A082109 -id:A082109 - cp's OEIS frontend

A083075 Square array read by antidiagonals: T(n,k) = (k(2k+3)^n + 1)/(k+1).

1, 1, 1, 1, 3, 1, 1, 5, 13, 1, 1, 7, 33, 63, 1, 1, 9, 61, 229, 313, 1, 1, 11, 97, 547, 1601, 1563, 1, 1, 13, 141, 1065, 4921, 11205, 7813, 1, 1, 15, 193, 1831, 11713, 44287, 78433, 39063, 1, 1, 17, 253, 2893, 23801, 128841, 398581, 549029, 195313, 1, 1, 19, 321

Offset: 0

Paul Barry, Apr 23 2003

			Array begins:
  1     1     1     1     1 ...
  1     3    13    63   313 ...
  1     5    33   229  1601 ...
  1     7    61   547  4921 ...
  1     9    97  1065 11713 ...
  ...

Nathaniel Johnston, Antidiagonals 0..100, flattened

Rows include A034478, A083076, A066443, A083077, A083078.
Columns include odds, A082109, A083079.
Diagonals include A083079, A083080, A083081, A083082.

T := proc(n,k) return (k*(2*k+3)^n+1)/(k+1): end: seq(seq(T(k,n-k),k=0..n),n=0..10); # Nathaniel Johnston, Jun 26 2011

A188646 Array of a(n)=a(n-1)*k-((k-1)/(k^n)) where a(0)=1 and k=(sqrt(x^2-1)+x)^2 for integers x>=1.

Original entry on oeis.org

1, 1, 1, 1, 13, 1, 1, 181, 33, 1, 1, 2521, 1121, 61, 1, 1, 35113, 38081, 3781, 97, 1, 1, 489061, 1293633, 234361, 9505, 141, 1, 1, 6811741, 43945441, 14526601, 931393, 20021, 193, 1, 1, 94875313, 1492851361, 900414901, 91267009, 2842841, 37441, 253, 1

Offset: 0

Charles L. Hohn, Apr 06 2011

Conjecture: Given function f(x, y)=(sqrt(x^2+y)+x)^2; constant k=f(x, y); and initial term a(0)=1; then for all integers x>=1 and y=[+-]1, k may be irrational, but sequence a(n)=a(n-1)*k-((k-1)/(k^n)) always produces integer sequences; y=-1 results shown here; y=1 results are A188647.

Also square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is (1/n) * T_{2*k+1}(n), with the Chebyshev polynomials of the first kind (type T). - Seiichi Manyama, Jan 01 2019

			Square array begins:
     | 0    1       2          3             4
-----+---------------------------------------------
   1 | 1,   1,      1,         1,            1, ...
   2 | 1,  13,    181,      2521,        35113, ...
   3 | 1,  33,   1121,     38081,      1293633, ...
   4 | 1,  61,   3781,    234361,     14526601, ...
   5 | 1,  97,   9505,    931393,     91267009, ...
   6 | 1, 141,  20021,   2842841,    403663401, ...
   7 | 1, 193,  37441,   7263361,   1409054593, ...
   8 | 1, 253,  64261,  16322041,   4145734153, ...
   9 | 1, 321, 103361,  33281921,  10716675201, ...
  10 | 1, 397, 158005,  62885593,  25028308009, ...
  11 | 1, 481, 231841, 111746881,  53861764801, ...
  12 | 1, 573, 328901, 188788601, 108364328073, ...
  13 | 1, 673, 453601, 305726401, 206059140673, ...
  14 | 1, 781, 610741, 477598681, 373481557801, ...
  15 | 1, 897, 805505, 723342593, 649560843009, ...
  ...

Row 1-8 give A000012, A001570, A077420, A302329, A302330, A302331, A302332, A253880.
Column 1 is A082109(n-1).
Cf. A188644, A188647 (f(x, y) as above with y=1).
Diagonal gives A322904.

A[n_, k_] := 1/n ChebyshevT[2k+1, n];
Table[A[n-k, k], {n, 1, 9}, {k, n-1, 0, -1}] // Flatten (* Jean-François Alcover, Jan 02 2019, after Seiichi Manyama *)

A(n,k) = 2 * A188644(n,k) - A(n,k-1).

A(n,k) = Sum_{j=0..k} binomial(2*k+1,2*j+1)*(n^2-1)^(k-j)*n^(2*j). - Seiichi Manyama, Jan 01 2019

Edited and extended by Seiichi Manyama, Jan 01 2019

A082105 Array A(n, k) = (kn)^2 + 4(k*n) + 1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 13, 13, 1, 1, 22, 33, 22, 1, 1, 33, 61, 61, 33, 1, 1, 46, 97, 118, 97, 46, 1, 1, 61, 141, 193, 193, 141, 61, 1, 1, 78, 193, 286, 321, 286, 193, 78, 1, 1, 97, 253, 397, 481, 481, 397, 253, 97, 1, 1, 118, 321, 526, 673, 726, 673, 526, 321, 118, 1

Offset: 0

Paul Barry, Apr 03 2003

			Array, A(n, k), begins as:
  1,  1,   1,   1,   1,   1, ... A000012;
  1,  6,  13,  22,  33,  46, ... A028872;
  1, 13,  33,  61,  97, 141, ... A082109;
  1, 22,  61, 118, 193, 286, ... ;
  1, 33,  97, 193, 321, 481, ... ;
  1, 46, 141, 286, 481, 726, ... ;
Triangle, T(n, k), begins as:
  1;
  1,  1;
  1,  6,   1;
  1, 13,  13,   1;
  1, 22,  33,  22,   1;
  1, 33,  61,  61,  33,   1;
  1, 46,  97, 118,  97,  46,   1;
  1, 61, 141, 193, 193, 141,  61,  1;
  1, 78, 193, 286, 321, 286, 193, 78,  1;

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

Cf. A016897, A028872, A047673, A082043, A082046, A082106, A082107, A082109.

[(k*(n-k))^2 + 4*(k*(n-k)) + 1: k in [0..n], n in [0..13]]; // G. C. Greubel, Dec 22 2022

T[n_, k_]:= (k*(n-k))^2 + 4*(k*(n-k)) + 1;
Table[T[n,k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 22 2022 *)

def A082105(n,k): return (k*(n-k))^2 + 4*(k*(n-k)) + 1
flatten([[A082105(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Dec 22 2022

A(n, k) = (k*n)^2 + 4*(k*n) + 1 (square array).
A(n, n) = T(2*n, n) = A082106(n) (main diagonal).
T(n, k) = A(n-k, k) (number triangle).
Sum_{k=0..n} T(n, k) = A082107(n) (diagonal sums).
T(n, n-1) = A028872(n-1), n >= 1.
T(n, n-2) = A082109(n-2), n >= 2.
From G. C. Greubel, Dec 22 2022: (Start)
Sum_{k=0..n} (-1)^k * T(n, k) = ((1+(-1)^n)/2)*A016897(n-1).
T(2*n+1, n+1) = A047673(n+1), n >= 0.
T(n, n-k) = T(n, k). (End)

A028874 Primes of form k^2 - 3.

Original entry on oeis.org

13, 61, 97, 193, 397, 673, 1021, 1153, 1597, 1933, 2113, 3361, 4093, 4621, 6397, 7393, 7741, 8461, 9601, 12097, 12541, 13921, 15373, 16381, 18493, 19597, 20161, 21313, 26893, 29581, 36097, 37633, 40801, 42433, 43261, 47521, 48397

Offset: 1

Patrick De Geest

nonn

Also primes equal to the product of two consecutive odd numbers (A000466) minus 2. - Giovanni Teofilatto, Feb 11 2010

All terms are of the form 6m + 1. - Zak Seidov, May 01 2014

			61 is prime and equal to 8^2 - 3, so it is in the sequence.
67 is prime but it's 8^2 + 3 = 9^2 - 14, so it is not in the sequence.
9^2 - 3 = 78 but it's composite, so it's not in the sequence either.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Patrick De Geest, Palindromic Quasipronics of the form n(n+x).
R. J. Mathar, Solutions to the exponential Diophantine 1 + p_1^x + p_2^y + p_3^z = w^2 for distinct primes p_1, p_2, p_3, 2022.
Eric Weisstein's World of Mathematics, Near-Square Prime.

Cf. A002476 (Primes of form 6m + 1), A028871, A028872, A028873.

Primes terms in A082109. Subsequence of A068228. - Klaus Purath, Jan 09 2023

[a: n in [2..300] | IsPrime(a) where a is n^2-3 ]; // Vincenzo Librandi, Nov 08 2014

Select[Range[2, 250]^2 - 3, PrimeQ] (* Harvey P. Dale, Aug 07 2013 *)
Select[Table[n^2 - 3, {n, 2, 300}], PrimeQ] (* Vincenzo Librandi, Nov 08 2014 *)

select(isprime, vector(100,n,n^2-3)) \\ Charles R Greathouse IV, Nov 19 2014

A028872 INTERSECT A000040. - Klaus Purath, Dec 07 2020

a(n) = A028873(n)^2 - 3. - Amiram Eldar, Mar 01 2025

A082107 Diagonal sums of number array A082105.

Original entry on oeis.org

1, 2, 8, 28, 79, 190, 406, 792, 1437, 2458, 4004, 6260, 9451, 13846, 19762, 27568, 37689, 50610, 66880, 87116, 112007, 142318, 178894, 222664, 274645, 335946, 407772, 491428, 588323, 699974, 828010, 974176, 1140337, 1328482, 1540728

Offset: 0

Paul Barry, Apr 03 2003

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A082043, A082045, A082046, A082105, A082106, A082109, A082110.

[(n^5+20*n^3+9*n+30)/30: n in [0..50]]; // G. C. Greubel, Dec 22 2022

LinearRecurrence[{6,-15,20,-15,6,-1}, {1,2,8,28,79,190}, 51] (* G. C. Greubel, Dec 22 2022 *)

[(n^5+20*n^3+9*n+30)/30 for n in range(51)] # G. C. Greubel, Dec 22 2022

a(n) = (n^5 + 20*n^3 + 9*n + 30)/30.
G.f.: (1-4*x+11*x^2-10*x^3+6*x^4)/(1-x)^6 . - R. J. Mathar, Mar 27 2019
E.g.f.: (1/30)*(30 +30*x +75*x^2 +45*x^3 +10*x^4 +x^5)*exp(x). - G. C. Greubel, Dec 22 2022

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

A082112 a(n) = 4n^2 + 10n + 1.

Original entry on oeis.org

1, 15, 37, 67, 105, 151, 205, 267, 337, 415, 501, 595, 697, 807, 925, 1051, 1185, 1327, 1477, 1635, 1801, 1975, 2157, 2347, 2545, 2751, 2965, 3187, 3417, 3655, 3901, 4155, 4417, 4687, 4965, 5251, 5545, 5847, 6157, 6475, 6801, 7135, 7477, 7827, 8185, 8551

Offset: 0

Paul Barry, Apr 04 2003

A row 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 (3,-3,1).

Cf. A082108, A082109, A082110.

[4*n^2 + 10*n + 1 : n in [0..50]]; // Wesley Ivan Hurt, Dec 22 2021

Table[n +(n+1)^2 -4, {n,1,200, 2}] (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *)
LinearRecurrence[{3,-3,1},{1,15,37},50] (* Harvey P. Dale, Dec 18 2014 *)

a(n)=4*n^2+10*n+1 \\ Charles R Greathouse IV, Jun 17 2017

[4*n^2+10*n+1 for n in range(51)] # G. C. Greubel, Dec 22 2022

a(n) = a(n-1) + 8*n + 6 (with a(0)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: (1+12*x-5*x^2) / (1-x)^3. - R. J. Mathar, Dec 03 2014
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Wesley Ivan Hurt, Dec 22 2021
E.g.f.: (1 + 14*x + 4*x^2)*exp(x). - G. C. Greubel, Dec 22 2022

A083079 4th column of number array A083075.

Original entry on oeis.org

1, 63, 229, 547, 1065, 1831, 2893, 4299, 6097, 8335, 11061, 14323, 18169, 22647, 27805, 33691, 40353, 47839, 56197, 65475, 75721, 86983, 99309, 112747, 127345, 143151, 160213, 178579, 198297, 219415, 241981, 266043, 291649, 318847, 347685

Offset: 0

Paul Barry, Apr 23 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A082109.

[8*n^3+28*n^2+26*n+1: n in [0..30]]; // Vincenzo Librandi, Nov 12 2011

Table[8 n^3 + 28 n^2 + 26 n + 1, {n, 0, 40}] (* or *) LinearRecurrence[ {4,-6,4,-1},{1,63,229,547},40] (* Harvey P. Dale, May 14 2011 *)

a(n) = 8*n^3 + 28*n^2 + 26*n + 1 = (n*(2n+3)^3 + 1)/(n+1).
G.f.: (1 + 59*x - 17*x^2 + 5*x^3)/(1-x)^4. - Harvey P. Dale, May 14 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=63, a(2)=229, a(3)=547. - Harvey P. Dale, May 14 2011

A227786 Take squares larger than 1, subtract 3 from even squares and 2 from odd squares; a(n) = a(n-1) + A168276(n+1) (with a(1) = 1).

Original entry on oeis.org

1, 7, 13, 23, 33, 47, 61, 79, 97, 119, 141, 167, 193, 223, 253, 287, 321, 359, 397, 439, 481, 527, 573, 623, 673, 727, 781, 839, 897, 959, 1021, 1087, 1153, 1223, 1293, 1367, 1441, 1519, 1597, 1679, 1761, 1847, 1933, 2023, 2113, 2207, 2301, 2399, 2497, 2599, 2701

Offset: 1

Antti Karttunen, Jul 31 2013

nonn

Conjecture: from n>=2 onward, a(n) gives the positions of 2's in A227761.

a(29) = 897 = 3*13*23 is the first term which is neither prime nor semiprime, that is, has more than two prime divisors.

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

Bisections: A082109, A073577. Cf. also A227761.

a(n) = A000290(n+1) - 2 - (n mod 2).
a(1)=1, and for n>1, a(n) = a(n-1)+A168276(n+1).
a(n) = (1/2) * (2*n^2 + 4*n -3 + (-1)^n) = 2*A116940(n-1) + 1. a(n-1) = 2*ceiling(n^2/2) - 3 = 2*A000985(n) - 3. G.f.: x*(-x^3 - x^2 + 5*x + 1)/((1-x)^3 * (1+x)). - Ralf Stephan, Aug 10 2013

A033596 a(n) = (n^2 - 1)*(n^2 - 3).

Original entry on oeis.org

3, 0, 3, 48, 195, 528, 1155, 2208, 3843, 6240, 9603, 14160, 20163, 27888, 37635, 49728, 64515, 82368, 103683, 128880, 158403, 192720, 232323, 277728, 329475, 388128, 454275, 528528, 611523, 703920, 806403, 919680, 1044483, 1181568, 1331715, 1495728, 1674435

Offset: 0

N. J. A. Sloane

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. A000290 (n^2), A000583 (n^4), A082109.

[(n^2-1)*(n^2-3) : n in [0..30]]; // Wesley Ivan Hurt, Oct 30 2014

A033596:=n->(n^2-1)*(n^2-3): seq(A033596(n), n=0..30); # Wesley Ivan Hurt, Oct 30 2014

Table[(n^2 - 1)*(n^2 - 3), {n, 0, 30}] (* or *)
CoefficientList[Series[3 (1 - 5 x + 11 x^2 + x^3)/(1 - x)^5, {x, 0, 30}], x] (* Wesley Ivan Hurt, Oct 30 2014 *)
LinearRecurrence[{5,-10,10,-5,1},{3,0,3,48,195},40] (* Harvey P. Dale, Nov 20 2024 *)

vector(31, n, my(m=n-1); (m^2-1)*(m^2-3)) \\ G. C. Greubel, Mar 05 2020

[(n^2-1)*(n^2-3) for n in (0..30)] # G. C. Greubel, Mar 05 2020

From Wesley Ivan Hurt, Oct 30 2014: (Start)
G.f.: 3*(1 -5*x +11*x^2 +x^3)/(1-x)^5.
a(n) = 5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4) +a(n-5).
a(n) = (n^2-1)*(n^2-3) = (A000290(n)-1)*(A000290(n)-3) = A000583(n) - A082109(n+1). (End)
E.g.f.: (3 - 3*x + 3*x^2 + 6*x^3 + x^4)*exp(x). - G. C. Greubel, Mar 05 2020

cp's OEIS Frontend

A083075 Square array read by antidiagonals: T(n,k) = (k*(2*k+3)^n + 1)/(k+1).

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A188646 Array of a(n)=a(n-1)*k-((k-1)/(k^n)) where a(0)=1 and k=(sqrt(x^2-1)+x)^2 for integers x>=1.

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A082105 Array A(n, k) = (k*n)^2 + 4*(k*n) + 1, read by antidiagonals.

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A028874 Primes of form k^2 - 3.

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A082107 Diagonal sums of number array A082105.

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A082106 Main diagonal of number array A082105.

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A082112 a(n) = 4*n^2 + 10*n + 1.

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A083075 Square array read by antidiagonals: T(n,k) = (k(2k+3)^n + 1)/(k+1).

A082105 Array A(n, k) = (kn)^2 + 4(k*n) + 1, read by antidiagonals.

A082112 a(n) = 4n^2 + 10n + 1.