A082044 -id:A082044 - cp's OEIS frontend

A120062 Number of triangles with integer sides a <= b <= c having integer inradius n.

1, 5, 13, 18, 15, 45, 24, 45, 51, 52, 26, 139, 31, 80, 110, 89, 33, 184, 34, 145, 185, 103, 42, 312, 65, 96, 140, 225, 36, 379, 46, 169, 211, 116, 173, 498, 38, 123, 210, 328, 44, 560, 60, 280, 382, 134, 64, 592, 116, 228, 230, 271, 47, 452, 229, 510, 276, 134, 54

Offset: 1

Hugo Pfoertner, Jun 11 2006

It is conjectured that the longest possible side c of a triangle with integer sides and inradius n is given by A057721(n) = n^4 + 3*n^2 + 1.
For n >= 1, a(n) >= 1 because triangle (a, b, c) = (n^2 + 2, n^4 + 2*n^2 + 1, n^4 + 3*n^2 + 1) has inradius n. - David W. Wilson, Jun 17 2006
Previous name was "Number of triangles with integer sides a<=bA362669); so, now effectively, a(10) = 52. - Bernard Schott, Apr 24 2023

			a(1)=1: {3,4,5} is the only triangle with integer sides and inradius 1.
a(2)=5: {5,12,13}, {6,8,10}, {6,25,29}, {7,15,20}, {9,10,17} are the only triangles with integer sides and inradius 2.
a(4)=A120252(1)+A120252(2)+A120252(4)=1+4+13 because 1, 2 and 4 are the factors of 4. The 1 primitive triangle with inradius n=1 is (3,4,5). The 4 primitive triangles with n=2 are (5,12,13), (9,10,17), (7,15,20), (6,25,29). The 13 primitive triangles with n=4 are (13,14,15), (15,15,24), (11,25,30), (15,26,37), (10,35,39), (9,40,41), (33,34,65), (25,51,74), (9,75,78), (11,90,97), (21,85,104), (19,153,170), (18,289,305). (Primitive means GCD(a, b, c, n)=1.)

David W. Wilson, Table of n, a(n) for n = 1..10000
Alan F. Beardon and Paul Stephenson, The Heron parameters of a triangle, Mathematical Gazette May 8, 2014.
Frank M Jackson, Mathematica program
Thomas Mautsch, Additional terms

Cf. A078644 [Pythagorean triangles with inradius n], A057721 [n^4+3*n^2+1].
Let S(n) be the set of triangles with integer sides a<=b<=c and inradius n. Then:
A120062(n) gives number of triangles in S(n).
A120261(n) gives number of triangles in S(n) with gcd(a, b, c) = 1.
A120252(n) gives number of triangles in S(n) with gcd(a, b, c, n) = 1.
A005408(n) = 2n+1 gives shortest short side a of triangles in S(n).
A120064(n) gives shortest middle side b of triangles in S(n).
A120063(n) gives shortest long side c of triangles in S(n).
A120570(n) gives shortest perimeter of triangles in S(n).
A120572(n) gives smallest area of triangles in S(n).
A058331(n) = 2n^2+1 gives longest short side a of triangles in S(n).
A082044(n) = n^4+2n^2+1 gives longest middle side b of triangles in S(n).
A057721(n) = n^4+3n^2+1 gives longest long side c of triangles in S(n).
A120571(n) = 2n^4+6n^2+4 gives longest perimeter of triangles in S(n).
A120573(n) = gives largest area of triangles in S(n).
Cf. A120252 [primitive triangles with integer inradius], A120063 [minimum of longest sides], A057721 [maximum of longest sides], A120064 [minimum of middle sides], A082044 [maximum of middle sides], A005408 [minimum of shortest sides], A058331 [maximum of shortest sides], A007237 [number of triangles with integer sides and area = n times perimeter].
Cf. A362669, A362670.

Mathematica
```
(* See link above. *)
```

The even-numbered terms are given by a(2*n)=A007237(n).

a(n) = Sum_{k|n} A120252(k).

More terms from Graeme McRae and Hugo Pfoertner, Jun 12 2006

Name corrected by Bernard Schott, Apr 24 2023

A114949 a(n) = n^2 + 6.

Original entry on oeis.org

6, 7, 10, 15, 22, 31, 42, 55, 70, 87, 106, 127, 150, 175, 202, 231, 262, 295, 330, 367, 406, 447, 490, 535, 582, 631, 682, 735, 790, 847, 906, 967, 1030, 1095, 1162, 1231, 1302, 1375, 1450, 1527, 1606, 1687, 1770, 1855, 1942, 2031, 2122, 2215, 2310, 2407, 2506

Offset: 0

Cino Hilliard, Feb 21 2006

2/a(n) = R(n)/r, n >= 0, with R(n) the n-th radius of the counterclockwise Pappus chain of the arbelos with semicircle radii r, r1 = 2r/3, r2 = r - r1 = r/3. See the MathWorld link for such a Pappus chain. The clockwise chain companion has circle radii R'(n)/r = 2/A222465(n), n >= 0. - Wolfdieter Lang, Mar 01 2013

			The arbelos chain defined in a comment above has circle radii [1/3, 2/7, 1/5, 2/15, 1/11, 2/31, 1/21, 2/55, 1/35, 2/87, 1/53,...], for n >= 0. - _Wolfdieter Lang_, Mar 01 2013

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pappus chain.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A002522, A016742, A059100, A082044, A087475, A114964 (see comment), A117619, A117950, A117951, A222465.

A114949:=n->n^2+6: seq(A114949(n), n=0..100); # Wesley Ivan Hurt, Apr 28 2017

Range[0, 49]^2 + 6 (* Alonso del Arte, Jan 30 2013 *)

From R. J. Mathar, May 17 2009: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -(6 - 11*x + 7*x^2)/(x - 1)^3. (End)
a(n) = 2*n + a(n - 1) - 1, with n > 0, a(0)=6. - Vincenzo Librandi, Nov 13 2010
a(n) = A000290(n) + 6. - Omar E. Pol, Mar 02 2013
a(n) = ((n-2)^3 + (n-1)^3 + n^3 + (n+1)^3 + (n+2)^3)/(5*n) for n>=1. - Bruno Berselli, May 12 2014
For n >= 1, a(n) = (A016742(n) + A082044(n) - 1)/A000290(n). - Bruce J. Nicholson, Apr 19 2017
From Amiram Eldar, Nov 02 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(6)*Pi*coth(sqrt(6)*Pi))/12.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(6)*Pi*cosech(sqrt(6)*Pi))/12. (End)
From Amiram Eldar, Feb 05 2024: (Start)
Product_{n>=0} (1 - 1/a(n)) = sqrt(5/6)*sinh(sqrt(5)*Pi)/sinh(sqrt(6)*Pi).
Product_{n>=0} (1 + 1/a(n)) = sqrt(7/6)*sinh(sqrt(7)*Pi)/sinh(sqrt(6)*Pi). (End)
E.g.f.: exp(x)*(6 + x + x^2). - Elmo R. Oliveira, Jan 17 2025

A221573 T(n,k)=Number of 0..k arrays of length n with each element differing from at least one neighbor by something other than 1.

Original entry on oeis.org

0, 0, 2, 0, 5, 2, 0, 10, 9, 4, 0, 17, 26, 25, 6, 0, 26, 59, 100, 57, 10, 0, 37, 114, 289, 342, 141, 16, 0, 50, 197, 676, 1293, 1210, 345, 26, 0, 65, 314, 1369, 3734, 5913, 4240, 853, 42, 0, 82, 471, 2500, 8991, 20944, 26911, 14898, 2097, 68, 0, 101, 674, 4225, 19014

Offset: 1

R. H. Hardin Jan 20 2013

Table starts
...0.....0.......0........0.........0..........0...........0............0
...2.....5......10.......17........26.........37..........50...........65
...2.....9......26.......59.......114........197.........314..........471
...4....25.....100......289.......676.......1369........2500.........4225
...6....57.....342.....1293......3734.......8991.......19014........36497
..10...141....1210.....5913.....20944......59705......145800.......317233
..16...345....4240....26911....117104.....395641.....1116400......2754635
..26...853...14898...122621....655198....2622817.....8550512.....23923281
..42..2097...52306...558547...3665306...17385993....65485386....207761745
..68..5149..183684..2544357..20505052..115249117...501533796...1804315029
.110.12633..645006.11590169.114711980..763966685..3841097940..15669633131
.178.31013.2264978.52796369.641737294.5064207645.29417832750.136083460405

			Some solutions for n=6 k=4
..2....2....3....1....0....0....2....1....2....2....3....1....4....3....1....4
..0....0....0....1....4....0....2....3....0....4....0....3....4....3....3....4
..1....1....4....3....4....2....0....4....0....0....4....2....3....0....1....2
..3....1....2....1....3....3....2....4....0....3....3....2....1....3....0....0
..3....4....1....1....0....0....0....3....1....2....1....4....0....3....3....0
..0....1....4....1....3....0....4....0....3....4....3....4....2....3....0....0

R. H. Hardin, Table of n, a(n) for n = 1..2080

Column 1 is A006355
Row 2 is A002522
Row 4 is A082044

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = 3*a(n-1) -2*a(n-2) +4*a(n-4)
k=3: a(n) = 3*a(n-1) +2*a(n-2) -a(n-3) +a(n-4)
k=4: a(n) = 5*a(n-1) -3*a(n-2) +a(n-3) +15*a(n-4) +3*a(n-5) for n>6
k=5: a(n) = 5*a(n-1) +3*a(n-2) +9*a(n-4) +6*a(n-5) +3*a(n-6)
k=6: a(n) = 7*a(n-1) -4*a(n-2) +6*a(n-3) +26*a(n-4) +10*a(n-5) +16*a(n-6) +12*a(n-8)
k=7: a(n) = 7*a(n-1) +4*a(n-2) +5*a(n-3) +20*a(n-4) +20*a(n-5) +23*a(n-6) -6*a(n-7) +3*a(n-8)
Empirical for row n:
n=2: a(n) = 1*n^2 + 1
n=3: a(n) = 1*n^3 - 1*n^2 + 3*n - 1
n=4: a(n) = 1*n^4 + 2*n^2 + 1
n=5: a(n) = 1*n^5 + 1*n^4 - 2*n^3 + 12*n^2 - 15*n + 9 for n>2
n=6: a(n) = 1*n^6 + 2*n^5 - 5*n^4 + 24*n^3 - 41*n^2 + 50*n - 31 for n>3
n=7: a(n) = 1*n^7 + 3*n^6 - 7*n^5 + 29*n^4 - 41*n^3 + 45*n^2 - 33*n + 19 for n>2

A082043 Square array, A(n, k) = (kn)^2 + 2k*n + 1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 25, 16, 1, 1, 25, 49, 49, 25, 1, 1, 36, 81, 100, 81, 36, 1, 1, 49, 121, 169, 169, 121, 49, 1, 1, 64, 169, 256, 289, 256, 169, 64, 1, 1, 81, 225, 361, 441, 441, 361, 225, 81, 1, 1, 100, 289, 484, 625, 676, 625, 484, 289, 100, 1

Offset: 0

Paul Barry, Apr 03 2003

			Array, A(n, k), begins as:
  1,   1,   1,    1,    1,    1,    1,    1,     1, ... A000012;
  1,   4,   9,   16,   25,   36,   49,   64,    81, ... A000290;
  1,   9,  25,   49,   81,  121,  169,  225,   289, ... A016754;
  1,  16,  49,  100,  169,  256,  361,  484,   625, ... A016778;
  1,  25,  81,  169,  289,  441,  625,  841,  1089, ... A016814;
  1,  36, 121,  256,  441,  676,  961, 1296,  1681, ... A016862;
  1,  49, 169,  361,  625,  961, 1369, 1849,  2401, ... A016922;
  1,  64, 225,  484,  841, 1296, 1849, 2500,  3249, ... A016994;
  1,  81, 289,  625, 1089, 1681, 2401, 3249,  4225, ... A017078;
  1, 100, 361,  784, 1369, 2116, 3025, 4096,  5329, ... A017174;
  1, 121, 441,  961, 1681, 2601, 3721, 5041,  6561, ... A017282;
  1, 144, 529, 1156, 2025, 3136, 4489, 6084,  7921, ... A017402;
  1, 169, 625, 1369, 2401, 3721, 5329, 7225,  9409, ... A017534;
  1, 196, 729, 1600, 2809, 4356, 6241, 8464, 11025, ... ;
Antidiagonals, T(n, k), begin as:
  1;
  1,   1;
  1,   4,   1;
  1,   9,   9,   1;
  1,  16,  25,  16,   1;
  1,  25,  49,  49,  25,   1;
  1,  36,  81, 100,  81,  36,   1;
  1,  49, 121, 169, 169, 121,  49,   1;
  1,  64, 169, 256, 289, 256, 169,  64,   1;
  1,  81, 225, 361, 441, 441, 361, 225,  81,   1;
  1, 100, 289, 484, 625, 676, 625, 484, 289, 100,  1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Rows include A000290, A016754, A016778, A016814, A016862, A016922, A016994, A017078, A017174, A017282, A017402, A017534.
Diagonals include A000583, A058031, A062938, A082044 (main diagonal).
Diagonal sums (row sums if viewed as number triangle) are A082045.
Cf. A082039, A082045, A082046, A082105.

A082043:= func< n,k | (k*(n-k))^2 +2*k*(n-k) +1 >;
[A082043(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 24 2022

T[n_, k_]:= (k*(n-k))^2 +2*k*(n-k) +1;
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 24 2022 *)

def A082043(n,k): return (k*(n-k))^2 +2*k*(n-k) +1
flatten([[A082043(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Dec 24 2022

A(n, k) = (k*n)^2 + 2*k*n + 1 (square array).
T(n, k) = (k*(n-k))^2 + 2*k*(n-k) + 1 (number triangle).
A(k, n) = A(n, k).
T(n, n-k) = T(n, k).
A(n, n) = T(2*n, n) = A082044(n).
A(n, n-1) = T(2*n+1, n-1) = A058031(n), n >= 1.
A(n, n-2) = T(2*(n-1), n) = A000583(n-1), n >= 2.
A(n, n-3) = T(2*n-3, n) = A062938(n-3), n >= 3.
Sum_{k=0..n} T(n, k) = A082045(n) (diagonal sums).
Sum_{k=0..n} (-1)^k * T(n, k) = (1/4)*(1+(-1)^n)*(2 - 3*n). - G. C. Greubel, Dec 24 2022

A331040 Numerator of squared radius of inscribed circle of a triangle with integer sides i <= j <= k, such that the number of triangles with this radius sets a new record. Denominators are A331041.

Original entry on oeis.org

1, 35, 3, 7, 3, 15, 8, 35, 55, 63, 95, 119, 135, 56, 231, 255, 80, 351, 455, 495, 855, 216, 224, 1071, 360

Offset: 1

Hugo Pfoertner, Jan 11 2020

The radius rho of the inscribed circle of a triangle (a,b,c) is rho = sqrt((s-a)*(s-b)*(s-c)/s), with s=(a+b+c)/2. For given integer values of a <= b and a rational target value r2 of the squared incircle radius, c is given by the two positive real roots of the polynomial P(a,b,x,r2) = x^3 - x^2 * (a+b) + x * (4*r2-(b-a)^2) + (a+b)^3 + 4*(a+b)*(r2-a*b). P(a,b,x,r2) = 0 may have 0, 1 or 2 positive integer solutions.

The potential ranges of the side lengths of the triangles can be determined in analogy to the ranges for the case of integer radii of the incircles, see A120062 for the relevant formulas and sequences.

			b(1) = a(1)/A331041(1) = 1/12: Triangle (1,1,1) has the least possible radius of incircle = sqrt(1/12).
b(2) = a(2)/A331041(2) = 35/52: Triangles (2,18,19) and (3,4,6) are the first occurrence of more than one triangle with the same radius of their incircles. rho = sqrt(35/52) in both cases.
b(3) = a(3)/A331041(3) = 3/4: Triangles are (2,7,7), (3,3,3), and (3,5,7).
b(4) = a(4)/A331041(4) = 7/4: (3,12,12), (3,22,23), (4,5,6), (5,18,22), (6,11,16) are the A331043(4) = 5 triangles with rho^2 = b(4).
b(15) = 231/4 includes the rare case, where two distinct integer solutions for the same pair of sides a and b exist: (20,37,38) and (20,37,39), both with rho^2=231/4 and thus contributing 2 of the A331043(15)=84 triangles with this squared radius of the incircle.

Hugo Pfoertner, Illustration of side lengths for terms a(13)-a(25).

Cf. A057721, A082044, A120062, A120572, A331012.

Cf. A331041 (corresponding denominators), A331042 (4*a(n)/A331041(n)), A331043 (records of numbers of triangles).

\\ Only suitable for demonstration of initial terms
rh2(a,b,c)={my(s=(a+b+c)/2);(s-a)*(s-b)*(s-c)/s};
lim_a(x)=ceil(4*(x^2+2));
lim_b(x)=ceil(4*(x^4+2*x^2+1));
target=35/4; v=vector(333938); n=0;
for(a=1,lim_a(sqrt(target)), for(b=a,lim_b(sqrt(target)), for(c=b,a+b-1, f=rh2(a,b,c);v[n++]=f)));
v=vecsort(v); print("A331040 A331041 A331043"); print(numerator(v[1])," ",denominator(v[1])," ",1); m=0; mm=0; for(k=2,#v, if(v[k]>target,break); if(v[k]==v[k-1], m++; if(m>mm&&v[k+1]>v[k], print(numerator(v[k])," ",denominator(v[k])," ",m);mm=m),m=1));

A082106 Main diagonal of number array A082105.

Original entry on oeis.org

1, 6, 33, 118, 321, 726, 1441, 2598, 4353, 6886, 10401, 15126, 21313, 29238, 39201, 51526, 66561, 84678, 106273, 131766, 161601, 196246, 236193, 281958, 334081, 393126, 459681, 534358, 617793, 710646, 813601, 927366, 1052673, 1190278, 1340961, 1505526, 1684801

Offset: 0

Paul Barry, Apr 03 2003

4*a(n) can be written as (n^2 + 2*n + 1)^2 + (n^2 - 2*n + 1)^2 + (n^2 - 2*n - 1)^2 + (n^2 + 2*n - 1)^2. - Bruno Berselli, Jun 20 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A082047, A082044, A082105, A082107, A082109.

[(n^2+2)^2 -3: n in [0..40]]; // G. C. Greubel, Dec 22 2022

Table[n^4+4n^2+1,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,6,33,118,321},40] (* Harvey P. Dale, Dec 06 2012 *)

[(n^2+2)^2 -3 for n in range(41)] # G. C. Greubel, Dec 22 2022

a(n) = n^4 + 4*n^2 + 1.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Dec 06 2012
G.f.: (1 + x + 13*x^2 + 3*x^3 + 6*x^4)/(1 - x)^5. - Bruno Berselli, Jun 20 2014
E.g.f.: (1 + 5*x + 11*x^2 + 6*x^3 + x^4)*exp(x). - G. C. Greubel, Dec 22 2022
Sum_{n>=0} 1/a(n) = 1/2 + (Pi/4)*((1/sqrt(2)+1/sqrt(6))*coth(sqrt(2-sqrt(3))*Pi) - (1/sqrt(2)-1/sqrt(6))*coth(sqrt(2+sqrt(3))*Pi)). - Amiram Eldar, Jan 08 2023

A120063 Shortest side c of all integer-sided triangles with sides a<=b<=c and inradius n.

Original entry on oeis.org

5, 10, 12, 15, 25, 24, 35, 30, 36, 39, 55, 45, 65, 63, 53, 60, 85, 68, 95, 75, 77, 88, 115, 85, 125, 130, 108, 105, 145, 106, 155, 120, 132, 170, 137, 135, 185, 190, 156, 150, 205, 154, 215, 165, 159, 230, 235, 170, 245, 195, 204, 195, 265, 204, 200, 195, 228, 290

Offset: 1

Hugo Pfoertner, Jun 13 2006

nonn

Terms a(11),..., a(100) computed by Thomas Mautsch (mautsch(AT)ethz.ch).
Empirically, 2*sqrt(3) < a(n)/n <= 5. The lower bound is provably tight, the upper bound seems to be achieved infinitely often, e.g, for prime n >= 5. It appears that a(p) = 5p for prime p != 3. - David W. Wilson, Jun 17 2006
Minimum of longest side occurring among all A120062(n) triangles having integer sides with integer inradius n.

			a(1)=5 because the only triangle with integer sides and inradius 1 is {3,4,5}; its longest side is 5.
a(2)=10: The triangles with inradius 2 are {5,12,13}, {6,8,10}, {6,25,29}, {7,15,20}, {9,10,17}. The minimum of their longest sides is min(13,10,29,20,17)=10.

Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.

David W. Wilson, Table of n, a(n) for n = 1..10000

See A120062 for sequences related to integer-sided triangles with integer inradius n.

Cf. A120062 [triangles with integer inradius], A120252 [primitive triangles with integer inradius], A057721 [maximum of longest sides], A058331 [maximum of shortest sides], A120064 [minimum of middle sides], A082044 [maximum of middle sides], A005408 [minimum of shortest sides], A007237.

A120064 Shortest side b of all integer-sided triangles with sides a<=b<=c and inradius n.

Original entry on oeis.org

4, 8, 10, 14, 20, 20, 28, 28, 30, 39, 44, 40, 52, 56, 50, 56, 68, 60, 76, 70, 70, 87, 92, 80, 100, 100, 90, 97, 116, 100, 124, 112, 110, 136, 120, 120, 148, 152, 130, 140, 164, 140, 172, 154, 150, 184, 188, 160, 196, 174, 170, 182, 212, 180, 196, 189, 190, 232, 236

Offset: 1

Hugo Pfoertner, Jun 13 2006

nonn

Terms a(11),..., a(100) computed by Thomas Mautsch (mautsch(AT)ethz.ch).

			a(1)=2 because the only triangle with integer sides a<=b<c and inradius 1 is {3,4,5}; its middle side is 4.
a(2)=8: The triangles with inradius 2 are {5,12,13}, {6,8,10}, {6,25,29}, {7,15,20}, {9,10,17}. The minimum of their middle sides is min(12,8,25,15,10)=8.

Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.

David W. Wilson, Table of n, a(n) for n = 1..10000

Cf. A120062 [triangles with integer inradius], A120252 [primitive triangles with integer inradius], A057721 [maximum of longest sides], A120063 [minimum of longest sides], A058331 [maximum of shortest sides], A082044 [maximum of middle sides], A005408 [minimum of shortest sides], A007237.

See A120062 for sequences related to integer-sided triangles with integer inradius n.

A244730 a(n) = 2*n^4.

Original entry on oeis.org

0, 2, 32, 162, 512, 1250, 2592, 4802, 8192, 13122, 20000, 29282, 41472, 57122, 76832, 101250, 131072, 167042, 209952, 260642, 320000, 388962, 468512, 559682, 663552, 781250, 913952, 1062882, 1229312, 1414562, 1620000, 1847042, 2097152, 2371842, 2672672

Offset: 0

Vincenzo Librandi, Jul 05 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000583, A141046, A219056.

Magma
```
[2*n^4: n in [0..40]];
```

I:=[0,2,32,162, 512]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]];

Table[2 n^4, {n, 0, 40}] (* or *) CoefficientList[Series[2(x + 11 x^2 + 11 x^3 + x^4)/(1 - x)^5, {x, 0, 40}], x]
LinearRecurrence[{5,-10,10,-5,1},{0,2,32,162,512},40] (* Harvey P. Dale, Jun 17 2022 *)

G.f.: 2*(x + 11*x^2 + 11*x^3 + x^4)/(1 - x)^5.
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>4.
a(n) = (A082044(n) + A099761(n+1)-2)/2. - Bruce J. Nicholson, Jun 12 2017

A082113 a(n) = n^4 + 5*n^2 + 1.

Original entry on oeis.org

1, 7, 37, 127, 337, 751, 1477, 2647, 4417, 6967, 10501, 15247, 21457, 29407, 39397, 51751, 66817, 84967, 106597, 132127, 162001, 196687, 236677, 282487, 334657, 393751, 460357, 535087, 618577, 711487, 814501, 928327, 1053697, 1191367

Offset: 0

Paul Barry, Apr 04 2003

Main diagonal of number array A082110.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A082044, A082047, A082106.

[n^4+5*n^2+1: n in [0..40]]; // G. C. Greubel, Dec 22 2022

Table[n^4+5n^2+1,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,7,37,127,337},40] (* Harvey P. Dale, May 16 2019 *)

[n^4+5*n^2+1 for n in range(41)] # G. C. Greubel, Dec 22 2022

a(n) = n^4 + 5*n^2 + 1.
G.f.: (1+2*x+12*x^2+2*x^3+7*x^4) / (1-x)^5. - R. J. Mathar, Dec 03 2014
E.g.f.: (1 + 6*x + 12*x^2 + 6*x^3 + x^4)*exp(x). - G. C. Greubel, Dec 22 2022

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