A212656 -id:A212656 - cp's OEIS frontend

A158493 a(n) = 20*n^2 + 1.

1, 21, 81, 181, 321, 501, 721, 981, 1281, 1621, 2001, 2421, 2881, 3381, 3921, 4501, 5121, 5781, 6481, 7221, 8001, 8821, 9681, 10581, 11521, 12501, 13521, 14581, 15681, 16821, 18001, 19221, 20481, 21781, 23121, 24501, 25921, 27381, 28881, 30421, 32001, 33621, 35281

Offset: 0

Vincenzo Librandi, Mar 21 2009

The identity (20*n^2 + 1)^2 - (100*n^2 + 10)*(2*n)^2 = 1 can be written as a(n)^2 - A158492(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 21 2012

Sequence found by reading the segment (1, 21) together with the line from 21, in the direction 21, 81, ..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. - Omar E. Pol, Nov 05 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158492, A195162, A212656.

I:=[1, 21, 81]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 21 2012

LinearRecurrence[{3, -3, 1}, {1, 21, 81}, 50] (* Vincenzo Librandi, Feb 21 2012 *)
20*Range[0,50]^2+1 (* Harvey P. Dale, Aug 06 2025 *)

for(n=0, 40, print1(20*n^2 + 1", ")); \\ Vincenzo Librandi, Feb 21 2012

From Vincenzo Librandi, Feb 21 2012: (Start)
G.f.: -(1 + 18*x + 21*x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 06 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/(2*sqrt(5)))*Pi/(2*sqrt(5)) + 1)/2.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(2*sqrt(5)))*Pi/(2*sqrt(5)) + 1)/2. (End)
From Elmo R. Oliveira, Jan 25 2025: (Start)
E.g.f.: exp(x)*(1 + 20*x + 20*x^2).
a(n) = A212656(2*n). (End)

Edited by N. J. A. Sloane, Oct 12 2009

A374378 Iterated rascal triangle R2: T(n,k) = Sum_{m=0..2} binomial(n-k,m)*binomial(k,m).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 19, 15, 6, 1, 1, 7, 21, 31, 31, 21, 7, 1, 1, 8, 28, 46, 53, 46, 28, 8, 1, 1, 9, 36, 64, 81, 81, 64, 36, 9, 1, 1, 10, 45, 85, 115, 126, 115, 85, 45, 10, 1, 1, 11, 55, 109, 155, 181, 181, 155, 109, 55, 11, 1

Offset: 0

Kolosov Petro, Jul 06 2024

Triangle T(n,k) is the second triangle R2 among the rascal-family triangles; A374452 is triangle R3; A077028 is triangle R1.
Triangle T(n,k) equals Pascal's triangle A007318 through row 2i+1, i=2 (i.e., row 5).
Triangle T(n,k) equals Pascal's triangle A007318 through column i, i=2 (i.e., column 2).

			Triangle begins:
--------------------------------------------------
k=     0   1   2   3    4    5    6   7   8   9 10
--------------------------------------------------
n=0:   1
n=1:   1   1
n=2:   1   2   1
n=3:   1   3   3   1
n=4:   1   4   6   4    1
n=5:   1   5  10  10    5    1
n=6:   1   6  15  19   15    6    1
n=7:   1   7  21  31   31   21    7   1
n=8:   1   8  28  46   53   46   28   8   1
n=9:   1   9  36  64   81   81   64  36   9   1
n=10:  1  10  45  85  115  126  115  85  45  10  1

Kolosov Petro, Table of n, a(n) for n = 0..1325
Amelia Gibbs and Brian K. Miceli, Two Combinatorial Interpretations of Rascal Numbers, arXiv:2405.11045 [math.CO], 2024.
Jena Gregory, Brandt Kronholm, and Jacob White, Iterated rascal triangles, Aequationes mathematicae, 2023.
Jena Gregory, Iterated rascal triangles, Theses and Dissertations. 1050., The University of Texas Rio Grande Valley, 2022.
Philip K. Hotchkiss, Student Inquiry and the Rascal Triangle, arXiv:1907.07749 [math.HO], 2019.
Philip K. Hotchkiss, Generalized Rascal Triangles, Journal of Integer Sequences, Vol. 23, 2020.
Petro Kolosov, Identities in Iterated Rascal Triangles, 2024.

Cf. A077028, A007318, A000027, A000217, A005448, A056108, A212656, A000292, A095667, A095666, A006261.

t[n_, k_]:=Sum[Binomial[n - k, m]*Binomial[k, m], {m, 0, 2}]; Column[Table[t[n, k], {n, 0, 12}, {k, 0, n}], Center]

T(n,k) = 1 + k*(n-k) + (1/4)*(k-1)*k*(n-k-1)*(n-k).
Row sums give A006261(n).
Diagonal T(n+1, n) gives A000027(n).
Diagonal T(n+2, n) gives A000217(n).
Diagonal T(n+3, n) gives A005448(n).
Diagonal T(n+4, n) gives A056108(n).
Diagonal T(n+5, n) gives A212656(n).
Column k=3 difference binomial(n+6, 3) - T(n+6, 3) gives C(n+3,3)=A007318(n+3,3).
Column k=4 difference binomial(n+7, 4) - T(n+7, 4) gives fifth column of (1,4)-Pascal triangle A095667.
G.f.: (1 + 3*x^4*y^2 - (2*x + 3*x^3*y)*(1 + y) + x^2*(1 + 5*y + y^2))/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Jul 09 2024

A158445 a(n) = 25*n^2 + 5.

Original entry on oeis.org

30, 105, 230, 405, 630, 905, 1230, 1605, 2030, 2505, 3030, 3605, 4230, 4905, 5630, 6405, 7230, 8105, 9030, 10005, 11030, 12105, 13230, 14405, 15630, 16905, 18230, 19605, 21030, 22505, 24030, 25605, 27230, 28905, 30630, 32405, 34230, 36105, 38030, 40005, 42030

Offset: 1

Vincenzo Librandi, Mar 19 2009

The identity (10*n^2 + 1)^2 - (25*n^2 + 5)*(2*n)^2 = 1 can be written as A158187(n)^2 - a(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158187, A212656.

I:=[30, 105, 230]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];

Table[25n^2+5,{n,50}]
LinearRecurrence[{3,-3,1},{30,105,230},50] (* Harvey P. Dale, Mar 21 2025 *)

PARI
```
a(n) = 25*n^2 + 5.
```

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f: 5*x*(6+3*x+x^2)/(1-x)^3.
From Amiram Eldar, Mar 05 2023: (Start)
Sum_{n>=1} 1/a(n) = (coth(Pi/sqrt(5))*Pi/sqrt(5) - 1)/10.
Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/sqrt(5))*Pi/sqrt(5))/10. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 5*(exp(x)*(5*x^2 + 5*x + 1) - 1).
a(n) = 5*A212656(n). (End)

A212707 Semiprimes of the form 5*n^2 + 1.

Original entry on oeis.org

6, 21, 46, 321, 501, 721, 1126, 2206, 2881, 3646, 3921, 4501, 7606, 10581, 11521, 13521, 14581, 15681, 16246, 18001, 19846, 20481, 21781, 23806, 24501, 27381, 30421, 32001, 38721, 40501, 42321, 48021, 61606, 64981, 72001, 79381, 83206, 89781, 106581, 121681

Offset: 1

Jonathan Vos Post, May 24 2012

This is to A137530 (primes of form 1+5n^2) as semiprimes A001358 are to primes A000040. Since Z[sqrt(-5)] is not a unique factorization domain, some numbers of form 1+5n^2 are primes in Z but composite in Z[sqrt(-5)]; some values in this sequence are semiprimes in Z but have a different number than 2 of prime factors in Z[sqrt(-5)].

			a(6) = 721 = 1 + 5*(12^2) = 7 * 103.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A001222, A001358, A137530, A212656 (5*n^2 + 1).

IsSemiprime:= func; [s: n in [1..180] | IsSemiprime(s) where s is 5*n^2 + 1]; // Vincenzo Librandi, Sep 22 2012

SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; Select[Table[5*n^2 + 1, {n, 200}], SemiPrimeQ] (* T. D. Noe, May 24 2012 *)
Select[Table[5*n^2 + 1, {n, 180}], PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 22 2012 *)

A212656 INTERSECTION A001358.

{k such that 5*n^2 + 1 for a natural number n, and bigomega(k) = A001222(k) = 2}.

Extended by T. D. Noe, May 24 2012

A243813 Table read by antidiagonals: T(n,k) is the curvature (truncated to integer) of a circle in a variation of nested Pappus chains (see Comments for details).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 9, 1, 1, 1, 1, 3, 13, 1, 1, 1, 1, 2, 5, 19, 1, 1, 1, 1, 1, 3, 7, 25, 1, 1, 1, 1, 1, 2, 4, 9, 33, 1, 1, 1, 1, 1, 1, 2, 5, 11, 41, 1, 1, 1, 1, 1, 1, 2, 3, 6, 14, 51, 1, 1, 1, 1, 1, 1, 1, 2, 4, 7, 17, 61, 1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 9, 21

Offset: 0

Kival Ngaokrajang, Jun 11 2014

Refer to the construction rule used in A243618. For this case, the curvature is defined by (-1/k, 1/(k-1), 1), the circle radius will diverge to infinity (zero curvature). The integral curvatures appearing as periodic, i.e., 2, 6, 6, 10, 30, 42, 28, 12, ..., = A083482(k-1). The integral curvatures seem to align as some sequence, e.g., 3, 7, 13, 21, 31, 43, ..., = A002061(k) and 9, 25, 49, ..., = A016754(k-1). See illustration.

			Table begins:
  n/k  2   3   4   5   6   7  ...
   0   1   1   1   1   1   1  ...
   1   1   1   1   1   1   1  ...
   2   3   1   1   1   1   1  ...
   3   5   2   1   1   1   1  ...
   4   9   3   2   1   1   1  ...
   5  13   5   3   2   1   1  ...
   6  19   7   4   2   2   1  ...
   7  25   9   5   3   2   2  ...
   8  33  11   6   4   3   2  ...
   9  41  14   7   5   3   2  ...
  10  51  17   9   6   4   3  ...
  11  61  21  11   7   5   3  ...
  12  73  25  13   8   5   4  ...
  ...

Kival Ngaokrajang, Illustration for k = 2..7

Cf. A243618, A083482, A002061, A016754.
Cf. Column 1 = A080827(n), column 2 = A056827(n) + 1.
Cf. Integral curvature in column 1..6: [A058331, A227776, A056107, A212656, A158558, A158604].

cp's OEIS Frontend

A158493 a(n) = 20*n^2 + 1.

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A374378 Iterated rascal triangle R2: T(n,k) = Sum_{m=0..2} binomial(n-k,m)*binomial(k,m).

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A158445 a(n) = 25*n^2 + 5.

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A212707 Semiprimes of the form 5*n^2 + 1.

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Magma

Mathematica

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Extensions

A243813 Table read by antidiagonals: T(n,k) is the curvature (truncated to integer) of a circle in a variation of nested Pappus chains (see Comments for details).

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