A085787 -id:A085787 - cp's OEIS frontend

A316672 Numbers k for which 120*k + 169 is a square.

-1, 0, 1, 3, 10, 14, 17, 22, 36, 43, 48, 56, 77, 87, 94, 105, 133, 146, 155, 169, 204, 220, 231, 248, 290, 309, 322, 342, 391, 413, 428, 451, 507, 532, 549, 575, 638, 666, 685, 714, 784, 815, 836, 868, 945, 979, 1002, 1037, 1121, 1158, 1183, 1221, 1312, 1352, 1379, 1420

Offset: 1

Bruno Berselli, Jul 10 2018

All terms of A303305 belong to this sequence.

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

Cf. A051869, A303305.
Subsequence of A047283.
Cf. Numbers k for which 8*(2*h+1)*k + (2*h-1)^2 is a square: A000217 (h=0), A001318 (h=1), A085787 (h=2), A118277 (h=3), A195160 (h=4), A195313 (h=5), A277082 (h=6), this sequence (h=7), A303813 (h=8), A303298 (h=9); A303815 (h=13).

[k: k in [0..1500] | IsSquare(120*k+169)];

select(k->issqr(120*k+169),[$-1..1500]); # Muniru A Asiru, Jul 10 2018

LinearRecurrence[{1, 0, 0, 2, -2, 0, 0, -1, 1}, {-1, 0, 1, 3, 10, 14, 17, 22, 36}, 60]

isok(n) = issquare(120*n+169); \\ Michel Marcus, Jul 11 2018

Vec(x*(-1 + x + x^2 + 2*x^3 + 9*x^4 + 2*x^5 + x^6 + x^7 - x^8)/((1 + x)^2*(1 - x)^3*(1 + x^2)^2) + O(x^40)) \\ Colin Barker, Jul 18 2018

print([k for k in (0..1500) if is_square(120*k+169)])

O.g.f.: x*(-1 + x + x^2 + 2*x^3 + 9*x^4 + 2*x^5 + x^6 + x^7 - x^8)/((1 + x)^2*(1 - x)^3*(1 + x^2)^2).
a(n) = a(1-n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9).
a(n) = (30*n^2 - 2*(15 + 3*(-1)^n + 10*i^(n*(n+1)))*n + 2*(5 + (-1)^n)*i^(n*(n+1)) + 3*(-1)^n - 79)/64, with i = sqrt(-1). Therefore:
a(4*k+1) = (3*k + 2)*(5*k - 1)/2;
a(4*k+2) = k*(15*k + 13)/2, first bisection of A303305;
a(4*k+3) = (k + 1)*(15*k + 2)/2, second bisection of A303305 (see A051869);
a(4*k+4) = (3*k + 1)*(5*k + 6)/2.

A274579 Values of k such that 2k+1 and 5k+1 are both triangular numbers.

Original entry on oeis.org

0, 1, 7, 27, 540, 2002, 10660, 39501, 779247, 2887450, 15372280, 56960982, 1123674201, 4163701465, 22166817667, 82137697110, 1620337419162, 6004054625647, 31964535704101, 118442502272205, 2336525434757970, 8657842606482076, 46092838318496542

Offset: 1

Colin Barker, Jun 29 2016

Intersection of A074377 and A085787.

			7 is in the sequence because 2*7+1 = 15, 5*7+1 = 36, and 15 and 36 are both triangular numbers.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1442,-1442,0,0,-1,1).

Cf. A000217, A074377, A085787, A124174.

concat(0, Vec(x^2*(1+6*x+20*x^2+513*x^3+20*x^4+6*x^5+x^6)/((1-x)*(1+6*x-x^2)*(1-6*x-x^2)*(1+38*x^2+x^4)) + O(x^30)))

isok(n) = ispolygonal(2*n+1, 3) && ispolygonal(5*n+1, 3); \\ Michel Marcus, Jun 29 2016

G.f.: x^2*(1+6*x+20*x^2+513*x^3+20*x^4+6*x^5+x^6) / ((1-x)*(1+6*x-x^2)*(1-6*x-x^2)*(1+38*x^2+x^4)).

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

A247643 a(n) = ( 10n(n+1)+(2n+1)(-1)^n+7 )/8.

Original entry on oeis.org

1, 3, 9, 15, 27, 37, 55, 69, 93, 111, 141, 163, 199, 225, 267, 297, 345, 379, 433, 471, 531, 573, 639, 685, 757, 807, 885, 939, 1023, 1081, 1171, 1233, 1329, 1395, 1497, 1567, 1675, 1749, 1863, 1941, 2061, 2143, 2269, 2355, 2487, 2577, 2715, 2809, 2953, 3051

Offset: 0

N. J. A. Sloane, Sep 23 2014

From Paul Curtz, Jan 01 2020: (Start)
In the following pentagonal spiral of odd numbers
                     101
                  99  61  63
              97  59  31  33  65
          95  57  29  11  13  35  67
      93  55  27   9   1   3  15  37  69
        91  53  25   7   5  17  39  71
          89  51  23  21  19  41  73
            87  49  47  45  43  75
              85  83  81  79  77
the terms of this sequence appear on the x axis. A062786 and A172043 are in the spiral as well. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

A diagonal of triangle in A247646.

Cf. A005408, A062786, A085787, A090772, A172043.

Maple
```
f:=n->(10*n*(n+1)+(2*n+1)*(-1)^n+7)/8;
```

Table[(10 n (n + 1) + (2 n + 1) (-1)^n + 7)/8, {n, 0, 60}] (* Vincenzo Librandi, Sep 26 2014 *)

Vec(-(x^4+2*x^3+4*x^2+2*x+1) / ((x-1)^3*(x+1)^2) + O(x^100)) \\ Colin Barker, Sep 25 2014

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Colin Barker, Sep 25 2014
G.f.: -(x^4+2*x^3+4*x^2+2*x+1) / ((x-1)^3*(x+1)^2). - Colin Barker, Sep 25 2014
From Paul Curtz, Jan 01 2020: (Start)
a(n) = 1 + 2*A085787(n).
a(n+1) = a(n-1) + A090772(n+1). (End)
E.g.f.: (1/4)*((1 + x)*(4 + 5*x)*cosh(x) + (3 + x*(11 + 5*x))*sinh(x)). - Stefano Spezia, Jan 01 2020

More terms from Colin Barker, Sep 25 2014

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

A274681 Numbers k such that 4*k + 1 is a triangular number.

Original entry on oeis.org

0, 5, 11, 26, 38, 63, 81, 116, 140, 185, 215, 270, 306, 371, 413, 488, 536, 621, 675, 770, 830, 935, 1001, 1116, 1188, 1313, 1391, 1526, 1610, 1755, 1845, 2000, 2096, 2261, 2363, 2538, 2646, 2831, 2945, 3140, 3260, 3465, 3591, 3806, 3938, 4163, 4301, 4536

Offset: 1

Colin Barker, Jul 02 2016

Also, numbers of the form m*(8*m + 3) for m = 0, -1, 1, -2, 2, -3, 3, ... - Bruno Berselli, Feb 26 2018

			5 is in the sequence since 4*5 + 1 = 21 is a triangular number (21 = 1 + 2 + 3 + 4 + 5 + 6). - _Michael B. Porter_, Jul 03 2016

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

Cf. A000217, A000096 (n+1), A074377 (2*n+1), A045943 (3*n+1), A085787 (5*n+1).
Cf. A057029.
Cf. similar sequences listed in A299645.

[(1-(-1)^n+2*(-4+(-1)^n)*n+8*n^2)/4: n in [1..80]]; // Wesley Ivan Hurt, Jul 02 2016

A274681:=n->(1-(-1)^n+2*(-4+(-1)^n)*n+8*n^2)/4: seq(A274681(n), n=1..100); # Wesley Ivan Hurt, Jul 02 2016

Rest@ CoefficientList[Series[x^2 (5 + 6 x + 5 x^2)/((1 - x)^3 (1 + x)^2), {x, 0, 48}], x] (* Michael De Vlieger, Jul 02 2016 *)
Select[Range[0,5000],OddQ[Sqrt[8(4#+1)+1]]&] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{0,5,11,26,38},50] (* Harvey P. Dale, Apr 21 2018 *)

PARI
```
isok(n) = ispolygonal(4*n+1, 3)
```

select(n->ispolygonal(4*n+1, 3), vector(10000, n, n-1))

concat(0, Vec(x^2*(5+6*x+5*x^2)/((1-x)^3*(1+x)^2) + O(x^100)))

def A274681(n): return (n>>1)*((n<<2)+(-1 if n&1 else -3)) # Chai Wah Wu, Mar 11 2025

G.f.: x^2*(5 + 6*x + 5*x^2) / ((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) for n>5.
a(n) = A057029(n) - 1.
a(n) = (1 - (-1)^n + 2*(-4 + (-1)^n)*n + 8*n^2)/4.
a(n) = (4*n^2 - 3*n)/2 for n even, a(n) = (4*n^2 - 5*n + 1)/2 for n odd.

A274757 Numbers k such that 6*k+1 is a triangular number (A000217).

Original entry on oeis.org

0, 9, 15, 42, 54, 99, 117, 180, 204, 285, 315, 414, 450, 567, 609, 744, 792, 945, 999, 1170, 1230, 1419, 1485, 1692, 1764, 1989, 2067, 2310, 2394, 2655, 2745, 3024, 3120, 3417, 3519, 3834, 3942, 4275, 4389, 4740, 4860, 5229, 5355, 5742, 5874, 6279, 6417

Offset: 1

Colin Barker, Jul 04 2016

Numbers of the type floor(3*m*(m+1)/4) for which floor(3*m*(m+1)/4) = 3*floor(m*(m+1)/4). A014601 lists the values of m. - Bruno Berselli, Jan 13 2017

Numbers of the form 3*k*(4*k + 1) for k in Z. - Peter Bala, Nov 21 2024

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

Cf. A000217, A014601.
Cf. A000096 (k+1), A074377 (2*k+1), A045943 (3*k+1), A274681 (4*k+1), A085787 (5*k+1).
Cf. similar sequences listed in A274830.

Table[3 (2 n - 1) (2 n + (-1)^n - 1)/4, {n, 1, 60}] (* Bruno Berselli, Jul 08 2016 *)
LinearRecurrence[{1,2,-2,-1,1},{0,9,15,42,54},50] (* Harvey P. Dale, Apr 13 2025 *)

PARI
```
isok(n) = ispolygonal(6*n+1, 3)
```

select(n->ispolygonal(6*n+1, 3), vector(7000, n, n-1))

concat(0, Vec(3*x^2*(3+2*x+3*x^2)/((1-x)^3*(1+x)^2) + O(x^60)))

G.f.: 3*x^2*(3 + 2*x + 3*x^2) / ((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) for n>5.
a(n) = 3*(2*n - 1)*(2*n + (-1)^n - 1)/4. Therefore:
a(n) = 3*n*(2*n - 1)/2 for n even,
a(n) = 3*(n-1)*(2*n - 1)/2 for n odd.

A274830 Numbers k such that 7*k+1 is a triangular number (A000217).

Original entry on oeis.org

0, 2, 5, 11, 17, 27, 36, 50, 62, 80, 95, 117, 135, 161, 182, 212, 236, 270, 297, 335, 365, 407, 440, 486, 522, 572, 611, 665, 707, 765, 810, 872, 920, 986, 1037, 1107, 1161, 1235, 1292, 1370, 1430, 1512, 1575, 1661, 1727, 1817, 1886, 1980, 2052, 2150, 2225

Offset: 1

Colin Barker, Jul 08 2016

From Peter Bala, Nov 21 2024: (Start)
Numbers of the form n*(7*n + 3)/2 for n in Z. Cf. A057570.
The sequence terms occur as the exponents in the expansion of Product_{n >= 1} (1 - x^(7*n)) * (1 + x^(7*n-2)) * (1 + x^(7*n-5)) = 1 + x^2 + x^5 + x^11 + x^17 + x^27 + x^36 + .... Cf. A363800. (End)

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

Cf. A000217, A057570, A363800.

Cf. similar sequences where k*n+1 is a triangular number: A000096 (k=1), A074377 (k=2), A045943 (k=3), A274681 (k=4), A085787 (k=5), A274757 (k=6).

Table[(14 (n - 1) n + (2 n - 1) (-1)^n + 1)/16, {n, 1, 60}] (* Bruno Berselli, Jul 08 2016 *)

select(n->ispolygonal(7*n+1, 3), vector(3000, n, n-1))

concat(0, Vec(x^2*(2+3*x+2*x^2)/((1-x)^3*(1+x)^2) + O(x^100)))

G.f.: x^2*(2 + 3*x + 2*x^2) / ((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) for n>5.
a(n) = (14*(n - 1)*n + (2*n - 1)*(-1)^n + 1)/16. Therefore:
a(n) = n*(7*n - 6)/8 for n even,
a(n) = (n - 1)*(7*n - 1)/8 for n odd.
E.g.f.: (x*(7*x -1)*cosh(x) + (7*x^2 + x + 1)*sinh(x))/8. - Stefano Spezia, Nov 26 2024

Edited by Bruno Berselli, Jul 08 2016

A303301 Square array T(n,k) read by antidiagonals upwards in which row n is obtained by taking the general formula for generalized n-gonal numbers: m((n - 2)m - n + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and n >= 5. Here n >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, -3, 0, 1, -2, 0, 0, 1, -1, 1, -8, 0, 1, 0, 2, -5, -3, 0, 1, 1, 3, -2, 0, -15, 0, 1, 2, 4, 1, 3, -9, -8, 0, 1, 3, 5, 4, 6, -3, -2, -24, 0, 1, 4, 6, 7, 9, 3, 4, -14, -15, 0, 1, 5, 7, 10, 12, 9, 10, -4, -5, -35, 0, 1, 6, 8, 13, 15, 15, 16, 6, 5, -20, -24, 0, 1, 7, 9, 16, 18, 21, 22, 16, 15, -5, -9, -48

Offset: 0

Omar E. Pol, Jun 08 2018

Note that the formula mentioned in the definition gives several kinds of numbers, for example:
Row 0 and row 1 give A317300 and A317301 respectively.
Row 2 gives A001057 (canonical enumeration of integers).
Row 3 gives 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6).
Row 4 gives A008794 (squares repeated) except the initial zero.
Finally, for n >= 5 row n gives the generalized k-gonal numbers (see Crossrefs section).

			Array begins:
------------------------------------------------------------------
n\m  Seq. No.    0   1  -1   2  -2   3   -3    4   -4    5   -5
------------------------------------------------------------------
0    A317300:    0,  1, -3,  0, -8, -3, -15,  -8, -24, -15, -35...
1    A317301:    0,  1, -2,  1, -5,  0,  -9,  -2, -14,  -5, -20...
2    A001057:    0,  1, -1,  2, -2,  3,  -3,   4,  -4,   5,  -5...
3   (A008795):   0,  1,  0,  3,  1,  6,   3,  10,   6,  15,  10...
4   (A008794):   0,  1,  1,  4,  4,  9,   9,  16,  16,  25,  25...
5    A001318:    0,  1,  2,  5,  7, 12,  15,  22,  26,  35,  40...
6    A000217:    0,  1,  3,  6, 10, 15,  21,  28,  36,  45,  55...
7    A085787:    0,  1,  4,  7, 13, 18,  27,  34,  46,  55,  70...
8    A001082:    0,  1,  5,  8, 16, 21,  33,  40,  56,  65,  85...
9    A118277:    0,  1,  6,  9, 19, 24,  39,  46,  66,  75, 100...
10   A074377:    0,  1,  7, 10, 22, 27,  45,  52,  76,  85, 115...
11   A195160:    0,  1,  8, 11, 25, 30,  51,  58,  86,  95, 130...
12   A195162:    0,  1,  9, 12, 28, 33,  57,  64,  96, 105, 145...
13   A195313:    0,  1, 10, 13, 31, 36,  63,  70, 106, 115, 160...
14   A195818:    0,  1, 11, 14, 34, 39,  69,  76, 116, 125, 175...
15   A277082:    0,  1, 12, 15, 37, 42,  75,  82, 126, 135, 190...
...

Alois P. Heinz, Antidiagonals n = 0..200 (first 45 antidiagonals from Robert G. Wilson v)

Columns 0..2 are A000004, A000012, A023445.
Column 3 gives A001477 which coincides with the row numbers.
Main diagonal gives A292551.
Row 0-2 gives A317300, A317301, A001057.
Row 3 gives 0 together with A008795.
Row 4 gives A008794.
For n >= 5, rows n gives the generalized n-gonal numbers: A001318 (n=5), A000217 (n=6), A085787 (n=7), A001082 (n=8), A118277 (n=9), A074377 (n=10), A195160 (n=11), A195162 (n=12), A195313 (n=13), A195818 (n=14), A277082 (n=15), A274978 (n=16), A303305 (n=17), A274979 (n=18), A303813 (n=19), A218864 (n=20), A303298 (n=21), A303299 (n=22), A303303 (n=23), A303814 (n=24), A303304 (n=25), A316724 (n=26), A316725 (n=27), A303812 (n=28), A303815 (n=29), A316729 (n=30).
Cf. A194801, A195152.
Cf. A317302 (a similar table but with polygonal numbers).

t[n_, r_] := PolygonalNumber[n, If[OddQ@ r, Floor[(r + 1)/2], -r/2]]; Table[ t[n - r, r], {n, 0, 11}, {r, 0, n}] // Flatten (* also *)
(* to view the square array *)  Table[ t[n, r], {n, 0, 15}, {r, 0, 10}] // TableForm (* Robert G. Wilson v, Aug 08 2018 *)

T(n,k) = A194801(n-3,k) if n >= 3.

A087348 a(n) = 10n^2 - 6n + 1.

Original entry on oeis.org

5, 29, 73, 137, 221, 325, 449, 593, 757, 941, 1145, 1369, 1613, 1877, 2161, 2465, 2789, 3133, 3497, 3881, 4285, 4709, 5153, 5617, 6101, 6605, 7129, 7673, 8237, 8821, 9425, 10049, 10693, 11357, 12041, 12745, 13469, 14213, 14977, 15761, 16565, 17389, 18233, 19097

Offset: 1

Charlie Marion, Oct 20 2003

Sequence found by reading the line from 5, in the direction 5, 29, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Jul 18 2012

			a(3)=73 since 73^2 = 48^2 + 55^2 = (4*12)^2 + (48 + 7)^2. See 1st formula.

Shawn A. Broyles, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000326, A033567, A033579, A056220, A085787, A153784.

Table[10*n^2 - 6*n + 1, {n, 50}] (* Paolo Xausa, Jul 18 2024 *)

a(n)=10*n^2-6*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n)^2 = A033579(n)^2 + A033567(n)^2 = (4*A000326(n))^2 + (A033579(n) + A056220(n-1))^2.
From Colin Barker, Jun 30 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(5 + 14*x + x^2)/(1-x)^3. (End)
a(n) = 1 + A153784(n). - Omar E. Pol, Jul 18 2012
E.g.f.: exp(x)*(10*x^2 + 4*x + 1) - 1. - Elmo R. Oliveira, Oct 31 2024

More terms from Ray Chandler, Oct 22 2003

cp's OEIS Frontend

A316672 Numbers k for which 120*k + 169 is a square.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A274579 Values of k such that 2*k+1 and 5*k+1 are both triangular numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A247643 a(n) = ( 10*n*(n+1)+(2*n+1)*(-1)^n+7 )/8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A274681 Numbers k such that 4*k + 1 is a triangular number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

PARI

Python

A274579 Values of k such that 2k+1 and 5k+1 are both triangular numbers.

A247643 a(n) = ( 10n(n+1)+(2n+1)(-1)^n+7 )/8.

A303301 Square array T(n,k) read by antidiagonals upwards in which row n is obtained by taking the general formula for generalized n-gonal numbers: m((n - 2)m - n + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and n >= 5. Here n >= 0.

A087348 a(n) = 10n^2 - 6n + 1.