A113541 -id:A113541 - cp's OEIS frontend

A261191 40-gonal numbers: a(n) = 38n(n-1)/2 + n.

0, 1, 40, 117, 232, 385, 576, 805, 1072, 1377, 1720, 2101, 2520, 2977, 3472, 4005, 4576, 5185, 5832, 6517, 7240, 8001, 8800, 9637, 10512, 11425, 12376, 13365, 14392, 15457, 16560, 17701, 18880, 20097, 21352, 22645, 23976, 25345, 26752, 28197, 29680, 31201

Offset: 0

Sergey Pavlov, Aug 11 2015

According to the common formula for the polygonal numbers: (s-2)*n*(n-1)/2 + n (here s = 40).
The 4th number of the sequence, 117, is also the 10th pentagonal number (see A000326). The next number of the series, 232, is also the 9th decagonal number (see A001107), while 576 is the 25th square number (see A000290). The 12th number of the sequence, 2101, is the 23rd 11-gonal number (see A051682).
From Bruno Berselli, Aug 21 2015: (Start)
a(n) and a(n) - 2*n + 1 provide the numbers m such that 19*m + 81 is a square.
Partial sums of the numbers of the type 38*h + 1 (quadrisections of A113541 and A151979). (End)

Colin Barker, Table of n, a(n) for n = 0..1000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

JavaScript
```
function a(n){return 38*n*(n-1)/2+n}
```

[n*(19*n-18): n in [0..45]]; // Vincenzo Librandi, Aug 12 2015

A261191:=n->38*n*(n-1)/2+n: seq(A261191(n), n=0..50); # Wesley Ivan Hurt, Aug 15 2015

Table[n (19 n - 18), {n, 0, 45}] (* Bruno Berselli, Aug 21 2015 *)

concat(0, Vec(-x*(37*x+1)/(x-1)^3 + O(x^100))) \\ Colin Barker, Aug 11 2015

first(m)=my(v=vector(m,i,i--;38*i*(i-1)/2+i));v; \\ Anders Hellström, Aug 13 2015

a(n) = n*(19*n - 18).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), for n > 2. - Colin Barker, Aug 11 2015
G.f.: -x*(37*x+1) / (x-1)^3. - Colin Barker, Aug 11 2015
E.g.f.: exp(x)*(x + 19*x^2). - Nikolaos Pantelidis, Feb 10 2023

A115272 Primes p such that p + 2, 18p^2 + 1, and 18(p+2)^2 + 1 are all primes.

Original entry on oeis.org

29, 107, 431, 1487, 1607, 2141, 5501, 10139, 10271, 17579, 22481, 23057, 27479, 32369, 36341, 36929, 38447, 55931, 57527, 69827, 75539, 78539, 79691, 81047, 81971, 84179, 86027, 89561, 93761, 102059, 112571, 113147, 118799, 119687

Offset: 1

Zak Seidov, Jan 19 2006

nonn

			a(1)=29 because 31, 18*29^2 + 1 = 15139, and 18*31^2 + 1 = 17299 are all primes.

Cf. A089001 (Numbers n such that 2*n^2 + 1 is prime),
A090612 (Numbers k such that the k-th prime is of the form 2*k^2+1),
A090698 (Primes of the form 2*n^2+1),
A113541 (Numbers n such that 18*n^2+1 is a multiple of 19).

[p: p in PrimesUpTo(200000)| IsPrime(p+2) and IsPrime(18*p^2+1) and IsPrime(18*(p+2)^2+1)] // Vincenzo Librandi, Nov 13 2010

More terms from Vincenzo Librandi, Mar 27 2010

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A261191 40-gonal numbers: a(n) = 38n(n-1)/2 + n.

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A115272 Primes p such that p + 2, 18p^2 + 1, and 18(p+2)^2 + 1 are all primes.

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cp's OEIS Frontend

A261191 40-gonal numbers: a(n) = 38*n*(n-1)/2 + n.

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A115272 Primes p such that p + 2, 18*p^2 + 1, and 18*(p+2)^2 + 1 are all primes.

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A261191 40-gonal numbers: a(n) = 38n(n-1)/2 + n.

A115272 Primes p such that p + 2, 18p^2 + 1, and 18(p+2)^2 + 1 are all primes.