A008865 -id:A008865 - cp's OEIS frontend

A225078 Numbers n such that n^2+1 and (n+1)^2-2 are both prime.

1, 2, 4, 6, 14, 20, 26, 36, 54, 74, 116, 120, 126, 130, 134, 160, 176, 204, 210, 230, 236, 256, 264, 284, 300, 314, 340, 386, 420, 440, 466, 490, 496, 544, 594, 636, 644, 714, 750, 760, 784, 816, 930, 950, 986, 1070, 1124, 1140, 1146, 1156, 1174, 1176, 1210

Offset: 1

César Aguilera, Apr 26 2013

nonn

Prime limits of the Legendré conjecture for a given n.

			n=2; n+1=3 ;n^2+1=5 and (n+1)^2-2=7.
n=490; n+1=491; n^2+1=240101 and (n+1)^2-2=241079.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Legendre's Conjecture
Wikipedia, Legendre's conjecture

Cf. A002522, A008865, A010051, A014085, A002496, A028871.

import Data.Function (on)
import Data.List (elemIndices)
a225078 n = a225078_list !! (n-1)
a225078_list = elemIndices 1 $
   zipWith ((*) `on` a010051') a002522_list a008865_list
-- Reinhard Zumkeller, May 06 2013

Select[Range[2000], PrimeQ[#^2 + 1] && PrimeQ[(# + 1)^2 - 2] &] (* T. D. Noe, May 06 2013 *)

A241201 a(n) is the least r such that there are n+2 consecutive increasing terms in the r-th row of Pascal's triangle (binomial(r,*)) which satisfy a polynomial of degree n.

Original entry on oeis.org

7, 14, 62, 31, 339, 1022

Offset: 1

T. D. Noe, Apr 21 2014

Old definition: "Numbers k such that n+2 consecutive terms of binomial(n,k) satisfy a polynomial relation of degree n for some k in the range 0 <= k <= n/2.".

Is this sequence finite?

			a(1) = 7 because the 3 terms 7, 21, 35 are linear.

Cf. A008865 (binomial(n,k) has 3 consecutive terms in a linear relation).
Cf. A062730 (3 terms in arithmetic progression in Pascal's triangle).
Cf. A241199, A241200 (similar, but quadratic).
Cf. A241202 (position of the first of terms).

t = Table[k = 1; While[b = Binomial[k, Range[0, k/2]]; d = Differences[b, n + 1]; ! MemberQ[d, 0], k++]; {k, Position[d, 0, 1, 1][[1, 1]] - 1}, {n, 6}]; Transpose[t][[1]]

Definition clarified by Don Reble, Dec 14 2020

A241202 Beginning of a polynomial relation of degree n in n+2 terms in the first half of Pascal's triangle. See A241201.

Original entry on oeis.org

1, 2, 26, 9, 149, 489

Offset: 1

T. D. Noe, Apr 21 2014

Is this sequence finite?

Cf. A008865 (binomial(n,k) has 3 consecutive terms in a linear relation).
Cf. A062730 (3 terms in arithmetic progression in Pascal's triangle).
Cf. A241199, A241200 (similar, but quadratic).

t = Table[k = 1; While[b = Binomial[k, Range[0, k/2]]; d = Differences[b, n + 1]; ! MemberQ[d, 0], k++]; {k, Position[d, 0, 1, 1][[1, 1]] - 1}, {n, 6}]; Transpose[t][[2]]

A293620 Numbers k such that f(k), f(k+1) and f(k+2) are all primes, where f(k) = (2k+1)^2 - 2 (A073577).

Original entry on oeis.org

1, 2, 16, 58, 149, 177, 534, 681, 954, 1045, 1052, 1255, 1367, 1563, 2046, 2074, 2515, 2557, 2564, 2788, 3586, 3593, 3908, 4062, 4552, 5252, 5371, 5385, 6400, 6729, 7443, 7478, 9305, 9375, 9942, 10355, 10411, 10726, 10740, 11286, 11545, 11559, 11832, 11965

Offset: 1

Amiram Eldar, Oct 13 2017

nonn

Sierpiński proved that under Schinzel's hypothesis H this sequence is infinite.
Sierpiński showed that the only quadruple of consecutive primes of the form (2k+1)^2 - 2 are for k = 1 (i.e., 1 and 2 are the only consecutive terms in this sequence).
Numbers k such that the 3 consecutive integers k, k+1 and k+2 belong to A088572. - Michel Marcus, Oct 13 2017

			The first triples are: k = 1: (7, 23, 47), k = 2: (23, 47, 79), k = 16: (1087, 1223, 1367).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wacław Sierpiński, Remarque sur la distribution de nombres premiers, Matematički Vesnik, Vol. 2(17), Issue 31 (1965), pp. 77-78.
Eric Weisstein's World of Mathematics, Near-Square Prime.
Wikipedia, Schinzel's hypothesis H.

Cf. A088572, A008865, A028870, A028871, A073577, A088572.

Select[Range[10^4], AllTrue[{(2#+1)^2-2, (2#+3)^2-2, (2#+5)^2-2},PrimeQ] &]
SequencePosition[Table[If[PrimeQ[(2k+1)^2-2],1,0],{k,12000}],{1,1,1}][[;;,1]] (* Harvey P. Dale, Feb 09 2025 *)

f(n) = 4*n^2 + 4*n - 1;
isok(n) = isprime(f(n)) && isprime(f(n+1)) && isprime(f(n+2)); \\ Michel Marcus, Oct 13 2017

A316649 Triangle read by rows in which T(n,k) is the number of length k chains from (0,0) to (n,n) of the poset [n] X [n] ordered by the product order, 0 <= k <= 2n, n>=0.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 7, 12, 6, 0, 1, 14, 55, 92, 70, 20, 0, 1, 23, 153, 471, 780, 720, 350, 70, 0, 1, 34, 336, 1584, 4251, 7002, 7238, 4592, 1638, 252, 0, 1, 47, 640, 4210, 16175, 39733, 65226, 72660, 54390, 26250, 7392, 924, 0, 1, 62, 1107, 9596, 49225, 164898, 380731, 623576, 732618, 614700, 360162, 140184, 32604, 3432

Offset: 0

Geoffrey Critzer, Jul 09 2018

			Triangle begins:
  1;
  0, 1,  2;
  0, 1,  7,  12,    6;
  0, 1, 14,  55,   92,   70,   20;
  0, 1, 23, 153,  471,  780,  720,  350,   70;
  0, 1, 34, 336, 1584, 4251, 7002, 7238, 4592, 1638, 252;
  ...

Alois P. Heinz, Rows n = 0..100, flattened

Columns k=0-2 give: A000007, A057427, A008865(n+1) for n>0.
Row sums give A052141.
T(n,n) gives A108628(n-1) for n>0.
T(n,2n) gives A000984.
Cf. A007318.

b:= proc(n, m) option remember; expand(`if`(n+m=0, 1, add(add(
     `if`(i+j=0, 0, b(sort([n-i, m-j])[])*x), j=0..m), i=0..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..8);  # Alois P. Heinz, Jul 10 2018

Join[{{1}},Table[a =Sort[Level[Table[Table[{i, j}, {i, 0, n}], {j, 0, n}], {2}]];f[list1_, list2_] :=Boole[(list1 - list2)[[1]] < 1 \[And] (list1 - list2)[[2]] < 1];m = Table[Table[f[a[[l]], a[[k]]], {k, 1, Length[a]}], {l, 1, Length[a]}];Prepend[Table[
     MatrixPower[m - IdentityMatrix[(n + 1)^2], k][[1, (n + 1)^2]], {k, 1, 2 n}], 0], {n, 1, 7}]] // Grid

A349427 a(n) = ((n+1)^2 - 2) * binomial(2*n-2,n-1) / 2.

Original entry on oeis.org

0, 1, 7, 42, 230, 1190, 5922, 28644, 135564, 630630, 2892890, 13117676, 58903572, 262303132, 1159666900, 5094808200, 22259364120, 96773942790, 418882316490, 1805951924700, 7758285404100, 33221013445620, 141830949914940, 603876402587640, 2564713671647400

Offset: 0

Ilya Gutkovskiy, Nov 17 2021

Cf. A000108, A000217, A000984, A001700, A001793, A002457, A002544, A008865, A037966, A088218, A127736.

Table[((n + 1)^2 - 2) Binomial[2 n - 2, n - 1]/2, {n, 0, 24}]
nmax = 24; CoefficientList[Series[x (1 - x) (1 - 2 x)/(1 - 4 x)^(5/2), {x, 0, nmax}], x]

a(n) = ((n+1)^2 - 2) * binomial(2*n-2,n-1) / 2 \\ Andrew Howroyd, Nov 20 2021

G.f.: x * (1 - x) * (1 - 2*x) / (1 - 4*x)^(5/2).
E.g.f.: x * exp(2*x) * (2 * (1 + x) * BesselI(0,2*x) + (1 + 2*x) * BesselI(1,2*x)) / 2.
a(n) = n * ((n+1)^2 - 2) * Catalan(n-1) / 2.
a(n) = Sum_{k=0..n} binomial(n,k)^2 * A000217(k).
a(n) ~ 2^(2*n-3) * n^(3/2) / sqrt(Pi).
D-finite with recurrence (n-1)*(n^2-2)*a(n) -2*(2*n-3)*(n^2+2*n-1)*a(n-1)=0. - R. J. Mathar, Mar 06 2022

A363365 Array read by ascending antidiagonals: A(1, k) = k; for n > 1, A(n, k) = (k + 1)*A(n-1, k) + k + 1 - n, with k > 0.

Original entry on oeis.org

1, 2, 2, 3, 7, 3, 4, 21, 14, 4, 5, 62, 57, 23, 5, 6, 184, 228, 117, 34, 6, 7, 549, 911, 586, 207, 47, 7, 8, 1643, 3642, 2930, 1244, 333, 62, 8, 9, 4924, 14565, 14649, 7465, 2334, 501, 79, 9, 10, 14766, 58256, 73243, 44790, 16340, 4012, 717, 98, 10

Offset: 1

Stefano Spezia, May 29 2023

			The array begins:
  1,   2,    3,     4,     5, ...
  2,   7,   14,    23,    34, ...
  3,  21,   57,   117,   207, ...
  4,  62,  228,   586,  1244, ...
  5, 184,  911,  2930,  7465, ...
  6, 549, 3642, 14649, 44790, ...
  ...

Cf. A000027 (n=1 or k=1), A008865, A051846 (diagonal), A064017 (k=9), A353094 (k=2), A353095 (k=3), A353096 (k=4), A353097 (k=5), A353098 (k=6), A353099 (k=7), A353100 (k=8), A363366 (antidiagonal sums).

A[n_,k_]:=((k-1)*(k+1)^(n+1)+k*n-k^2+1)/k^2; Table[A[n-k+1,k],{n,10},{k,n}]//Flatten (* or *)
A[n_,k_]:=SeriesCoefficient[x*(k-(k+1)*x)/((1-x)^2*(1-(k+1)*x)),{x,0,n}]; Table[A[n-k+1,k],{n,10},{k,n}]//Flatten (* or *)
A[n_,k_]:=n!SeriesCoefficient[Exp[x]((k^2-1)(Exp[k x]-1)+k x)/k^2,{x,0,n}]; Table[A[n-k+1,k],{n,10},{k,n}]//Flatten

A(n, k) = ((k - 1)*(k + 1)^(n+1) + k*n - k^2 + 1)/k^2.
O.g.f. of k-th column: x*(k - (k + 1)*x)/((1 - x)^2*(1 - (k + 1)*x)).
E.g.f. of k-th column: exp(x)*((k^2 - 1)*(exp(k*x) - 1) + k*x)/k^2.
A(2, n) = A008865(n+1).

A386206 Triangle read by rows: T(n,k) = n^2 - k, with 0 <= k <= n.

Original entry on oeis.org

0, 1, 0, 4, 3, 2, 9, 8, 7, 6, 16, 15, 14, 13, 12, 25, 24, 23, 22, 21, 20, 36, 35, 34, 33, 32, 31, 30, 49, 48, 47, 46, 45, 44, 43, 42, 64, 63, 62, 61, 60, 59, 58, 57, 56, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 100, 99, 98, 97, 96, 95, 94, 93, 92, 91, 90

Offset: 0

Stefano Spezia, Jul 15 2025

			The triangle begins as:
   0;
   1,  0;
   4,  3,  2;
   9,  8,  7,  6;
  16, 15, 14, 13, 12;
  25, 24, 23, 22, 21, 20;
  36, 35, 34, 33, 32, 31, 30;
  49, 48, 47, 46, 45, 44, 43, 42;
  64, 63, 62, 61, 60, 59, 58, 57, 56;
  ...

Vincenzo Librandi, Table of n, a(n) for n = 0..3320 (rows 0..80)

Cf. A000290 (k=0), A002414 (row sums), A005563, A008865, A028347 (k=4), A028872 (k=3), A028875 (k=5), A279019 (diagonal).

[[n^2-k: k in [0..n]]: n in [0..9]]; // Vincenzo Librandi, Jul 17 2025

T[n_,k_]:=n^2-k; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten

G.f.: x*(1 + x + 2*x*y^2 + 5*x^3*y^2 - x^2*y*(4 + 5*y))/((1 - x)^3*(1 - x*y)^3).
T(n,1) = A005563(n-1) for n > 0.
T(n,2) = A008865(n) for n > 1.

A386481 Array read by upward antidiagonals: T(k,n) = 1 (k = 0, n >= 0), T(k,n) = binomial(n,2)k^2 + n(k-1) + 1 (k >= 1, n >= 0).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 7, 4, 1, 1, 4, 14, 16, 7, 1, 1, 5, 23, 34, 29, 11, 1, 1, 6, 34, 58, 63, 46, 16, 1, 1, 7, 47, 88, 109, 101, 67, 22, 1, 1, 8, 62, 124, 167, 176, 148, 92, 29, 1, 1, 9, 79, 166, 237, 271, 259, 204, 121, 37, 1, 1, 10, 98, 214, 319, 386, 400, 358, 269, 154, 46, 1, 1, 11, 119, 268, 413, 521, 571, 554, 473, 343, 191, 56, 1

Offset: 0

N. J. A. Sloane, Aug 11 2025

T(k,n) is the maximum number of regions the plane can be divided into by drawing n k-armed long-legged V's.

			Array begins (the rows are T(0,n>=0),, T(1,n>=0), T(2,n>=0), ...):
   1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   1, 1, 2, 4, 7, 11, 16, 22, 29, ...
   1, 2, 7, 16, 29, 46, 67, 92, 121, ...
   1, 3, 14, 34, 63, 101, 148, 204, 269, ...
   1, 4, 23, 58, 109, 176, 259, 358, 473, ...
   1, 5, 34, 88, 167, 271, 400, 554, 733, ...
   1, 6, 47, 124, 237, 386, 571, 792, 1049, ...
   1, 7, 62, 166, 319, 521, 772, 1072, 1421, ...
    ...
The first few antidiagonals are:
    1,
    1, 1,
    1, 1, 1,
    1, 2, 2, 1,
    1, 3, 7, 4, 1,
    1, 4, 14, 16, 7, 1,
    1, 5, 23, 34, 29, 11, 1,
    1, 6, 34, 58, 63, 46, 16, 1,
    1, 7, 47, 88, 109, 101, 67, 22, 1,
     ...

David O. H. Cutler and Neil J. A. Sloane, Cutting a pancake with an exotic knife, Paper in preparation, Sep 05 2025

N. J. A. Sloane, Illustration for T(3,2) = 14.
N. J. A. Sloane, Illustration for T(3,3) = 34 (a 3-armed V is also known as a Wu).

This is a companion to the array A386478.

The rows and columns include A000124, A130883, A140064, A383464, and A008865.

A163159 Fibonacci numbers F such that F^2-2 is prime.

Original entry on oeis.org

2, 3, 5, 13, 21, 55, 89, 233, 987, 1597, 5702887, 1836311903, 99194853094755497, 26925748508234281076009, 184551825793033096366333, 468340976726457153752543329995929, 30010821454963453907530667147829489881, 1188518561323126046432205871807859915657177

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 21 2009

nonn

In condensed representation: The A000045(i) at i = 3, 4, 5, 7, 8, 10, 11, 13, 16, 17, 34, 46,... [R. J. Mathar, Jul 25 2009]

			2^2-2=2. 3^2-2=7. 5^2-2=23.

Cf. A000045, A028871, A163158

f[n_]:=Fibonacci[n]; f2[n_]:=f[n]^2-2; lst={};Do[If[PrimeQ[f2[n]],AppendTo[lst, f[n]]],{n,6!}];lst
Select[Fibonacci[Range[400]],PrimeQ[#^2-2]&] (* Harvey P. Dale, Oct 21 2011 *)

{A000045(i): A008865(A000045(i)) in A000040}. [R. J. Mathar, Jul 25 2009]

More terms from Harvey P. Dale, Oct 21 2011

cp's OEIS Frontend

A225078 Numbers n such that n^2+1 and (n+1)^2-2 are both prime.

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Haskell

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A241201 a(n) is the least r such that there are n+2 consecutive increasing terms in the r-th row of Pascal's triangle (binomial(r,*)) which satisfy a polynomial of degree n.

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A241202 Beginning of a polynomial relation of degree n in n+2 terms in the first half of Pascal's triangle. See A241201.

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A293620 Numbers k such that f(k), f(k+1) and f(k+2) are all primes, where f(k) = (2k+1)^2 - 2 (A073577).

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A316649 Triangle read by rows in which T(n,k) is the number of length k chains from (0,0) to (n,n) of the poset [n] X [n] ordered by the product order, 0 <= k <= 2n, n>=0.

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Maple

Mathematica

A349427 a(n) = ((n+1)^2 - 2) * binomial(2*n-2,n-1) / 2.

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A363365 Array read by ascending antidiagonals: A(1, k) = k; for n > 1, A(n, k) = (k + 1)*A(n-1, k) + k + 1 - n, with k > 0.

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A386206 Triangle read by rows: T(n,k) = n^2 - k, with 0 <= k <= n.

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Magma

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A386481 Array read by upward antidiagonals: T(k,n) = 1 (k = 0, n >= 0), T(k,n) = binomial(n,2)k^2 + n(k-1) + 1 (k >= 1, n >= 0).