 1

Bruno Berselli, Nov 14 2012

Equivalently, numbers m such that 20*m+1 is a square.
Also, integer values of h*(h+1)/5.
More generally, for the numbers of the form n*(k*n+1) with n in A001057, we have:
. generating function (offset 1): x^2*(k-1+2*x+(k-1)*x^2)/((1+x)^2*(1-x)^3);
. n-th term: b(n) = (2*k*n*(n-1)+(k-2)*(-1)^n*(2*n-1)+k-2)/8;
. first differences: (n-1)*((-1)^n*(k-2)+k)/2;
. b(2n+1)-b(2n) = 2*n (independent from k);
. (4*k)*b(n)+1 = (2*k*n+(k-2)*(-1)^n-k)^2/4.

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

Subsequence of A011858.
Cf. A090771: square roots of 20*a(n)+1 (see the first comment).
Cf. numbers of the form n*(k*n+1) with n in A001057: k=0, A001057; k=1, A110660; k=2, A000217; k=3, A152749; k=4, A074378; k=5, this sequence; k=6, A036498; k=7, A219191; k=8, A154260.
Cf. similar sequences listed in A219257.

k:=5; f:=func; [0] cat [f(n*m): m in [-1,1], n in [1..25]];

I:=[0,4,6,18,22]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

Rest[Flatten[{# (5 # - 1), # (5 # + 1)} & /@ Range[0, 25]]]
CoefficientList[Series[2 x (2 + x + 2 x^2) / ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,4,6,18,22},50] (* Harvey P. Dale, Jan 21 2015 *)

G.f.: 2*x^2*(2 + x + 2*x^2)/((1 + x)^2*(1 - x)^3).
a(n) = a(-n+1) = (10*n*(n-1) + 3*(-1)^n*(2*n - 1) + 3)/8.
a(n) = 2*A057569(n) = A008851(n+1)*A047208(n)/5.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Harvey P. Dale, Jan 21 2015
Sum_{n>=2} 1/a(n) = 5 - sqrt(1+2/sqrt(5))*Pi. - Amiram Eldar, Mar 15 2022
a(n) = A132356(n-1)/2, n >= 1. - Bernard Schott, Mar 15 2022

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

A220082 Numbers k such that 10*k-1 is a square.

Original entry on oeis.org

1, 5, 17, 29, 53, 73, 109, 137, 185, 221, 281, 325, 397, 449, 533, 593, 689, 757, 865, 941, 1061, 1145, 1277, 1369, 1513, 1613, 1769, 1877, 2045, 2161, 2341, 2465, 2657, 2789, 2993, 3133, 3349, 3497, 3725, 3881, 4121, 4285, 4537, 4709, 4973, 5153, 5429, 5617, 5905

Offset: 1

Bruno Berselli, Dec 05 2012

Equivalently, numbers of the form m*(10*m+6)+1, where m=0,-1,1,-2,2,-3,3,...

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

Cf. A085787, A132356 (numbers n such that 10*n+1 is a square).

Cf. numbers n such that k*n-1 is a square: A002522 (k=1), A001844 (k=2), A062317 (k=5).

[n: n in [1..6000] | IsSquare(10*n-1)]; /* or (see the first comment): */ [1] cat [m*(10*m+6)+1: m in [-n,n], n in [1..24]];

I:=[1,5,17,29,53]; [n le 5 select I[n] else Self(n-1) +2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Aug 18 2013

A220082:=proc(q)
local n;
for n from 1 to q do if type(sqrt(10*n-1), integer) then print(n);
fi; od; end:
A220082(1000); # Paolo P. Lava, Feb 19 2013

Select[Range[0, 6000], IntegerQ[Sqrt[10 # - 1]] &]
CoefficientList[Series[(1 + 4 x + 10 x^2 + 4 x^3 + x^4) / ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{1,5,17,29,53},50] (* Harvey P. Dale, Nov 19 2023 *)

G.f.: x*(1+4*x+10*x^2+4*x^3+x^4)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = (10*n*(n-1)-(2*n-1)*(-1)^n+3)/4.
For the definition: 10*a(n)-1 = ((10*n-(-1)^n-5)/2)^2.
a(n) = A212570(n)-A212570(n-1) = 4*A085787(n-1)+1 = A132356(n-1)-(2*n-1)*(-1)^n.

cp's OEIS Frontend

A090771 Numbers that are congruent to {1, 9} mod 10.

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A219257 Numbers k such that 11*k+1 is a square.

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

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A273366 a(n) = 10*n^2 + 10*n + 2.

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A071176 Smallest k such that the concatenation of n and k is a square (decimal notation).

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

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A219190 Numbers of the form k*(5*k+1), where k = 0,-1,1,-2,2,-3,3,...

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

A273366 a(n) = 10n^2 + 10n + 2.

A219190 Numbers of the form k(5k+1), where k = 0,-1,1,-2,2,-3,3,...