A154428 -id:A154428 - cp's OEIS frontend

A017281 a(n) = 10*n + 1.

1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 201, 211, 221, 231, 241, 251, 261, 271, 281, 291, 301, 311, 321, 331, 341, 351, 361, 371, 381, 391, 401, 411, 421, 431, 441, 451, 461, 471, 481, 491, 501, 511, 521, 531

Offset: 0

N. J. A. Sloane

Equals [1, 2, 3, ...] convolved with [1, 9, 0, 0, 0, ...]. - Gary W. Adamson, May 30 2009
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=10, (i>1), A[i,i-1] = -1, and A[i,j]=0 otherwise. Then, for n>=2, a(n-1) = -coeff(charpoly(A,x),x^(n-1)). - Milan Janjic, Feb 21 2010
Positive integers with last decimal digit = 1. - Wesley Ivan Hurt, Jun 17 2015
Also the number of (not necessarily maximal) cliques in the 2n-crossed prism graph. - Eric W. Weisstein, Nov 29 2017
From Martin Renner, May 28 2024: (Start)
Also number of squares in a grid cross with equally long arms and a width of two points (cf. A017113), e.g. for n = 2 there are nine squares of size 1 unit of area, four of size 2, two of size 5, four of size 8 and two of size 13, thus a total of 21 squares.
    · ·           · ·           · ·           · ·           * ·
    · ·           · ·           * ·           * ·           · ·
* * · · · ·   · · * · · ·   · · · · * ·   · · · · · ·   · · · · · *
* * · · · ·   · * · * · ·   · * · · · ·   * · · · * ·   * · · · · ·
    · ·           * ·           · *           · ·           · ·
    · ·           · ·           · ·           * ·           · *
The possible areas of the squares are given by ceiling(k^2/2) for 1 <= k <= 2*n+1, cf. A000982. In general, there are 4*n + 1 squares with one unit area to be found in the cross, cf. A016813, for n > 0 always four squares of even area and two squares of odd area > 1. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Milan Janjic, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Crossed Prism Graph
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A093645 (column 1).
Subsequence of A034709, together with A017293, A017329, A139222, A139245, A139249, A139264, A139279 and A139280.
Cf. A048161, A154428.
Cf. A005408, A016813, A016921, A017533, A161700, A128470, A158057, A161705, A161709, A161714.
Cf. A030430 (primes).
Cf. A272914, first comment. [Bruno Berselli, May 26 2016]

List([0..60], n-> 10*n+1 ); # G. C. Greubel, Sep 18 2019

a017281 = (+ 1) . (* 10)
a017281_list = [1,11..]  -- Reinhard Zumkeller, Apr 16 2012

[10*n+1 : n in [0..60]]; // Zaki Khandaker, May 16 2015

A017281:=n->10*n + 1; seq(A017281(n), n=0..80); # Wesley Ivan Hurt, Jan 29 2014

f[n_] := FromDigits[IntegerDigits[n^2, n + 1]]; Array[f, 60] (* Robert G. Wilson v, Apr 14 2009 *)
Range[1, 1000, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
(* From Eric W. Weisstein, Nov 29 2017: (Start) *)
Table[10n+1, {n, 0, 60}]
10*Range[0, 60] + 1
LinearRecurrence[{2, -1}, {11, 21}, {0, 60}]
CoefficientList[Series[(1+9x)/(1-x)^2, {x, 0, 60}], x] (* End *)

Vec((1+9*x)/(1-x)^2 + O(x^80)) \\ Michel Marcus, Jun 17 2015

[10*n+1 for n in (0..60)] # G. C. Greubel, Sep 18 2019

G.f.: (1+9*x)/(1-x)^2.
a(n) = 20*n - a(n-1) - 8, with a(0)=1. - Vincenzo Librandi, Nov 20 2010
a(n) = 2*a(n-1) - a(n-2), for n > 2. - Wesley Ivan Hurt, Jun 17 2015
E.g.f.: (1 + 10*x)*exp(x). - G. C. Greubel, Sep 18 2019

A048161 Primes p such that q = (p^2 + 1)/2 is also a prime.

Original entry on oeis.org

3, 5, 11, 19, 29, 59, 61, 71, 79, 101, 131, 139, 181, 199, 271, 349, 379, 409, 449, 461, 521, 569, 571, 631, 641, 661, 739, 751, 821, 881, 929, 991, 1031, 1039, 1051, 1069, 1091, 1129, 1151, 1171, 1181, 1361, 1439, 1459, 1489, 1499, 1531, 1709, 1741, 1811, 1831, 1901

Offset: 1

Harvey Dubner (harvey(AT)dubner.com)

Primes which are a leg of an integral right triangle whose hypotenuse is also prime.
It is conjectured that there are an infinite number of such triangles.
The Pythagorean triple {p, (p^2 - 1)/2, (p^2 + 1)/2} corresponds to {a(n), A067755(n), A067756(n)}. - Lekraj Beedassy, Oct 27 2003
There is no Pythagorean triangle all of whose sides are prime numbers. Still there are Pythagorean triangles of which the hypotenuse and one side are prime numbers, for example, the triangles (3,4,5), (11,60,61), (19,180,181), (61,1860,1861), (71,2520,2521), (79,3120,3121). [Sierpiński]
We can always write p=(Y+1)^2-Y^2, with Y=(p-1)/2, therefore q=(Y+1)^2+Y^2. - Vincenzo Librandi, Nov 19 2010
p^2 and p^2+1 are semiprimes; p^2 are squares in A070552 Numbers n such that n and n+1 are products of two primes. - Zak Seidov, Mar 21 2011

			For p=11, (p^2+1)/2=61; p=61, (p^2+1)/2=1861.
For p(1)=3, the right triangle 3, 4, 5 is the smallest where 5=(3*3+1)/2.
For p(10)=101, the right triangle is 101, 5100, 5101 where 5101=(101*101+1)/2.

Wacław Sierpiński, Pythagorean triangles, Dover Publications, Inc., Mineola, NY, 2003, p. 6. MR2002669

T. D. Noe, Table of n, a(n) for n = 1..10000
H. Dubner and T. Forbes, Prime Pythagorean triangles, Journal of Integer Sequences, Vol. 4(2001), #01.2.3.

Cf. A067755, A067756. Complement in primes of A094516.
Cf. A017281, A154428, A010051, A065091, A005383.
Cf. A048270, A048295, A308635, A308636. Primes contained in A002731.

a048161 n = a048161_list !! (n-1)
a048161_list = [p | p <- a065091_list, a010051 ((p^2 + 1) `div` 2) == 1]
-- Reinhard Zumkeller, Aug 26 2012

[p: p in PrimesInInterval(3, 2000) | IsPrime((p^2+1) div 2)]; // Vincenzo Librandi, Dec 31 2013

a := proc (n) if isprime(n) = true and type((1/2)*n^2+1/2, integer) = true and isprime((1/2)*n^2+1/2) = true then n else end if end proc: seq(a(n), n = 1 .. 2000) # Emeric Deutsch, Jan 18 2009

Select[Prime[Range[200]], PrimeQ[(#^2 + 1)/2] &] (* Stefan Steinerberger, Apr 07 2006 *)
a[ n_] := Module[{p}, If[ n < 1, 0, p = a[n - 1]; While[ (p = NextPrime[p]) > 0, If[ PrimeQ[(p*p + 1)/2], Break[]]]; p]]; (* Michael Somos, Nov 24 2018 *)

{a(n) = my(p); if( n<1, 0, p = a(n-1) + (n==1); while(p = nextprime(p+2), if( isprime((p*p+1)/2), break)); p)}; /* Michael Somos, Mar 03 2004 */

from sympy import isprime, nextprime; p = 2
while p < 1901: p = nextprime(p); print(p, end = ', ') if isprime((p*p+1)//2) else None # Ya-Ping Lu, Apr 24 2025

A000035(a(n))*A010051(a(n))*A010051((a(n)^2+1)/2) = 1. - Reinhard Zumkeller, Aug 26 2012

More terms from David W. Wilson

cp's OEIS Frontend

A017281 a(n) = 10*n + 1.

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