A158558 -id:A158558 - cp's OEIS frontend

A300656 Triangle read by rows: T(n,k) = 30k^2(n-k)^2 + 1; n >= 0, 0 <= k <= n.

1, 1, 1, 1, 31, 1, 1, 121, 121, 1, 1, 271, 481, 271, 1, 1, 481, 1081, 1081, 481, 1, 1, 751, 1921, 2431, 1921, 751, 1, 1, 1081, 3001, 4321, 4321, 3001, 1081, 1, 1, 1471, 4321, 6751, 7681, 6751, 4321, 1471, 1, 1, 1921, 5881, 9721, 12001, 12001, 9721, 5881, 1921, 1

Offset: 0

Kolosov Petro, Mar 10 2018

From Kolosov Petro, Apr 12 2020: (Start)
Let A(m, r) = A302971(m, r) / A304042(m, r).
Let L(m, n, k) = Sum_{r=0..m} A(m, r) * k^r * (n - k)^r.
Then T(n, k) = L(2, n, k).
Fifth power can be expressed as row sum of triangle T(n, k).
T(n, k) is symmetric: T(n, k) = T(n, n-k). (End)

			Triangle begins:
--------------------------------------------------------------------------
k=    0     1     2      3      4      5      6      7     8     9    10
--------------------------------------------------------------------------
n=0:  1;
n=1:  1,    1;
n=2:  1,   31,    1;
n=3:  1,  121,  121,     1;
n=4:  1,  271,  481,   271,     1;
n=5:  1,  481, 1081,  1081,   481,     1;
n=6:  1,  751, 1921,  2431,  1921,   751,     1;
n=7:  1, 1081, 3001,  4321,  4321,  3001,  1081,     1;
n=8:  1, 1471, 4321,  6751,  7681,  6751,  4321,  1471,    1;
n=9:  1, 1921, 5881,  9721, 12001, 12001,  9721,  5881, 1921,    1;
n=10: 1, 2431, 7681, 13231, 17281, 18751, 17281, 13231, 7681, 2431,   1;

Kolosov Petro, Rows n = 0..2078 of triangle, flattened.
Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.
Petro Kolosov, Polynomial identity involving binomial theorem and Faulhaber's formula, 2023.
Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.

Various cases of L(m, n, k): A287326(m=1), This sequence (m=2), A300785(m=3). See comments for L(m, n, k).
Row sums give the nonzero terms of A002561.
Cf. A000584, A287326, A007318, A077028, A294317, A068236, A302971, A304042, A002561, A258807, A158558, A094053, A024003, A316349.

T:=Flat(List([0..9],n->List([0..n],k->30*k^2*(n-k)^2+1))); # Muniru A Asiru, Oct 24 2018

[[30*k^2*(n-k)^2+1: k in [0..n]]: n in [0..12]]; // G. C. Greubel, Dec 14 2018

a:=(n,k)->30*k^2*(n-k)^2+1: seq(seq(a(n,k),k=0..n),n=0..9); # Muniru A Asiru, Oct 24 2018

T[n_, k_] := 30 k^2 (n - k)^2 + 1; Column[
Table[T[n, k], {n, 0, 10}, {k, 0, n}], Center] (* Kolosov Petro, Apr 12 2020 *)
f[n_]:=Table[SeriesCoefficient[(1 + 26 y + 336 y^2 + 326 y^3 + 31 y^4 + x^2 (1 + 116 y + 486 y^2 + 116 y^3 + y^4) + x (-2 - 82 y - 882 y^2 - 502 y^3 + 28 y^4))/((-1 + x)^3 (-1 + y)^5), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n}]; Flatten[Array[f, 11, 0]] (* Stefano Spezia, Oct 30 2018 *)

t(n, k) = 30*k^2*(n-k)^2+1
trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))
/* Print initial 9 rows of triangle as follows */ trianglerows(9)

[[30*k^2*(n-k)^2+1 for k in range(n+1)] for n in range(12)] # G. C. Greubel, Dec 14 2018

From Kolosov Petro, Apr 12 2020: (Start)
T(n, k) = 30 * k^2 * (n-k)^2 + 1.
T(n, k) = 30 * A094053(n,k)^2 + 1.
T(n, k) = A158558((n-k) * k).
T(n+2, k) = 3*T(n+1, k) - 3*T(n, k) + T(n-1, k), for n >= k.
Sum_{k=1..n} T(n, k) = A000584(n).
Sum_{k=0..n-1} T(n, k) = A000584(n).
Sum_{k=0..n} T(n, k) = A002561(n).
Sum_{k=1..n-1} T(n, k) = A258807(n).
Sum_{k=1..n-1} T(n, k) = -A024003(n), n > 1.
Sum_{k=1..r} T(n, k) = A316349(2,r,0)*n^0 - A316349(2,r,1)*n^1 + A316349(2,r,2)*n^2. (End)
G.f.: (1 + 26*y + 336*y^2 + 326*y^3 + 31*y^4 + x^2*(1 + 116*y + 486*y^2 + 116*y^3 + y^4) + x*(-2 - 82*y - 882*y^2 - 502*y^3 + 28*y^4))/((-1 + x)^3*(-1 + y)^5). - Stefano Spezia, Oct 30 2018

A158557 a(n) = 225*n^2 + 15.

Original entry on oeis.org

15, 240, 915, 2040, 3615, 5640, 8115, 11040, 14415, 18240, 22515, 27240, 32415, 38040, 44115, 50640, 57615, 65040, 72915, 81240, 90015, 99240, 108915, 119040, 129615, 140640, 152115, 164040, 176415, 189240, 202515, 216240, 230415, 245040, 260115, 275640, 291615

Offset: 0

Vincenzo Librandi, Mar 21 2009

The identity (30*n^2 + 1)^2 - (225*n^2 + 15)*(2*n)^2 = 1 can be written as A158558(n)^2 - a(n)*A005843(n)^2 = 1.

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, A158558.

I:=[15, 240, 915]; [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 14 2012

LinearRecurrence[{3, -3, 1}, {15, 240, 915}, 50] (* Vincenzo Librandi, Feb 14 2012 *)
225*Range[0,40]^2+15 (* Harvey P. Dale, Apr 06 2019 *)

for(n=0, 22, print1(225*n^2 + 15", ")); \\ Vincenzo Librandi, Feb 14 2012

G.f.: 15*(1 + 13*x + 16*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 09 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(15))*Pi/sqrt(15) + 1)/30.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(15))*Pi/sqrt(15) + 1)/30. (End)
E.g.f.: 15*exp(x)*(1 + 15*x + 15*x^2). - Elmo R. Oliveira, Jan 15 2025

Comment rewritten, a(0) added by R. J. Mathar, Oct 16 2009

A158672 a(n) = 900*n^2 + 30.

Original entry on oeis.org

30, 930, 3630, 8130, 14430, 22530, 32430, 44130, 57630, 72930, 90030, 108930, 129630, 152130, 176430, 202530, 230430, 260130, 291630, 324930, 360030, 396930, 435630, 476130, 518430, 562530, 608430, 656130, 705630, 756930, 810030, 864930, 921630, 980130, 1040430

Offset: 0

Vincenzo Librandi, Mar 24 2009

The identity (60*n^2 + 1)^2 - (900*n^2 + 30)*(2*n)^2 = 1 can be written as A158673(n)^2 - a(n)*A005843(n)^2 = 1.

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, A158558, A158673.

I:=[30, 930, 3630]; [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 19 2012

LinearRecurrence[{3, -3, 1}, {30, 930, 3630}, 50] (* Vincenzo Librandi, Feb 19 2012 *)
900*Range[0,40]^2+30 (* Harvey P. Dale, May 02 2025 *)

for(n=0, 40, print1(900*n^2 + 30", ")); \\ Vincenzo Librandi, Feb 19 2012

G.f.: -30*(1 + 28*x + 31*x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 20 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(30))*Pi/sqrt(30) + 1)/60.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(30))*Pi/sqrt(30) + 1)/60. (End)
From Elmo R. Oliveira, Jan 15 2025: (Start)
E.g.f.: 30*exp(x)*(1 + 30*x + 30*x^2).
a(n) = 30*A158558(n). (End)

Comment rewritten, a(0) added and formula replaced by R. J. Mathar, Oct 22 2009

A283867 Numbers n such that 30n^2 - 1 and 30n^2 + 1 are (twin) primes.

Original entry on oeis.org

1, 3, 10, 14, 18, 38, 62, 73, 116, 118, 143, 183, 221, 232, 242, 330, 333, 413, 430, 455, 470, 496, 507, 533, 538, 556, 606, 622, 645, 675, 687, 701, 720, 777, 792, 819, 846, 879, 881, 895, 913, 1000, 1019, 1030, 1092, 1155, 1214, 1238, 1253, 1261, 1313, 1337, 1350, 1407, 1418, 1429, 1431

Offset: 1

Juri-Stepan Gerasimov, Mar 17 2017

nonn

			3 is in this sequence because 30*3^2 - 1 = 269 and 30*3^2 + 1 = 271 are twin primes.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000290, A001097, A077800, A158558, A158560.

[n: n in [1..1500] | IsPrime(30*n^2-1) and IsPrime(30*n^2+1)];

Select[Range@ 1431, PrimeQ[30*#^2 + 1] && PrimeQ[30*#^2 - 1] &] (* Indranil Ghosh, Mar 17 2017 *)

is(n)=isprime(30*n^2-1) && isprime(30*n^2+1) \\ Charles R Greathouse IV, Mar 17 2017

from sympy import isprime
[i for i in range(1, 1501) if isprime(30*i**2 - 1) and isprime(30*i**2 + 1)] # Indranil Ghosh, Mar 17 2017

a(n) >> n log^2 n. - Charles R Greathouse IV, Mar 17 2017

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

A300656 Triangle read by rows: T(n,k) = 30*k^2*(n-k)^2 + 1; n >= 0, 0 <= k <= n.

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A158557 a(n) = 225*n^2 + 15.

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A158672 a(n) = 900*n^2 + 30.

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A283867 Numbers n such that 30*n^2 - 1 and 30*n^2 + 1 are (twin) primes.

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Magma

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Python

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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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A300656 Triangle read by rows: T(n,k) = 30k^2(n-k)^2 + 1; n >= 0, 0 <= k <= n.

A283867 Numbers n such that 30n^2 - 1 and 30n^2 + 1 are (twin) primes.