A054000 -id:A054000 - cp's OEIS frontend

A132356 a(2k) = k(10k+2), a(2k+1) = 10k^2 + 18k + 8, with k >= 0.

0, 8, 12, 36, 44, 84, 96, 152, 168, 240, 260, 348, 372, 476, 504, 624, 656, 792, 828, 980, 1020, 1188, 1232, 1416, 1464, 1664, 1716, 1932, 1988, 2220, 2280, 2528, 2592, 2856, 2924, 3204, 3276, 3572, 3648, 3960, 4040, 4368, 4452, 4796, 4884, 5244, 5336, 5712

Offset: 0

Mohamed Bouhamida, Nov 08 2007

X values of solutions to the equation 10*X^3 + X^2 = Y^2.
Polygonal number connection: 2*H_n + 6S_n, where H_n is the n-th hexagonal number and S_n is the n-th square number. This is the base formula that is expanded upon to achieve the full series. See contributing formula below. - William A. Tedeschi, Sep 12 2010
Equivalently, numbers of the form 2*h*(5*h+1), where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ... . - Bruno Berselli, Feb 02 2017

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. A054000, A056220, A087475, A028347, A046092, A132209.
Cf. numbers m such that k*m+1 is a square: A005563 (k=1), A046092 (k=2), A001082 (k=3), A002378 (k=4), A036666 (k=5), A062717 (k=6), A132354 (k=7), A000217 (k=8), A132355 (k=9), A219257 (k=11), A152749 (k=12), A219389 (k=13), A219390 (k=14), A204221 (k=15), A074378 (k=16), A219394 (k=17), A219395 (k=18), A219396 (k=19), A219190 (k=20), A219391 (k=21), A219392 (k=22), A219393 (k=23), A001318 (k=24), A219259 (k=25), A217441 (k=26), A219258 (k=27), A219191 (k=28).
Cf. A220082 (numbers k such that 10*k-1 is a square).

CoefficientList[Series[4*x*(2*x^2 + x + 2)/((1 - x)^3*(1 + x)^2), {x, 0, 50}], x] (* G. C. Greubel, Jun 12 2017 *)
LinearRecurrence[{1,2,-2,-1,1},{0,8,12,36,44},50] (* Harvey P. Dale, Dec 15 2023 *)

my(x='x+O('x^50)); concat([0], Vec(4*x*(2*x^2+x+2)/((1-x)^3*(1+x)^2))) \\ G. C. Greubel, Jun 12 2017

a(n) = n^2 + n + 6*((n+1)\2)^2 \\ Charles R Greathouse IV, Sep 11 2022

G.f.: 4*x*(2*x^2+x+2)/((1-x)^3*(1+x)^2). - R. J. Mathar, Apr 07 2008
a(n) = 10*x^2 - 2*x, where x = floor(n/2)*(-1)^n for n >= 1. - William A. Tedeschi, Sep 12 2010
a(n) = ((2*n+1-(-1)^n)*(10*(2*n+1)-2*(-1)^n))/16. - Luce ETIENNE, Sep 13 2014
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4. - Chai Wah Wu, May 24 2016
Sum_{n>=1} 1/a(n) = 5/2 - sqrt(1+2/sqrt(5))*Pi/2. - Amiram Eldar, Mar 15 2022
a(n) = n^2 + n + 6*ceiling(n/2)^2. - Ridouane Oudra, Aug 06 2022

More terms from Max Alekseyev, Nov 13 2009

A090238 Triangle T(n, k) read by rows. T(n, k) is the number of lists of k unlabeled permutations whose total length is n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 24, 16, 6, 1, 0, 120, 72, 30, 8, 1, 0, 720, 372, 152, 48, 10, 1, 0, 5040, 2208, 828, 272, 70, 12, 1, 0, 40320, 14976, 4968, 1576, 440, 96, 14, 1, 0, 362880, 115200, 33192, 9696, 2720, 664, 126, 16, 1, 0, 3628800, 996480, 247968, 64704, 17312, 4380, 952, 160, 18, 1

Offset: 0

Philippe Deléham, Jan 23 2004, Jun 14 2007

T(n,k) is the number of lists of k unlabeled permutations whose total length is n. Unlabeled means each permutation is on an initial segment of the positive integers. Example: with dashes separating permutations, T(3,2) = 4 counts 1-12, 1-21, 12-1, 21-1. - David Callan, Nov 29 2007
For n > 0, -Sum_{i=0..n} (-1)^i*T(n,i) is the number of indecomposable permutations A003319. - Peter Luschny, Mar 13 2009
Also the convolution triangle of the factorial numbers for n >= 1. - Peter Luschny, Oct 09 2022

			Triangle begins:
  1;
  0,       1;
  0,       2,      1;
  0,       6,      4,      1;
  0,      24,     16,      6,     1;
  0,     120,     72,     30,     8,     1;
  0,     720,    372,    152,    48,    10,     1;
  0,    5040,   2208,    828,   272,    70,    12,    1;
  0,   40320,  14976,   4968,  1576,   440,    96,   14,   1;
  0,  366880, 115200,  33192,  9696,  2720,   664,  126,  16,   1;
  0, 3628800, 996480, 247968, 64704, 17312,  4380,  952, 160,  18,  1;
  ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 171, #34.

Another version: A059369.
Row sums: A051296, A003319 (n>0).
Diagonals: A000007, A000142, A059371, A000012, A005843, A054000.
Cf. A084938.

T := proc(n,k) option remember; if n=0 and k=0 then return 1 fi;
if n>0 and k=0 or k>0 and n=0 then return 0 fi;
T(n-1,k-1)+(n+k-1)*T(n-1,k)/k end:
for n from 0 to 10 do seq(T(n,k),k=0..n) od; # Peter Luschny, Mar 03 2016
# Uses function PMatrix from A357368.
PMatrix(10, factorial); # Peter Luschny, Oct 09 2022

T[n_, k_] := T[n, k] = T[n-1, k-1] + ((n+k-1)/k)*T[n-1, k]; T[0, 0] = 1; T[, 0] = T[0, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2018 *)

T(n, k) is given by [0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
T(n, k) = T(n-1, k-1) + ((n+k-1)/k)*T(n-1, k); T(0, 0)=1, T(n, 0)=0 if n > 0, T(0, k)=0 if k > 0.
G.f. for the k-th column: (Sum_{i>=1} i!*t^i)^k = Sum_{n>=k} T(n, k)*t^n.
Sum_{k=0..n} T(n, k)*binomial(m, k) = A084938(m+n, m). - Philippe Deléham, Jan 31 2004
T(n, k) = Sum_{j>=0} A090753(j)*T(n-1, k+j-1). - Philippe Deléham, Feb 18 2004
From Peter Bala, May 27 2017: (Start)
Conjectural o.g.f.: 1/(1 + t - t*F(x)) = 1 + t*x + (2*t + t^2)*x^2 + (6*t + 4*t^2 + t^3)*x^3 + ..., where F(x) = Sum_{n >= 0} n!*x^n.
If true then a continued fraction representation of the o.g.f. is 1 - t + t/(1 - x/(1 - t*x - x/(1 - 2*x/(1 - 2*x/(1 - 3*x/(1 - 3*x/(1 - 4*x/(1 - 4*x/(1 - ... ))))))))). (End)

New name using a comment from David Callan by Peter Luschny, Sep 01 2022

A067727 a(n) = 7n^2 + 14n.

Original entry on oeis.org

21, 56, 105, 168, 245, 336, 441, 560, 693, 840, 1001, 1176, 1365, 1568, 1785, 2016, 2261, 2520, 2793, 3080, 3381, 3696, 4025, 4368, 4725, 5096, 5481, 5880, 6293, 6720, 7161, 7616, 8085, 8568, 9065, 9576, 10101, 10640, 11193, 11760, 12341, 12936

Offset: 1

Robert G. Wilson v, Feb 05 2002

Positive numbers k such that 7*(7 + k) is a perfect square.

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

Cf. A186029.

Cf. numbers k such that k*(k + m) is a perfect square: A028560 (k=9), A067728 (k=8), A067726 (k=6), A067724 (k=5), A028347 (k=4), A067725 (k=3), A054000 (k=2), A005563 (k=1).

List([1..45], n-> 7*n*(n+2)); # G. C. Greubel, Sep 01 2019

[7*n*(n+2): n in [1..50]]; // Vincenzo Librandi, Jul 08 2012

seq(7*n*(n+2), n=1..45); # G. C. Greubel, Sep 01 2019

Select[ Range[15000], IntegerQ[ Sqrt[ 7(7 + # )]] & ]
CoefficientList[Series[7*(3-x)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 08 2012 *)
7*(Range[2,45]^2 -1) (* G. C. Greubel, Sep 01 2019 *)
LinearRecurrence[{3,-3,1},{21,56,105},50] (* Harvey P. Dale, Dec 07 2022 *)

a(n)= 7*n*(n+2) \\ Charles R Greathouse IV, Dec 07 2011

[7*n*(n+2) for n in (1..45)] # G. C. Greubel, Sep 01 2019

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 08 2012
G.f.: 7*x*(3-x)/(1-x)^3. - Vincenzo Librandi, Jul 08 2012
E.g.f.: 7*x*(3 + x)*exp(x). - G. C. Greubel, Sep 01 2019
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 3/28.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/28. (End)

Edited by Charles R Greathouse IV, Jul 25 2010

A100345 Triangle read by rows: T(n,k) = n*(n+k), 0 <= k <= n.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 9, 12, 15, 18, 16, 20, 24, 28, 32, 25, 30, 35, 40, 45, 50, 36, 42, 48, 54, 60, 66, 72, 49, 56, 63, 70, 77, 84, 91, 98, 64, 72, 80, 88, 96, 104, 112, 120, 128, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190

Offset: 0

Reinhard Zumkeller, Nov 18 2004

Distinct members (except 0) are in A071562. Numbers occurring at least twice are in A175040. - Franklin T. Adams-Watters, Apr 04 2010

			Triangle begins:
   0
   1   2
   4   6   8
   9  12  15  18
  16  20  24  28  32
  25  30  35  40  45  50
  36  42  48  54  60  66  72
  49  56  63  70  77  84  91  98
  64  72  80  88  96 104 112 120 128

Cf. A000290, A000384, A001105, A002378, A005563, A014107, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A046092, A054000, A071355, A071562, A085789, A098603, A175040.

Table[n(n+k),{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Aug 16 2018 *)

row(n) = vector(n+1, k, n*(n+k-1)); \\ Amiram Eldar, May 09 2025

T(n,0) = A000290(n).
T(n,1) = A002378(n) for n > 0.
T(n,2) = A005563(n) for n > 1.
T(n,3) = A028552(n) for n > 2.
T(n,4) = A028347(n+2) for n > 3.
T(n,5) = A028557(n) for n > 4.
T(n,6) = A028560(n) for n > 5.
T(n,7) = A028563(n) for n > 6.
T(n,8) = A028566(n) for n > 7.
T(n,9) = A028569(n) for n > 8.
T(n,10) = A098603(n) for n > 9.
T(n,n-5) = A071355(n-4) for n > 4.
T(n,n-4) = A054000(n-1) for n > 3.
T(n,n-3) = A014107(n) for n > 2.
T(n,n-2) = A046092(n-1) for n > 1.
T(n,n-1) = A000384(n) for n > 0.
T(n,n) = A001105(n).
Row sums give A085789 for n > 0.
G.f.: x*(1 + 2*y + 6*x^3*y^2 - 3*x^2*y*(1 + 2*y) + x*(1 - 3*y + 2*y^2))/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Jul 03 2025

A285165 Triangle read by rows: T(n,k) is the number of c-nets with n-k inner vertices and k outer vertices, 3 <= n, 2 <= k <= n-1.

Original entry on oeis.org

1, 1, 1, 7, 6, 1, 73, 56, 16, 1, 879, 640, 208, 30, 1, 11713, 8256, 2848, 560, 48, 1, 167423, 115456, 41216, 9440, 1240, 70, 1, 2519937, 1710592, 624384, 156592, 25864, 2408, 96, 1, 39458047, 26468352, 9812992, 2613664, 496944, 61712, 4256, 126, 1, 637446145, 423641088, 158883840, 44169600, 9234368, 1377600, 132480, 7008, 160, 1, 10561615871, 6966960128, 2636197888, 756712960, 169378560, 28663040, 3430528, 261648, 10920, 198, 1

Offset: 3

Gheorghe Coserea, Apr 12 2017

			Triangle starts:
n\k  [2]       [3]       [4]      [5]      [6]     [7]    [8]   [9]  [10]
[3]  1;
[4]  1,        1;
[5]  7,        6,        1;
[6]  73,       56,       16,      1;
[7]  879,      640,      208,     30,      1;
[8]  11713,    8256,     2848,    560,     48,     1
[9]  167423,   115456,   41216,   9440,    1240,   70,    1;
[10] 2519937,  1710592,  624384,  156592,  25864,  2408,  96,   1;
[11] 39458047, 26468352, 9812992, 2613664, 496944, 61712, 4256, 126, 1;
[12] ...

Gheorghe Coserea, Rows n=3..203, flattened
M. Bodirsky, C. Groepl, D. Johannsen and M. Kang, A Direct Decomposition of 3-connected Planar Graphs, conference paper (FPSAC05).

Cf. A290326.

Columns k=2-9 give: A106651(k=2), A285166(k=3), A285167(k=4), A285168(k=5), A285169(k=6), A285170(k=7), A285171(k=8), A285172(k=9).

x='x; y='y;
system("wget http://oeis.org/A106651/a106651.txt");
Fy = read("a106651.txt");
A106651_ser(N) = {
  my(y0 = 1 + O(x^N), y1=0, n=1);
  while(n++,
    y1 = y0 - subst(Fy, y, y0)/subst(deriv(Fy, y), y, y0);
    if (y1 == y0, break()); y0 = y1);
  y0;
};
z='z; t='t; u='u; c0='c0;
r1 = 2*t*u + 2*t^2*u + 2*t*u^2 + 2*t^2*u^2;
r2 = 4*t^2 + 4*t^3 + 4*t^2*u + 4*t^3*u;
r3 = -4*t^2 - 4*t^3 - 2*t*u - 6*t^2*u - 4*t^3*u - 2*t*u^2 - 2*t^2*u^2;
r4 = 2*t + 2*t^2 + 4*t^3 - u + t*u + 4*t^3*u + u^2 + t*u^2 - 2*t^2*u^2;
r5 = -2*t - 2*t^2 - 4*t^3 - 4*t*u - 2*t^2*u - 4*t^3*u + 2*t^2*u^2;
r6 = u + 2*t*u + 2*t^2*u - t*u^2;
Fz = r1*z^2 + (r3*c0 + r4)*z + r2*c0^2 + r5*c0 + r6;
seq(N) = {
  N += 10; my(z0 = 1 + O(t^N) + O(u^N), z1=0, n=1,
  Fz = subst(Fz, 'c0, subst(A106651_ser(N), 'x, 't)));
  while(n++,
    z1 = z0 - subst(Fz, z, z0)/subst(deriv(Fz, z) , z, z0);
    if (z1 == z0, break()); z0 = z1);
  vector(N-10, n, vector(n, k, polcoeff(polcoeff(z0, n-k), k-1)));
};
concat(seq(11))

A106651(n) = T(n,2) = Sum_{k=3..n-1} T(n,k), for n>=4.

T(n,n-2) = A054000(n-3) for n>= 5, T(n,n-3) = 8*A006325(n-3) for n>=6. - Gheorghe Coserea, Apr 19 2017

A067724 a(n) = 5n^2 + 10n.

Original entry on oeis.org

15, 40, 75, 120, 175, 240, 315, 400, 495, 600, 715, 840, 975, 1120, 1275, 1440, 1615, 1800, 1995, 2200, 2415, 2640, 2875, 3120, 3375, 3640, 3915, 4200, 4495, 4800, 5115, 5440, 5775, 6120, 6475, 6840, 7215, 7600, 7995, 8400, 8815, 9240, 9675

Offset: 1

Robert G. Wilson v, Feb 05 2002

Positive numbers m such that 5*(5 + m) is a perfect square.

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

Cf. numbers k such that k*(k + m) is a perfect square: A028560 (k=9), A067728 (k=8), A067727 (k=7), A067726 (k=6), A028347 (k=4), A067725 (k=3), A054000 (k=2), A067998 (k=1).

Cf. A055998.

[5*n*(n+2): n in [1..50]]; // Vincenzo Librandi, Jul 08 2012

Select[Range[10000], IntegerQ[ Sqrt[5 (5 + # )]] &]
CoefficientList[Series[5 (3 - x)/(1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 08 2012 *)
Table[5n^2+10n,{n,60}] (* or *) LinearRecurrence[{3,-3,1},{15,40,75},60] (* Harvey P. Dale, May 22 2018 *)

a(n)=5*n*(n+2) \\ Charles R Greathouse IV, Dec 07 2011

From Vincenzo Librandi, Jul 08 2012: (Start)
G.f.: 5*x*(3 - x)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = A055998(3*n) + A055998(n). - Bruno Berselli, Sep 23 2016
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 3/20.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/20. (End)
E.g.f.: 5*exp(x)*x*(3 + x). - Stefano Spezia, Oct 01 2023

A067726 a(n) = 6n^2 + 12n.

Original entry on oeis.org

18, 48, 90, 144, 210, 288, 378, 480, 594, 720, 858, 1008, 1170, 1344, 1530, 1728, 1938, 2160, 2394, 2640, 2898, 3168, 3450, 3744, 4050, 4368, 4698, 5040, 5394, 5760, 6138, 6528, 6930, 7344, 7770, 8208, 8658, 9120, 9594, 10080, 10578, 11088, 11610

Offset: 1

Robert G. Wilson v, Feb 05 2002

Positive numbers k such that 6*(6 + k) is a perfect square.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Leo Tavares, Illustration: Star Cut Hexagons
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. numbers k such that k*(k + m) is a perfect square: A028560 (k=9), A067728 (k=8), A067727 (k=7), A067724 (k=5), A028347 (k=4), A067725 (k=3), A054000 (k=2), A005563 (k=1).

Cf. A003154, A003215.

List([1..45], n-> 6*n*(n+2)); # G. C. Greubel, Sep 01 2019

[6*n*(n+2): n in [1..50]]; // Vincenzo Librandi, Jul 08 2012

seq(6*n*(n+2), n=1..45); # G. C. Greubel, Sep 01 2019

Select[ Range[15000], IntegerQ[ Sqrt[ 6(6 + # )]] & ]
CoefficientList[Series[6*(3-x)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 08 2012 *)
6*(Range[2, 45]^2 -1) (* G. C. Greubel, Sep 01 2019 *)
LinearRecurrence[{3,-3,1},{18,48,90},60] (* Harvey P. Dale, May 10 2022 *)

a(n)=6*n*(n+2) \\ Charles R Greathouse IV, Dec 07 2011

[6*n*(n+2) for n in (1..45)] # G. C. Greubel, Sep 01 2019

G.f.: 6*x*(3 - x)/(1 - x)^3. - Vincenzo Librandi, Jul 08 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 08 2012
E.g.f.: 6*x*(3 + x)*exp(x). - G. C. Greubel, Sep 01 2019
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/8.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/24. (End)
a(n) = A003215(2*n) - A003154(n). - Leo Tavares, May 20 2023
a(n) = 6*A005563(n). - Hugo Pfoertner, May 24 2023

A067721 Least number k such that k (k + n) is a perfect square, or 0 if impossible.

Original entry on oeis.org

1, 0, 0, 1, 0, 4, 2, 9, 1, 3, 8, 25, 4, 36, 18, 1, 2, 64, 6, 81, 16, 4, 50, 121, 1, 20, 72, 9, 36, 196, 2, 225, 4, 11, 128, 1, 12, 324, 162, 13, 5, 400, 8, 441, 100, 3, 242, 529, 1, 63, 40, 17, 144, 676, 18, 9, 7, 19, 392, 841, 4, 900, 450, 1, 8, 16, 22, 1089, 256, 23, 2, 1225

Offset: 0

Robert G. Wilson v, Feb 05 2002

nonn

Impossible only for 1, 2 and 4. k equals 1 when n is in A005563. k equals 2 when n is in A054000.

Let k*(k+n)= c*c, gcd(n,k,c)=1 . Then primitive triples (n,k,c) are of the form : 1) n is prime. (n,k,c)=( p, (p*p-2*p+1)/4, (p*p-1)/4 ) 2) n=(c/t)*(c/t)- t*t, n is not a prime, t positive integer. (n,k,c)=( (c/t)*(c/t)- t*t, t*t, c ). [Ctibor O. Zizka, May 04 2009]

			a(7) = 9 because 9 (7+9) = 144 = 12^2.

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Carmine Suriano)

Cf. A005563, A007913, A054000, A067632, A076942.

Do[k = 1; While[ !IntegerQ[ Sqrt[ k (k + n)]], k++ ]; Print[k], {n, 5, 75} ]

from itertools import takewhile
from collections import deque
from sympy import divisors
def A067721(n): return ((a:=next(iter(deque((d for d in takewhile(lambda d:d>2) if n else 1 # Chai Wah Wu, Aug 21 2024

A268581 a(n) = 2n^2 + 8n + 5.

Original entry on oeis.org

5, 15, 29, 47, 69, 95, 125, 159, 197, 239, 285, 335, 389, 447, 509, 575, 645, 719, 797, 879, 965, 1055, 1149, 1247, 1349, 1455, 1565, 1679, 1797, 1919, 2045, 2175, 2309, 2447, 2589, 2735, 2885, 3039, 3197, 3359, 3525, 3695, 3869, 4047, 4229, 4415, 4605

Offset: 0

Juri-Stepan Gerasimov, Apr 10 2016

Also, numbers m such that 2*m + 6 is a square.

All the terms end with a digit in {5, 7, 9}, or equivalently, are congruent to {5, 7, 9} mod 10. - Stefano Spezia, Aug 05 2021

Stefano Spezia, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. numbers n such that 2*n + k is a perfect square: A093328 (k=-6), A097080 (k=-5), no sequence (k=-4), A051890 (k=-3), A058331 (k=-2), A001844 (k=-1), A001105 (k=0), A046092 (k=1), A056222 (k=2), A142463 (k=3), A054000 (k=4), A090288 (k=5), this sequence (k=6), A059993 (k=7), A147973 (k=8), A139570 (k=9), no sequence (k=10), A222182 (k=11), A152811 (k=12), A181570 (k=13).

Magma
```
[2*n^2+8*n+5: n in [0..60]];
```
Magma
```
[n: n in [0..6000] | IsSquare(2*n+6)];
```

Table[2 n^2 + 8 n + 5, {n, 0, 50}] (* Vincenzo Librandi, Apr 13 2016 *)
LinearRecurrence[{3,-3,1},{5,15,29},50] (* Harvey P. Dale, Jan 18 2017 *)

lista(nn) = for(n=0, nn, print1(2*n^2+8*n+5, ", ")); \\ Altug Alkan, Apr 10 2016

[2*n^2 + 8*n + 5 for n in [0..46]] # Stefano Spezia, Aug 04 2021

From Vincenzo Librandi, Apr 13 2016: (Start)
G.f.: (5-x^2)/(1-x)^3.
a(n) = 2*(n+2)^2 - 3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
E.g.f.: exp(x)*(5 + 10*x + 2*x^2). - Stefano Spezia, Aug 03 2021

Changed offset from 1 to 0, adapted formulas and programs by Bruno Berselli, Apr 13 2016

A271625 a(n) = = 2*(n+1)^2 - 5.

Original entry on oeis.org

3, 13, 27, 45, 67, 93, 123, 157, 195, 237, 283, 333, 387, 445, 507, 573, 643, 717, 795, 877, 963, 1053, 1147, 1245, 1347, 1453, 1563, 1677, 1795, 1917, 2043, 2173, 2307, 2445, 2587, 2733, 2883, 3037, 3195, 3357, 3523, 3693, 3867, 4045, 4227, 4413, 4603, 4797, 4995, 5197, 5403, 5613, 5827

Offset: 1

Juri-Stepan Gerasimov, Apr 11 2016

Numbers n such that 2*n + 10 is a perfect square.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A201713.

Numbers h such that 2*h + k is a perfect square: A294774 (k=-9), A255843 (k=-8), A271649 (k=-7), A093328 (k=-6), A097080 (k=-5), A271624 (k=-4), A051890 (k=-3), A058331 (k=-2), A001844 (k=-1), A001105 (k=0), A046092 (k=1), A056222 (k=2), A142463 (k=3), A054000 (k=4), A090288 (k=5), A268581 (k=6), A059993 (k=7), (-1)*A147973 (k=8), A139570 (k=9), this sequence (k=10), A222182 (k=11), A152811 (k=12), A181510 (k=13), A161532 (k=14), no sequence (k=15).

Magma
```
[ 2*n^2 + 4*n - 3: n in [1..60]];
```

[ n: n in [1..6000] | IsSquare(2*n+10)];

Table[2 n^2 + 4 n - 3, {n, 53}] (* Michael De Vlieger, Apr 11 2016 *)
LinearRecurrence[{3,-3,1},{3,13,27},60] (* Harvey P. Dale, Jun 08 2023 *)
2*Range[2,60]^2 -5 (* G. C. Greubel, Jan 21 2025 *)

x='x+O('x^99); Vec(x*(3+4*x-3*x^2)/(1-x)^3) \\ Altug Alkan, Apr 11 2016

def A271625(n): return 2*pow(n+1,2) - 5
print([A271625(n) for n in range(1,61)]) # G. C. Greubel, Jan 21 2025

G.f.: x*(3 + 4*x - 3*x^2)/(1 - x)^3. - Ilya Gutkovskiy, Apr 11 2016
Sum_{n>=1} 1/a(n) = 13/30 - Pi*cot(sqrt(5/2)*Pi)/(2*sqrt(10)) = 0.5627678459924... . - Vaclav Kotesovec, Apr 11 2016
From Elmo R. Oliveira, Nov 17 2024: (Start)
E.g.f.: exp(x)*(2*x^2 + 6*x - 3) + 3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
a(n) = 2*A000290(n+1) - 5. - G. C. Greubel, Jan 21 2025

Name simplified by G. C. Greubel, Jan 21 2025

cp's OEIS Frontend

A132356 a(2*k) = k*(10*k+2), a(2*k+1) = 10*k^2 + 18*k + 8, with k >= 0.

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A090238 Triangle T(n, k) read by rows. T(n, k) is the number of lists of k unlabeled permutations whose total length is n.

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A067727 a(n) = 7*n^2 + 14*n.

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A100345 Triangle read by rows: T(n,k) = n*(n+k), 0 <= k <= n.

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A285165 Triangle read by rows: T(n,k) is the number of c-nets with n-k inner vertices and k outer vertices, 3 <= n, 2 <= k <= n-1.

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A067724 a(n) = 5*n^2 + 10*n.

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A067726 a(n) = 6*n^2 + 12*n.

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A132356 a(2k) = k(10k+2), a(2k+1) = 10k^2 + 18k + 8, with k >= 0.

A067727 a(n) = 7n^2 + 14n.

A067724 a(n) = 5n^2 + 10n.

A067726 a(n) = 6n^2 + 12n.

A268581 a(n) = 2n^2 + 8n + 5.