A176691 -id:A176691 - cp's OEIS frontend

A000051 a(n) = 2^n + 1.

2, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649, 4294967297, 8589934593

Offset: 0

N. J. A. Sloane

Same as Pisot sequence L(2,3).
Length of the continued fraction for Sum_{k=0..n} 1/3^(2^k). - Benoit Cloitre, Nov 12 2003
See also A004119 for a(n) = 2a(n-1)-1 with first term = 1. - Philippe Deléham, Feb 20 2004
From the second term on (n>=1), in base 2, these numbers present the pattern 1000...0001 (with n-1 zeros), which is the "opposite" of the binary 2^n-2: (0)111...1110 (cf. A000918). - Alexandre Wajnberg, May 31 2005
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=5, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1)* charpoly(A,3). - Milan Janjic, Jan 27 2010
First differences of A006127. - Reinhard Zumkeller, Apr 14 2011
The odd prime numbers in this sequence form A019434, the Fermat primes. - David W. Wilson, Nov 16 2011
Pisano period lengths: 1, 1, 2, 1, 4, 2, 3, 1, 6, 4, 10, 2, 12, 3, 4, 1, 8, 6, 18, 4, ... . - R. J. Mathar, Aug 10 2012
Is the mentioned Pisano period lengths (see above) the same as A007733? - Omar E. Pol, Aug 10 2012
Only positive integers that are not 1 mod (2k+1) for any k>1. - Jon Perry, Oct 16 2012
For n >= 1, a(n) is the total length of the segments of the Hilbert curve after n iterations. - Kival Ngaokrajang, Mar 30 2014
Frénicle de Bessy (1657) proved that a(3) = 9 is the only square in this sequence. - Charles R Greathouse IV, May 13 2014
a(n) is the number of distinct possible sums made with at most two elements in {1,...,a(n-1)} for n > 0. - Derek Orr, Dec 13 2014
For n > 0, given any set of a(n) lattice points in R^n, there exist 2 distinct members in this set whose midpoint is also a lattice point. - Melvin Peralta, Jan 28 2017
Also the number of independent vertex sets, irredundant sets, and vertex covers in the (n+1)-star graph. - Eric W. Weisstein, Aug 04 and Sep 21 2017
Also the number of maximum matchings in the 2(n-1)-crossed prism graph. - Eric W. Weisstein, Dec 31 2017
Conjecture: For any integer n >= 0, a(n) is the permanent of the (n+1) X (n+1) matrix with M(j, k) = -floor((j - k - 1)/(n + 1)). This conjecture is inspired by the conjecture of Zhi-Wei Sun in A036968. - Peter Luschny, Sep 07 2021

Paul Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 75.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 46, 60, 244.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 141.

Ivan Panchenko, Table of n, a(n) for n = 0..100
E. R. Berlekamp, A contribution to mathematical psychometrics, Unpublished Bell Labs Memorandum, Feb 08 1968 [Annotated scanned copy]
Bakir Farhi, Summation of Certain Infinite Lucas-Related Series, J. Int. Seq., Vol. 22 (2019), Article 19.1.6.
Massimiliano Fasi and Gian Maria Negri Porzio, Determinants of Normalized Bohemian Upper Hessemberg Matrices, University of Manchester (England, 2019).
Bartomeu Fiol, Jairo Martínez-Montoya, and Alan Rios Fukelman, The planar limit of N=2 superconformal field theories, arXiv:2003.02879 [hep-th], 2020.
Bernard Frénicle de Bessy, Solutio duorum problematum circa numeros cubos et quadratos, (1657). Bibliothèque Nationale de Paris.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 114
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 362
Edouard Lucas, The Theory of Simply Periodic Numerical Functions, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.
Kival Ngaokrajang, Illustration of Hilbert curve for n = 1..5
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
D. C. Santos, E. A. Costa, and P. M. M. C. Catarino, On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence, Axioms 14, 203, (2025). See p. 1.
Amelia Carolina Sparavigna, On the generalized sums of Mersenne, Fermat, Cullen and Woodall Numbers, Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, Composition Operations of Generalized Entropies Applied to the Study of Numbers, International Journal of Sciences (2019) Vol. 8, No. 4, 87-92.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Eric Weisstein's World of Mathematics, Crossed Prism Graph.
Eric Weisstein's World of Mathematics, Cunningham Number.
Eric Weisstein's World of Mathematics, Fermat-Lucas Number.
Eric Weisstein's World of Mathematics, Hilbert curve.
Eric Weisstein's World of Mathematics, Independent Vertex Set.
Eric Weisstein's World of Mathematics, Irredundant Set.
Eric Weisstein's World of Mathematics, Matching Number.
Eric Weisstein's World of Mathematics, Maximum Independent Edge Set.
Eric Weisstein's World of Mathematics, Rudin-Shapiro Sequence.
Eric Weisstein's World of Mathematics, Star Graph.
Eric Weisstein's World of Mathematics, Vertex Cover.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Apart from the initial 1, identical to A094373.
See A008776 for definitions of Pisot sequences.
Column 2 of array A103438.
Cf. A007583 (a((n-1)/2)/3 for odd n).
Cf. A000079, A000225, A000918, A004119, A005126, A006217, A006521, A007583.
Cf. A007689, A007733, A019434, A024036, A027641, A027642, A034472, A034474.
Cf. A034491, A034524, A036968, A052539, A052944, A062394, A062395, A062396.
Cf. A062397, A063376, A063481, A074600 - A074624, A127904, A173786, A176691.
Cf. A178248, A194455, A228081, A323482.

a000051 = (+ 1) . a000079
a000051_list = iterate ((subtract 1) . (* 2)) 2
-- Reinhard Zumkeller, May 03 2012

[2^n+1: n in [0..40]]; // G. C. Greubel, Jan 18 2025

A000051:=-(-2+3*z)/(2*z-1)/(z-1); # Simon Plouffe in his 1992 dissertation
a := n -> add(binomial(n,k)*bernoulli(n-k,1)*2^(k+1)/(k+1),k=0..n); # Peter Luschny, Apr 20 2009

Table[2^n + 1, {n,0,40}]
2^Range[0,40] + 1 (* Eric W. Weisstein, Jul 17 2017 *)
LinearRecurrence[{3, -2}, {2, 3}, 40] (* Eric W. Weisstein, Sep 21 2017 *)

PARI
```
a(n)=2^n+1
```

first(n) = Vec((2 - 3*x)/((1 - x)*(1 - 2*x)) + O(x^n)) \\ Iain Fox, Dec 31 2017

def A000051(n): return (1<Chai Wah Wu, Dec 21 2022

a(n) = 2*a(n-1) - 1 = 3*a(n-1) - 2*a(n-2).
G.f.: (2-3*x)/((1-x)*(1-2*x)).
First differences of A052944. - Emeric Deutsch, Mar 04 2004
a(0) = 1, then a(n) = (Sum_{i=0..n-1} a(i)) - (n-2). - Gerald McGarvey, Jul 10 2004
Inverse binomial transform of A007689. Also, V sequence in Lucas sequence L(3, 2). - Ross La Haye, Feb 07 2005
a(n) = A127904(n+1) for n>0. - Reinhard Zumkeller, Feb 05 2007
Equals binomial transform of [2, 1, 1, 1, ...]. - Gary W. Adamson, Apr 23 2008
a(n) = A000079(n)+1. - Omar E. Pol, May 18 2008
E.g.f.: exp(x) + exp(2*x). - Mohammad K. Azarian, Jan 02 2009
a(n) = A024036(n)/A000225(n). - Reinhard Zumkeller, Feb 14 2009
From Peter Luschny, Apr 20 2009: (Start)
A weighted binomial sum of the Bernoulli numbers A027641/A027642 with A027641(1)=1 (which amounts to the definition B_{n} = B_{n}(1)).
a(n) = Sum_{k=0..n} C(n,k)*B_{n-k}*2^(k+1)/(k+1). (See also A052584.) (End)
a(n) is the a(n-1)-th odd number for n >= 1. - Jaroslav Krizek, Apr 25 2009
From Reinhard Zumkeller, Feb 28 2010: (Start)
a(n)*A000225(n) = A000225(2*n).
a(n) = A173786(n,0). (End)
If p[i]=Fibonacci(i-4) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise, then, for n>=1, a(n-1)= det A. - Milan Janjic, May 08 2010
a(n+2) = a(n) + a(n+1) + A000225(n). - Ivan N. Ianakiev, Jun 24 2012
a(A006521(n)) mod A006521(n) = 0. - Reinhard Zumkeller, Jul 17 2014
a(n) = 3*A007583((n-1)/2) for n odd. - Eric W. Weisstein, Jul 17 2017
Sum_{n>=0} 1/a(n) = A323482. - Amiram Eldar, Nov 11 2020

A100314 Number of 2 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).

Original entry on oeis.org

1, 4, 8, 14, 24, 42, 76, 142, 272, 530, 1044, 2070, 4120, 8218, 16412, 32798, 65568, 131106, 262180, 524326, 1048616, 2097194, 4194348, 8388654, 16777264, 33554482, 67108916, 134217782, 268435512, 536870970, 1073741884, 2147483710, 4294967360, 8589934658

Offset: 0

Sergey Kitaev, Nov 13 2004

An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2, j1 < j2 and these elements are in the same relative order as those in the triple (x,y,z). In general, the number of m X n 0-1 matrices in question is given by 2^m + 2^n + 2*(n*m-n-m).

Arthur H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall, 1971.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Ronald Cools, Encyclopaedia of Cubature Formulas
S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. this sequence (m=2), A100315 (m=3), A100316 (m=4).
Cf. A000051, A000079, A001787, A002064, A005126, A005843.
Cf. A052548, A058331, A099003, A132750, A176691.
Row sums of A131830.

List([0..40],n->2^n+2*n); # Muniru A Asiru, Dec 21 2018

[2^n+2*n: n in [1..40]]; // Vincenzo Librandi, Oct 22 2011

a:= proc(n) 2^n + 2*n: end: seq(a(n),n=0..50); # Gary W. Adamson, Jul 20 2007

LinearRecurrence[{4,-5,2}, {1,4,8}, 34] (* Jean-François Alcover, Mar 19 2020 *)

makelist(2^n + 2*n, n, 0, 50); /* Franck Maminirina Ramaharo, Dec 19 2018 */

[2^n +2*n for n in range(41)] # G. C. Greubel, Feb 01 2023

a(n) = 2^n + 2*n.
From Gary W. Adamson, Jul 20 2007: (Start)
Binomial transform of (1, 3, 1, 1, 1, ...).
For n > 0, a(n) = 2*A005126(n-1). (End)
From R. J. Mathar, Jun 13 2008: (Start)
G.f.: 1 + 2*x*(2 -4*x +x^2)/((1-x)^2*(1-2*x)).
a(n+1)-a(n) = A052548(n). (End)
From Colin Barker, Oct 16 2013: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
G.f.: (1 - 3*x^2)/((1-x)^2*(1-2*x)). (End)
E.g.f.: exp(2*x) + 2*x*exp(x). - Franck Maminirina Ramaharo, Dec 19 2018
a(n) = A000079(n) + A005843(n). - Muniru A Asiru, Dec 21 2018

a(0)=1 prepended by Alois P. Heinz, Dec 21 2018

A176805 a(n) = 3^n + 3*n + 1.

Original entry on oeis.org

2, 7, 16, 37, 94, 259, 748, 2209, 6586, 19711, 59080, 177181, 531478, 1594363, 4783012, 14348953, 43046770, 129140215, 387420544, 1162261525, 3486784462, 10460353267, 31381059676, 94143178897, 282429536554, 847288609519, 2541865828408, 7625597485069, 22876792455046

Offset: 0

Jonathan Vos Post, Apr 26 2010

			a(7) = (3^7) + (3*7) + 1 = 2209 = 47^2.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Cf. A000244, A008585, A016777, A176691.

[3^n + 3*n + 1: n in [0..30]]; // Vincenzo Librandi, May 09 2011

a(n)=3^n+3*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 3^n + 3n + 1 = A000244(n) + A008585(n) + 1 = A000244(n) + A016777(n).
From R. J. Mathar, Apr 27 2010: (Start)
a(n)= 5*a(n-1) - 7*a(n-2) + 3*a(n-3).
G.f.: (1+x)*(5*x-2) / ( (3*x-1)*(x-1)^2 ). (End)
E.g.f.: exp(x)*(1 + exp(2*x) + 3*x). - Stefano Spezia, Aug 19 2024

Corrected (1 replaced by 2, 2209 inserted) by R. J. Mathar, Apr 27 2010

A194455 a(n) = 2^n + 3n + 1.

Original entry on oeis.org

2, 6, 11, 18, 29, 48, 83, 150, 281, 540, 1055, 2082, 4133, 8232, 16427, 32814, 65585, 131124, 262199, 524346, 1048637, 2097216, 4194371, 8388678, 16777289, 33554508, 67108943, 134217810, 268435541, 536871000, 1073741915, 2147483742, 4294967393, 8589934692, 17179869287

Offset: 0

Bruno Berselli, Sep 01 2011

Inverse binomial transform of this sequence: 2,4,1,1 (1 continued).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000051, A005126, A176691; A120845.
Cf. A062709 (first differences), A000079 (second and successive differences).
Cf. A146529 (differences between alternate terms, for n>2).
Cf. A086653, A115067.

Magma
```
[2^n+3*n+1: n in [0..31]];
```

Table[2^n + 3 n + 1, {n, 0, 40}] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{4,-5,2},{2,6,11},40] (* Harvey P. Dale, Oct 01 2014 *)

PARI
```
for(n=0, 31, print1(2^n+3*n+1", "));
```

G.f.: (2 - 2*x - 3*x^2)/((1 - 2*x)*(1 - x)^2).
a(n) = A086653(n) - 1  for n > 0.
Sum_{i=0..n} a(i) = A115067(n+1) + 2^(n+1).
a(n) = 3*a(n-1) - 2*a(n-2) - 3  for n > 1.
a(n)^2 = 2^(n+1)*(a(n-1) + 3) + (3*n + 1)^2  for n > 2.
E.g.f.: exp(x)*(1 + exp(x) + 3*x). - Stefano Spezia, May 06 2023

A176916 a(n) = 5^n + 5*n + 1.

Original entry on oeis.org

2, 11, 36, 141, 646, 3151, 15656, 78161, 390666, 1953171, 9765676, 48828181, 244140686, 1220703191, 6103515696, 30517578201, 152587890706, 762939453211, 3814697265716, 19073486328221, 95367431640726, 476837158203231, 2384185791015736, 11920928955078241, 59604644775390746

Offset: 0

Jonathan Vos Post, Apr 28 2010

			a(3) = 5^3 + 5*3 + 1 = 141.

Index entries for linear recurrences with constant coefficients, signature (7,-11,5).

Cf. A000351, A008587, A016861, A176691, A176805.

LinearRecurrence[{7,-11,5},{2,11,36},25] (* Stefano Spezia, Aug 19 2024 *)

a(n)=5^n+5*n+1 \\ Charles R Greathouse IV, Aug 23 2024

a(n) = A000351(n) + A008587(n) + 1 = A000351(n) + A016861(n).
From R. J. Mathar, Apr 29 2010: (Start)
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3).
G.f.: ( -2+3*x+19*x^2 ) / ( (5*x-1)*(x-1)^2 ). (End)
E.g.f.: exp(x)*(1 + exp(4*x) + 5*x). - Stefano Spezia, Aug 19 2024

First term corrected by several authors, Apr 29 2010

a(22)-a(24) from Stefano Spezia, Aug 19 2024

A176972 a(n) = 7^n + 7*n + 1.

Original entry on oeis.org

2, 15, 64, 365, 2430, 16843, 117692, 823593, 5764858, 40353671, 282475320, 1977326821, 13841287286, 96889010499, 678223072948, 4747561510049, 33232930569714, 232630513987327, 1628413597910576, 11398895185373277, 79792266297612142, 558545864083284155, 3909821048582988204

Offset: 0

Jonathan Vos Post, Apr 29 2010

			a(5) = 7^5 + 7*5 + 1 = 16843 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (9,-15,7).

Cf. A000420, A008589, A016993, A176691, A176805, A176916.

Cf. A370664.

[7^n + 7*n + 1: n in [0..25]]; // Vincenzo Librandi, May 06 2011

Table[7^n+7n+1,{n,0,20}] (* or *) LinearRecurrence[{9,-15,7},{2,15,64},20] (* Harvey P. Dale, Apr 17 2014 *)

a(n) = A000420(n) + A008589(n) + 1 = A000420(n) + A016993(n).
a(n) = 7*a(n-1) - 42*(n-1) + 1, with n > 0. For n=5, a(5) = 7*2430 - 42*4 + 1 = 16843. - Bruno Berselli, May 18 2010
From R. J. Mathar, May 22 2010: (Start)
a(n) = 9*a(n-1) - 15*a(n-2) + 7*a(n-3).
G.f.: (-2 + 3*x + 41*x^2) / ((7*x-1)*(x-1)^2). (End)
E.g.f.: exp(x)*(1 + exp(6*x) + 7*x). - Stefano Spezia, Aug 19 2024

A301634 Numbers k such that 2^k + 2*k + 1 is prime.

Original entry on oeis.org

0, 1, 5, 13, 65, 85, 229, 2005, 3875, 3919, 5417, 8819, 11899, 16668, 19445, 28242, 33407, 37918, 40594, 141251

Offset: 1

Seiichi Manyama, Mar 25 2018

Next term, if it exists, is greater than 50000. Terms up to 229 correspond to provable primes. The terms greater than or equal to 2005 correspond to probable primes. - Jon E. Schoenfield and Vaclav Kotesovec, Mar 27 2018

A163115 gives the primes.
Numbers k such that b^k + b*k + 1 is prime: this sequence (b=2), A171058 (b=3), A301635 (b=5).
Cf. A176691.

a:=k->`if`(isprime(2^k+2*k+1),k,NULL): seq(a(k),k=0..6000); # Muniru A Asiru, Mar 25 2018

Flatten[{0, Select[Range[5000], PrimeQ[2^# + 2*# + 1] &]}] (* Vaclav Kotesovec, Mar 25 2018 *)

for(n=0, 500, if(isprime(2^n+2*n+1), print1(n", ")))

a(9)-a(15) from Vaclav Kotesovec, Mar 25 2018
a(16), a(18)-a(19) from Jon E. Schoenfield, Mar 26 2018
a(17) inserted by and a(20) from Michael S. Branicky, Jun 23 2024

A321123 a(n) = 2^n + 2n^2 + 2n + 1.

Original entry on oeis.org

2, 7, 17, 33, 57, 93, 149, 241, 401, 693, 1245, 2313, 4409, 8557, 16805, 33249, 66081, 131685, 262829, 525049, 1049417, 2098077, 4195317, 8389713, 16778417, 33555733, 67110269, 134219241, 268437081, 536872653, 1073743685, 2147485633, 4294969409, 8589936837

Offset: 0

Franck Maminirina Ramaharo, Dec 17 2018

For n >= 2, a(n) is the number of evaluation points on the n-dimensional cube in Genz and Malik's degree 7 cubature rule.

Marius A. Burtea, Table of n, a(n) for n = 0..201
Ronald Cools, Encyclopaedia of Cubature Formulas
Ronald Cools and Philip Rabinowitz, Monomial cubature rules since "Stroud": a compilation, Journal of Computational and Applied Mathematics Vol. 48 (1993), 309-326.
Alan C. Genz and Awais A. Malik, Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region, Journal of Computational and Applied Mathematics Vol. 6 (1980), 295-302.
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Cf. A000051, A000079, A001787, A001844, A002064, A005126, A058331, A100314, A131830, A100314, A132750, A176691, A322593.

[2^n + 2*n^2 + 2*n + 1: n in [0..33]]; // Marius A. Burtea, Dec 28 2018

Table[2^n + 2*n^2 + 2*n + 1, {n, 0, 50}]

makelist(2^n + 2*n^2 + 2*n + 1, n, 0, 50);

a(n) = A000079(n) + A001844(n).
a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4), n >= 4.
G.f.: (2 - 3*x - 3*x^3)/((1 - 2*x)*(1 - x)^3).
E.g.f.: exp(2*x) + (1 + 4*x + 2*x^2)*exp(x).

A322593 a(n) = 2^n + 2*n^2 + 1.

Original entry on oeis.org

2, 5, 13, 27, 49, 83, 137, 227, 385, 675, 1225, 2291, 4385, 8531, 16777, 33219, 66049, 131651, 262793, 525011, 1049377, 2098035, 4195273, 8389667, 16778369, 33555683, 67110217, 134219187, 268437025, 536872595, 1073743625, 2147485571, 4294969345, 8589936771

Offset: 0

Franck Maminirina Ramaharo, Dec 18 2018

For n = 3..7, a(n) is the number of evaluating points on the n-dimensional sphere (also n-space with weight function exp(-r^2) or exp(-r)) in a degree 7 cubature rule.

Arthur H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall, 1971.

Marius A. Burtea, Table of n, a(n) for n = 0..200
Ronald Cools, Encyclopaedia of Cubature Formulas
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Cf. A000051, A000079, A001580, A001787, A002064, A005126, A058331, A100314, A131830, A100314, A132750, A176691, A321123.

[2^n + 2*n^2 + 1: n in [0..33]]; // Marius A. Burtea, Dec 28 2018

Table[2^n + 2*n^2 + 1, {n, 0, 50}]
LinearRecurrence[{5,-9,7,-2},{2,5,13,27},50] (* Harvey P. Dale, Mar 23 2021 *)

Maxima
```
makelist(2^n + 2*n^2 + 1, n, 0, 50);
```

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 4.
a(n) = a(n-1) + A100315(n-1), n >= 2.
G.f.: (2 - 5*x + 6*x^2 - 7*x^3)/((1 - 2*x)*(1 - x)^3)
E.g.f.: exp(2*x) + (1 + 2*x + 2*x^2)*exp(x).

A192764 Numbers k such that 2^(k-1)+2*k-1 is a prime number.

Original entry on oeis.org

1, 2, 6, 14, 66, 86, 230, 2006, 3876, 3920, 5418, 8820, 11900, 16669, 19446, 28243, 33408, 37919, 40595

Offset: 1

Juri-Stepan Gerasimov, Jul 09 2011

a(20) > 10^5 if it exists. - Michael S. Branicky, Aug 26 2024

Cf. A004051, A163115, A176691, A192436.

Select[Range[4000], PrimeQ[2^(# - 1) + 2# - 1] &] (* Alonso del Arte, Jul 09 2011 *)

for(n=1,10^6,if(ispseudoprime(2^(n-1)+2*n-1),print1(n,", ")));

a(13)-a(19) from Michael S. Branicky, Jul 14 2023

cp's OEIS Frontend

A000051 a(n) = 2^n + 1.

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A100314 Number of 2 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).

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A176805 a(n) = 3^n + 3*n + 1.

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A194455 a(n) = 2^n + 3n + 1.

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A176916 a(n) = 5^n + 5*n + 1.

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A176972 a(n) = 7^n + 7*n + 1.

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A321123 a(n) = 2^n + 2n^2 + 2n + 1.