A016802 -id:A016802 - cp's OEIS frontend

A141759 a(n) = 16n^2 + 32n + 15.

15, 63, 143, 255, 399, 575, 783, 1023, 1295, 1599, 1935, 2303, 2703, 3135, 3599, 4095, 4623, 5183, 5775, 6399, 7055, 7743, 8463, 9215, 9999, 10815, 11663, 12543, 13455, 14399, 15375, 16383, 17423, 18495, 19599, 20735, 21903, 23103, 24335, 25599

Offset: 0

Miklos Kristof, Sep 15 2008

Via the partial fraction decomposition 1/((4n+3)*(4n+5)) = (1/2) *(1/(4n+3) -1/(4n+5)) we find 2*Sum_{n>=0} (-1)^n/a(n) = 2*Sum_{n>=0} (-1)^n/( (4*n+3)*(4*n+5) ) = 1/3 -1/5 -1/7 +1/9 +1/11 -1/13 -1/15 +1/17 +1/19 -- ++ ... = (1/1 + 1/3 -1/5 -1/7 +1/9 +1/11 -1/13 -1/15 +1/17 +1/19 -- ++ ..)-1 = Sum_{n>=0} (-1)^n/A016813(n) + Sum_{n>=0} (-1)^n/A004767(n) -1 = -1 + Sum_{n>=0} b(n)/n^1 where b(n) = 1, 0, 1, 0, -1, 0, -1, 0 is a sequence with period length 8, one of the Dirichlet L-series modulo 8. The alternating sum becomes -1 +L(m=8,r=4,s=1) = Pi*sqrt(2)/4-1 = A093954 - 1.
Pi = 4 - 8*Sum(1/a(n)) noted by Bronstein-Semendjajew for the variant a(n) = (4n-1)*(4n+1) starting at n=1. - Frank Ellermann, Sep 18 2011
The identity (16*n^2-1)^2 - (64*n^2-8)*(2*n)^2 = 1 can be written as a(n)^2 - A158487(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 09 2012
Sequence found by reading the line from 15, in the direction 15, 63,... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012
Essentially the least common multiple of 4*n+1 and 4*n-1. - Colin Barker, Feb 11 2017

Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th German ed., 1965, ch. 4.1.8.
Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), pp. 980-981.

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

Cf. A005843, A074377, A093954, A133818, A158487.

[(4*n+3)*(4*n+5): n in [0..50]]; // Vincenzo Librandi, Sep 22 2011

A141759:=n->16*n^2 + 32*n + 15: seq(A141759(n), n=0..60);

LinearRecurrence[{3, -3, 1}, {15, 63, 143}, 50] (* Vincenzo Librandi, Feb 09 2012 *)

a(n)=n*(n+2)<<4+15 \\ Charles R Greathouse IV, Oct 27 2011

G.f.: (15+18*x-x^2)/(1-x)^3.
E.g.f.: (15+48*x+16*x^2)*exp(x).
a(n) = a(-n-2) = A016802(n+1) - 1. - Bruno Berselli, Sep 22 2011
From Amiram Eldar, Feb 04 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = Pi/(2*sqrt(2)) (A093954).
Product_{n>=0} (1 - 1/a(n)) = sin(Pi/(2*sqrt(2))). (End)

Formula indices corrected by R. J. Mathar, Jul 07 2009

A195323 a(n) = 22*n^2.

Original entry on oeis.org

0, 22, 88, 198, 352, 550, 792, 1078, 1408, 1782, 2200, 2662, 3168, 3718, 4312, 4950, 5632, 6358, 7128, 7942, 8800, 9702, 10648, 11638, 12672, 13750, 14872, 16038, 17248, 18502, 19800, 21142, 22528, 23958, 25432, 26950, 28512, 30118, 31768, 33462, 35200, 36982, 38808

Offset: 0

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 0, in the direction 0, 22, ..., in the square spiral whose vertices are the generalized tridecagonal numbers A195313. Semi-axis opposite to A195318 in the same spiral.

Surface area of a rectangular prism with dimensions n, 2n and 3n. - Wesley Ivan Hurt, Apr 10 2015

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

Bisection of A195149.
Cf. A016802, A033581, A033583, A135453, A139098, A144555, A195321, A195322.
Cf. A000290, A001105, A008604, A033584, A195313, A195318.

[22*n^2 : n in [0..40]]; // Vincenzo Librandi, Sep 20 2011

A195323:=n->22*n^2: seq(A195323(n), n=0..80); # Wesley Ivan Hurt, Apr 10 2015

22Range[0,50]^2 (* or *) LinearRecurrence[{3,-3,1},{0,22,88},50] (* Harvey P. Dale, Sep 19 2011 *)

vector(50,n,22*(n-1)^2) \\ Derek Orr, Apr 10 2015

a(n) = 22*A000290(n) = 11*A001105(n) = 2*A033584(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Sep 19 2011
G.f.: 22*x*(1+x)/(1-x)^3. - Wesley Ivan Hurt, Apr 10 2015
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 22*x*(1 + x)*exp(x).
a(n) = n*A008604(n) = A195149(2*n). (End)

A016910 a(n) = (6*n)^2.

Original entry on oeis.org

0, 36, 144, 324, 576, 900, 1296, 1764, 2304, 2916, 3600, 4356, 5184, 6084, 7056, 8100, 9216, 10404, 11664, 12996, 14400, 15876, 17424, 19044, 20736, 22500, 24336, 26244, 28224, 30276, 32400, 34596, 36864, 39204, 41616, 44100, 46656, 49284, 51984, 54756, 57600, 60516, 63504, 66564, 69696, 72900

Offset: 0

N. J. A. Sloane

Areas A of two classes of triangles with integer sides (a,b,c) where a = 9k, b=10k and c = 17k, or a = 3k, b = 25k and c = 26k for k=0,1,2,... These areas are given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) = (6k)^2, with the semiperimeter s = (a+b+c)/2. This sequence is a subsequence of A188158. - Michel Lagneau, Oct 11 2013

Sequence found by reading the line from 0, in the direction 0, 36, ..., in the square spiral whose vertices are the generalized 20-gonal numbers A218864. - Omar E. Pol, May 13 2018.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A089491, A188158, A243941.

Cf. similar sequences of the type k*n^2: A000290 (k=1), A001105 (k=2), A033428 (k=3), A016742 (k=4), A033429 (k=5), A033581 (k=6), A033582 (k=7), A139098 (k=8), A016766 (k=9), A033583 (k=10), A033584 (k=11), A135453 (k=12), A152742 (k=13), A144555 (k=14), A064761 (k=15), A016802 (k=16), A244630 (k=17), A195321 (k=18), A244631 (k=19), A195322 (k=20), A064762 (k=21), A195323 (k=22), A244632 (k=23), A195824 (k=24), A016850 (k=25), A244633 (k=26), A244634 (k=27), A064763 (k=28), A244635 (k=29), A244636 (k=30).

[(6*n)^2: n in [0..40]]; // Vincenzo Librandi, May 03 2011

(6*Range[0,40])^2 (* or *) LinearRecurrence[{3,-3,1},{0,36,144},40] (* Harvey P. Dale, Dec 25 2017 *)
Table[36 n^2, {n, 0, 45}] (* Omar E. Pol, Jun 07 2018 *)

a(n)=36*n^2 \\ Charles R Greathouse IV, Jun 10 2016

From Ilya Gutkovskiy, Jun 09 2016: (Start)
O.g.f.: 36*x*(1 + x)/(1 - x)^3.
E.g.f.: 36*x*(1 + x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
Sum_{n>=1} 1/a(n) = Pi^2/216 = A086726. (End)
Product_{n>=1} a(n)/A136017(n) = Pi/3. - Fred Daniel Kline, Jun 09 2016
a(n) = t(9*n) - 9*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(9*n) - 9*A000217(n). - Bruno Berselli, Aug 31 2017
a(n) = 36*A000290(n) = 18*A001105(n) = 12*A033428 = 9*A016742(n) = 6*A033581(n) = 4*A016766(n) = 3*A135453(n) = 2*A195321(n). - Omar E. Pol, Jun 07 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/432. - Amiram Eldar, Jun 27 2020
From Amiram Eldar, Jan 25 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/6)/(Pi/6).
Product_{n>=1} (1 - 1/a(n)) = sin(Pi/6)/(Pi/6) = 3/Pi (A089491). (End)

A195315 Centered 32-gonal numbers.

Original entry on oeis.org

1, 33, 97, 193, 321, 481, 673, 897, 1153, 1441, 1761, 2113, 2497, 2913, 3361, 3841, 4353, 4897, 5473, 6081, 6721, 7393, 8097, 8833, 9601, 10401, 11233, 12097, 12993, 13921, 14881, 15873, 16897, 17953, 19041, 20161, 21313, 22497, 23713, 24961, 26241, 27553, 28897, 30273

Offset: 1

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 1, in the direction 1, 33, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. Semi-axis opposite to A016802 in the same spiral.

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

Bisection of A195146.
Cf. A003154, A069129, A069133, A069190, A195314, A195316, A195317, A195318.
Cf. A016802, A074377.

[(16*n^2-16*n+1): n in [1..50]]; // Vincenzo Librandi, Sep 19 2011

Table[16*n^2 - 16*n + 1, {n, 1, 41}] (* Amiram Eldar, Feb 11 2022 *)
LinearRecurrence[{3,-3,1},{1,33,97},50] (* Harvey P. Dale, Feb 11 2024 *)

a(n)=16*n^2-16*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 16*n^2 - 16*n + 1.
G.f.: -x*(1 + 30*x + x^2)/(x-1)^3. - R. J. Mathar, Sep 18 2011
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(3)*Pi/4)/(8*sqrt(3)). - Amiram Eldar, Feb 11 2022
From Elmo R. Oliveira, Nov 14 2024: (Start)
E.g.f.: exp(x)*(16*x^2 + 1) - 1.
a(n) = 2*A069129(n) - 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A226008 a(0) = 0; for n>0, a(n) = denominator(1/4 - 4/n^2).

Original entry on oeis.org

0, 4, 4, 36, 1, 100, 36, 196, 16, 324, 100, 484, 9, 676, 196, 900, 64, 1156, 324, 1444, 25, 1764, 484, 2116, 144, 2500, 676, 2916, 49, 3364, 900, 3844, 256, 4356, 1156, 4900, 81, 5476, 1444, 6084, 400, 6724, 1764, 7396, 121, 8100

Offset: 0

Paul Curtz, May 22 2013

Numerators are in A225948.

Repeated terms of A016826 are in the positions 1, 2, 3, 6, 5, 10, ... (A043547).

			a(0) = (-1+1)^2 = 0, a(1) = (-3+5)^2 = 4, a(2) = (-1+3)^2 = 4.

Cf. A016754, A016802, A016826, A017090, A017138, A154615, A225948 (numerators).

Cf. A225975 (associated square roots).

[0] cat [Denominator(1/4-4/n^2): n in [1..50]]; // Bruno Berselli, May 23 2013

Join[{0},Table[Denominator[1/4 - 4/n^2], {n, 49}]] (* Alonso del Arte, May 22 2013 *)

a(n)    = 3*a(n-8) -3*a(n-16) +a(n-24).
a(8n)   = A016802(n), a(8n+4) = A016754(n).
a(4n)   = A154615(n).
a(4n+1) = A017090(n).
a(4n+2) = a(2n+1) = A016826(n); a(2n) = A061038(n).
a(4n+3) = A017138(n).
From Bruno Berselli, May 23 2013: (Start)
G.f.: x*(4 +4*x +36*x^2 +x^3 +100*x^4 +36*x^5 +196*x^6 +16*x^7 +312*x^8 +88*x^9 +376*x^10 +6*x^11 +376*x^12 +88*x^13 +312*x^14 +16*x^15 +196*x^16 +36*x^17 +100*x^18 +x^19 +36*x^20 +4*x^21 +4*x^22)/(1-x^8)^3.
a(n) = n^2*(6*cos(3*Pi*n/4)+6*cos(Pi*n/4)-54*cos(Pi*n/2)-219*(-1)^n+293)/128.
a(n+9) = a(n+1)*((n+9)/(n+1))^2. (End)
Sum_{n>=1} 1/a(n) = 19*Pi^2/96. - Amiram Eldar, Aug 14 2022

Edited by Bruno Berselli, May 23 2013

A244082 a(n) = 32*n^2.

Original entry on oeis.org

0, 32, 128, 288, 512, 800, 1152, 1568, 2048, 2592, 3200, 3872, 4608, 5408, 6272, 7200, 8192, 9248, 10368, 11552, 12800, 14112, 15488, 16928, 18432, 20000, 21632, 23328, 25088, 26912, 28800, 30752, 32768, 34848, 36992, 39200, 41472, 43808, 46208, 48672, 51200

Offset: 0

Wesley Ivan Hurt, Jun 19 2014

Geometric connections of a(n) to the area and perimeter of a square.
Area:
. half the area of a square with side 8n (cf. A008590);
. area of a square with diagonal 8n (cf. A008590);
. twice the area of a square with side 4n (cf. A008586);
. four times the area of a square with diagonal 4n (cf. A008586);
. eight times the area of a square with side 2n (cf. A005843);
. sixteen times the area of a square with diagonal 2n (cf. A005843);
. thirty two times the area of a square with side n (cf. A001477);
. sixty four times the area of a square with diagonal n (cf. A001477).
Perimeter:
. perimeter of a square with side 8n^2 (cf. A139098);
. twice the perimeter of a square with side 4n^2 (cf. A016742);
. four times the perimeter of a square with side 2n^2 (cf. A001105);
. eight times the perimeter of a square with side n^2 (cf. A000290).
Sequence found by reading the line from 0, in the direction 0, 32, ..., in the square spiral whose vertices are the generalized 18-gonal numbers. - Omar E. Pol, May 10 2018

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

Pentasection of A077221, A181900.
Cf. A000290, A001105, A010021, A016742, A016802, A139098.
Cf. A001477, A005843, A008586, A008590, A139098, A174312.

Magma
```
[32*n^2 : n in [0..50]];
```

A244082:=n->32*n^2; seq(A244082(n), n=0..50);

32 Range[0, 50]^2 (* or *)
Table[32 n^2, {n, 0, 50}] (* or *)
CoefficientList[Series[32 x (1 + x)/(1 - x)^3, {x, 0, 30}], x]

a(n)=32*n^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 32*x*(1+x)/(1-x)^3.
a(n) =  2 * A016802(n).
a(n) =  4 * A139098(n).
a(n) =  8 * A016742(n).
a(n) = 16 * A001105(n).
a(n) = 32 * A000290(n).
a(n) = A010021(n) - 2 for n > 0. - Bruno Berselli, Jun 24 2014
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Nov 19 2021
From Elmo R. Oliveira, Dec 02 2024: (Start)
E.g.f.: 32*x*(1 + x)*exp(x).
a(n) = n*A174312(n) = A139098(2*n). (End)

A176027 Binomial transform of A005563.

Original entry on oeis.org

0, 3, 14, 48, 144, 400, 1056, 2688, 6656, 16128, 38400, 90112, 208896, 479232, 1089536, 2457600, 5505024, 12255232, 27131904, 59768832, 131072000, 286261248, 622854144, 1350565888, 2919235584, 6291456000, 13522436096

Offset: 0

Paul Curtz, Dec 06 2010

The numbers appear on the diagonal of a table T(n,k), where the left column contains the elements of A005563, and further columns are recursively T(n,k) = T(n,k-1)+T(n-1,k-1):
....0....-1.....0.....0.....0.....0.....0.....0.....0.....0.
....3.....3.....2.....2.....2.....2.....2.....2.....2.....2.
....8....11....14....16....18....20....22....24....26....28.
...15....23....34....48....64....82...102...124...148...174.
...24....39....62....96...144...208...290...392...516...664.
...35....59....98...160...256...400...608...898..1290..1806.
...48....83...142...240...400...656..1056..1664..2562..3852.
...63...111...194...336...576...976..1632..2688..4352..6914.
...80...143...254...448...784..1360..2336..3968..6656.11008.
...99...179...322...576..1024..1808..3168..5504..9472.16128.
..120...219...398...720..1296..2320..4128..7296.12800.22272.
The second column is A142463, the third A060626, the fourth essentially A035008 and the fifth essentially A016802. Transposing the array gives A005563 and its higher order differences in the individual rows.

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A001792, A001793, A005563, A058396, A084266, A127276.

[2^(n-2)*n*(5+n) : n in [0..30]]; // Vincenzo Librandi, Oct 08 2011

LinearRecurrence[{6,-12,8},{0,3,14},30] (* Harvey P. Dale, Oct 19 2015 *)

a(n)=n*(n+5)<<(n-2) \\ Charles R Greathouse IV, Sep 21 2017

G.f.: x*(-3+4*x)/(2*x-1)^3. - R. J. Mathar, Dec 11 2010
a(n) = 2^(n-2)*n*(5+n). - R. J. Mathar, Dec 11 2010
a(n) = A127276(n) - A127276(n+1).
a(n+1)-a(n) = A084266(n+1).
a(n+2) = 16*A058396(n) for n > 0.
a(n) = 2*a(n-1) + A001792(n).
a(n) = A001793(n) - 2^(n-1) for n > 0. - Brad Clardy, Mar 02 2012
a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} (k+3) * C(n-1,i). - Wesley Ivan Hurt, Sep 20 2017
From Amiram Eldar, Aug 13 2022: (Start)
Sum_{n>=1} 1/a(n) = 1322/75 - 124*log(2)/5.
Sum_{n>=1} (-1)^(n+1)/a(n) = 132*log(3/2)/5 - 782/75. (End)

A017066 a(n) = (8*n)^2.

Original entry on oeis.org

0, 64, 256, 576, 1024, 1600, 2304, 3136, 4096, 5184, 6400, 7744, 9216, 10816, 12544, 14400, 16384, 18496, 20736, 23104, 25600, 28224, 30976, 33856, 36864, 40000, 43264, 46656, 50176, 53824, 57600, 61504, 65536, 69696, 73984, 78400, 82944, 87616, 92416, 97344, 102400

Offset: 0

N. J. A. Sloane

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

Cf. A000290, A008590, A016802, A152691, A244082.

[(8*n)^2: n in [0..35]]; // Vincenzo Librandi, Jul 10 2011

LinearRecurrence[{3, -3, 1}, {0, 64, 256}, 50] (* Vincenzo Librandi, Feb 10 2012 *)

a(n)=(8*n)^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: -64*x*(1+x)/(x-1)^3. - R. J. Mathar, Jul 14 2016
a(n) = A000290(8*n) = A008590(n)^2 = A000290(A008590(n)).
From Amiram Eldar, Jan 25 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/384.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/768.
Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/8)/(Pi/8).
Product_{n>=1} (1 - 1/a(n)) = sin(Pi/8)/(Pi/8) = 4*sqrt(2-sqrt(2))/Pi. (End)
From Elmo R. Oliveira, Dec 06 2024: (Start)
E.g.f.: 64*exp(x)*x*(1 + x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = n*A152691(n) = 2*A244082(n) = A016802(2*n). (End)

A108211 a(n) = 16*n^2 + 1.

Original entry on oeis.org

17, 65, 145, 257, 401, 577, 785, 1025, 1297, 1601, 1937, 2305, 2705, 3137, 3601, 4097, 4625, 5185, 5777, 6401, 7057, 7745, 8465, 9217, 10001, 10817, 11665, 12545, 13457, 14401, 15377, 16385, 17425, 18497, 19601, 20737, 21905, 23105, 24337, 25601, 26897, 28225, 29585

Offset: 1

Reinhard Zumkeller, Jun 15 2005

Area of a Maltese cross conventionally inscribed in a 5n X 5n-grid.
Areas of some other crosses, each made from unit squares, as shown in Weisstein's illustrations: Greek Cross = x-pentomino = 5. Latin Cross = 6. Saint Andrew's cross = crux decussata = 9. Saint Anthony's Cross = tau cross = crux commissa = 10. Gaullist Cross = cross of Lorraine or patriarchal cross = 13. Papal Cross = 22. - Jonathan Vos Post, Jun 18 2005
The identity (16*n^2 + 1)^2 - (64*n^2 + 8)*(2*n)^2 = 1 can be written as a(n)^2 - A158488(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 08 2012
Sequence found by reading the line from 17, in the direction 17, 65, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012
Conjecture: a(n) = floor(1/(1/(4*n) - log(2) + 1/(n+1) + 1/(n+2) + ... + 1/(2*n))). - Clark Kimberling, Sep 09 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Maltese Cross.
Eric Weisstein's World of Mathematics, Gaullist Cross.
Eric Weisstein's World of Mathematics, Greek Cross.
Eric Weisstein's World of Mathematics, Latin Cross.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002522, A005843, A016802, A053755, A074377, A158488.

I:=[17, 65, 145]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 08 2012

A108211:=n->16*n^2+1: seq(A108211(n), n=1..50); # Wesley Ivan Hurt, Sep 24 2014

LinearRecurrence[{3, -3, 1}, {17, 65, 145}, 40] (* Vincenzo Librandi, Feb 08 2012 *)
16*Range[50]^2+1 (* Harvey P. Dale, Jun 06 2025 *)

a(n)= 16*n^2+1 \\ Charles R Greathouse IV, Dec 23 2011

a(n) = A002522(4*n) = A016802(n) + 1.
G.f.: x*(17+14*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 08 2012
From Amiram Eldar, Jul 13 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi*coth(Pi/4)/8 - 1/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/2 - Pi*csch(Pi/4)/8. (End)
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/4)*sinh(Pi/sqrt(8)).
Product_{n>=1} (1 - 1/a(n)) = (Pi/4)*csch(Pi/4). (End)
From Elmo R. Oliveira, Jan 17 2025: (Start)
E.g.f.: exp(x)*(16*x^2 + 16*x + 1) - 1.
a(n) = A053755(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A303302 a(n) = 34*n^2.

Original entry on oeis.org

0, 34, 136, 306, 544, 850, 1224, 1666, 2176, 2754, 3400, 4114, 4896, 5746, 6664, 7650, 8704, 9826, 11016, 12274, 13600, 14994, 16456, 17986, 19584, 21250, 22984, 24786, 26656, 28594, 30600, 32674, 34816, 37026, 39304, 41650, 44064, 46546, 49096, 51714, 54400, 57154, 59976, 62866, 65824, 68850, 71944

Offset: 0

Omar E. Pol, May 13 2018

Sequence found by reading the line from 0, in the direction 0, 34, ..., in the square spiral whose vertices are the generalized 19-gonal numbers A303813.

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

Cf. A005843, A008599, A303813.

Cf. similar sequences of the type k*n^2: A000290 (k=1), A001105 (k=2), A033428 (k=3), A016742 (k=4), A033429 (k=5), A033581 (k=6), A033582 (k=7), A139098 (k=8), A016766 (k=9), A033583 (k=10), A033584 (k=11), A135453 (k=12), A152742 (k=13), A144555 (k=14), A064761 (k=15), A016802 (k=16), A244630 (k=17), A195321 (k=18), A244631 (k=19), A195322 (k=20), A064762 (k=21), A195323 (k=22), A244632 (k=23), A195824 (k=24), A016850 (k=25), A244633 (k=26), A244634 (k=27), A064763 (k=28), A244635 (k=29), A244636 (k=30), A244082 (k=32), this sequence (k=34), A016910 (k=36), A016982 (k=49), A017066 (k=64), A017162 (k=81), A017270 (k=100), A017390 (k=121), A017522 (k=144).

[34*n^2: n in [0..50]]; // Vincenzo Librandi Jun 07 2018

Table[34 n^2, {n, 0, 40}]
LinearRecurrence[{3,-3,1},{0,34,136},50] (* Harvey P. Dale, Jul 23 2018 *)

PARI
```
a(n) = 34*n^2;
```

concat(0, Vec(34*x*(1 + x) / (1 - x)^3 + O(x^40))) \\ Colin Barker, Jun 12 2018

a(n) = 34*A000290(n) = 17*A001105(n) = 2*A244630(n).
G.f.: 34*x*(1 + x)/(1 - x)^3. - Vincenzo Librandi, Jun 07 2018
From Elmo R. Oliveira, Dec 02 2024: (Start)
E.g.f.: 34*x*(1 + x)*exp(x).
a(n) = A005843(n)*A008599(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

cp's OEIS Frontend

A141759 a(n) = 16n^2 + 32n + 15.

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A195323 a(n) = 22*n^2.

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A016910 a(n) = (6*n)^2.

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A195315 Centered 32-gonal numbers.

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A226008 a(0) = 0; for n>0, a(n) = denominator(1/4 - 4/n^2).

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A244082 a(n) = 32*n^2.

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A176027 Binomial transform of A005563.

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