A056109 -id:A056109 - cp's OEIS frontend

A122430 Primes of the form 1+2n+3n^2.

17, 457, 617, 1009, 1777, 2081, 3137, 4409, 5897, 9521, 11657, 14009, 24481, 25577, 29009, 39217, 43441, 47881, 49409, 62497, 67801, 75209, 81017, 85009, 87041, 93281, 97561, 104161, 110977, 120401, 132721, 135257, 140401, 159161, 182041

Offset: 1

Zak Seidov, Oct 20 2006

nonn

3*a(n)-2 is a square (of the form (3*k+1)^2). - Vincenzo Librandi, Mar 15 2013

Also primes which are the sum of 2 consecutive pentagonal numbers (A000326). - Vicente Izquierdo Gomez, Aug 13 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A086285 (numbers n such that 1+2n+3n^2 is prime).

Subsequence of A056109.

[a: n in [1..300] | IsPrime(a) where a is 1+2*n+3*n^2]; // Vincenzo Librandi, Mar 15 2013

Select[Table[1 + 2 n + 3 n^2, {n, 500}], PrimeQ] (* Vincenzo Librandi, Mar 15 2013 *)

is(n)=isprime(n) && issquare(3*n-2,&n) && n%3==1 \\ Charles R Greathouse IV, Sep 23 2013

A131464 a(n) = 4n^3 - 3n^2 + 2*n - 1.

Original entry on oeis.org

2, 23, 86, 215, 434, 767, 1238, 1871, 2690, 3719, 4982, 6503, 8306, 10415, 12854, 15647, 18818, 22391, 26390, 30839, 35762, 41183, 47126, 53615, 60674, 68327, 76598, 85511, 95090, 105359, 116342, 128063, 140546, 153815, 167894, 182807, 198578, 215231, 232790

Offset: 1

Mohammad K. Azarian, Jul 26 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A084849, A056109, A056105, A056578.

[4*n^3-3*n^2+2*n-1: n in [1..40]]; // Vincenzo Librandi, Feb 12 2013

I:=[2, 23, 86, 215]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 12 2013

CoefficientList[Series[(2 + 15 x + 6 x^2 + x^3)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2013 *)
Table[4n^3-3n^2+2n-1,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{2,23,86,215},40] (* Harvey P. Dale, May 05 2018 *)

From Vincenzo Librandi, Feb 12 2013: (Start)
G.f.: x*(2 + 15*x + 6*x^2 + x^3)/(1 - x)^4.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). (End)
E.g.f.: 1 - exp(x)*(1 - 3*x - 9*x^2 - 4*x^3). - Stefano Spezia, Dec 06 2024

A238377 Row sums of triangle in A204028.

Original entry on oeis.org

1, 2, 6, 10, 17, 24, 34, 44, 57, 70, 86, 102, 121, 140, 162, 184, 209, 234, 262, 290, 321, 352, 386, 420, 457, 494, 534, 574, 617, 660, 706, 752, 801, 850, 902, 954, 1009, 1064, 1122, 1180

Offset: 0

Philippe Deléham, Feb 25 2014

			Triangle in A204028 begins:
  1;............................sum = 1
  1, 1;.........................sum = 2
  1, 4, 1;......................sum = 6
  1, 4, 4, 1;...................sum = 10
  1, 4, 7, 4, 1;................sum = 17
  1, 4, 7, 7, 4, 1;.............sum = 24
  1, 4, 7, 10, 7, 4, 1;.........sum = 34
  1, 4, 7, 10, 10, 7, 4, 1;.....sum = 44

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A056109, A049450, A204028.

G.f.: (1+2*x^2)/((1+x)*(1-x)^3).
a(2*n) = A056109(n).
a(2*n+1) = A049450(n+1) = 2*A000326(n+1).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), a(0) = 1, a(1) = 2, a(2) = 6, a(3) = 10.

Corrected by R. J. Mathar, Aug 08 2015

A113531 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6*n^5.

Original entry on oeis.org

1, 21, 321, 2005, 7737, 22461, 54121, 114381, 219345, 390277, 654321, 1045221, 1604041, 2379885, 3430617, 4823581, 6636321, 8957301, 11886625, 15536757, 20033241, 25515421, 32137161, 40067565, 49491697, 60611301, 73645521

Offset: 0

Jonathan Vos Post, Jan 12 2006

1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 is the derivative of 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 = (n^7 - 1)/(n-1). a(6) = 1 + 2*6 + 3*6^2 + 4*6^3 + 5*6^4 + 6*6^5 = 54121 is prime, the smallest prime in the sequence. The next is a(a(1)) = a(21) = 1 + 2*21 + 3*21^2 + 4*21^3 + 5*21^4 + 6*21^5 = 25515421. Then a(24) = 49491697.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

Cf. A000012, A005408, A056109, A056578, A056579.

With[{eq=Total[# n^(#-1)&/@Range[6]]},Table[eq,{n,0,30}]] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1,21,321,2005,7737,22461},30] (* Harvey P. Dale, Nov 02 2011 *)

for(n=0,50, print1(1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5, ", ")) \\ G. C. Greubel, Mar 15 2017

a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5.
O.g.f.: 2064/(-1+x)^4+3/(-1+x)+2040/(-1+x)^5+132/(-1+x)^2+720/(-1+x)^6+872/(-1+x)^3 . - R. J. Mathar, Feb 26 2008
a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6), a(0)=1, a(1)=21, a(2)=321, a(3)=2005, a(4)=7737, a(5)=22461. - Harvey P. Dale, Nov 02 2011

Corrected by R. J. Mathar, Feb 26 2008

A113532 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6n^5 + 7n^6.

Original entry on oeis.org

1, 28, 769, 7108, 36409, 131836, 380713, 937924, 2054353, 4110364, 7654321, 13446148, 22505929, 36167548, 56137369, 84557956, 124076833, 177920284, 249972193, 344857924, 468033241, 625878268, 825796489, 1076318788

Offset: 0

Jonathan Vos Post, Jan 12 2006

1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 + 7*n^6 is the derivative of 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 = (x^8 - 1)/(x-1). a(2) = 1 + 2*2 + 3*2^2 + 4*2^3 + 5*2^4 + 6*2^5 + 7*2^6 = 769 is prime. Other primes begin a(6) = 380713, a(12) = 22505929, a(26) = 2236055953, a(38) = 21562615273, a(44) = 51802781449, a(52) = 140712620569.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000012, A005408, A056109, A056578, A056579.

Table[1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6, {n,0,50}] (* or *) LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {1, 28, 769, 7108, 36409, 131836, 380713}, 50] (* G. C. Greubel, Mar 15 2017 *)

for(n=0,50, print1(1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6, ", ")) \\ G. C. Greubel, Mar 15 2017

a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6.

O.g.f.: -12636/(-1+x)^4 -4/(-1+x) -21480/(-1+x)^5 -309/(-1+x)^2 -16920/(-1+x)^6 -3342/(-1+x)^3-5040/(-1+x)^7 . - R. J. Mathar, Feb 26 2008

A130884 3n^3 + 2n^2 + n + 1.

Original entry on oeis.org

1, 7, 35, 103, 229, 431, 727, 1135, 1673, 2359, 3211, 4247, 5485, 6943, 8639, 10591, 12817, 15335, 18163, 21319, 24821, 28687, 32935, 37583, 42649, 48151, 54107, 60535, 67453, 74879, 82831, 91327, 100385, 110023, 120259, 131111

Offset: 0

Mohammad K. Azarian, Jul 26 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A084849, A056109, A056105.

[3*n^3+2*n^2+n+1: n in [0..35]]; // Vincenzo Librandi, Feb 12 2013

Table[3n^3+2n^2+n+1,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,7,35,103},40] (* Harvey P. Dale, Jan 17 2012 *)

G.f.: (1+13*x^2+x^3+3*x)/(-1+x)^4. - R. J. Mathar, Nov 14 2007

a(0)=1, a(1)=7, a(2)=35, a(3)=103, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)- a(n-4) [Harvey P. Dale, Jan 17 2012]

A130885 3n^3 - 2n^2 + n - 1.

Original entry on oeis.org

1, 17, 65, 163, 329, 581, 937, 1415, 2033, 2809, 3761, 4907, 6265, 7853, 9689, 11791, 14177, 16865, 19873, 23219, 26921, 30997, 35465, 40343, 45649, 51401, 57617, 64315, 71513, 79229, 87481, 96287, 105665, 115633, 126209, 137411

Offset: 1

Mohammad K. Azarian, Jul 26 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A084849; A056109; A056105.

[3*n^3-2*n^2+n-1: n in [1..40]]; // Vincenzo Librandi, Feb 12 2013

I:=[1, 17, 65, 163]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 12 2013

Table[3*n^3 - 2*n^2 + n - 1, {n, 1, 40}] (* Vincenzo Librandi, Feb 12 2013 *)
LinearRecurrence[{4,-6,4,-1},{1,17,65,163},40] (* Harvey P. Dale, Nov 21 2019 *)

G.f.: x*(1+13*x+3*x^2+x^3)/(-1+x)^4. - R. J. Mathar, Nov 14 2007

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Vincenzo Librandi, Feb 12 2013

A130886 4n^4 + 3n^3 + 2n^2 + n + 1.

Original entry on oeis.org

1, 11, 99, 427, 1253, 2931, 5911, 10739, 18057, 28603, 43211, 62811, 88429, 121187, 162303, 213091, 274961, 349419, 438067, 542603, 664821, 806611, 969959, 1156947, 1369753, 1610651, 1882011, 2186299, 2526077, 2904003

Offset: 0

Mohammad K. Azarian, Jul 26 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A084849, A056109, A056105.

[4*n^4+3*n^3+2*n^2+n+1: n in [0..40]]; // Vincenzo Librandi, Feb 12 2013

CoefficientList[Series[(1 + 6 x + 54 x^2 + 32 x^3 + 3 x^4)/(1 - x)^5, {x, 0, 35}], x] (* Vincenzo Librandi, Feb 12 2013 *)

G.f.: (1 + 6*x + 54*x^2 + 32*x^3 + 3*x^4)/(1 - x)^5. - Vincenzo Librandi, Feb 12 2013

A131466 a(n) = 5n^4 - 4n^3 + 3n^2 - 2n + 1.

Original entry on oeis.org

1, 3, 57, 319, 1065, 2691, 5713, 10767, 18609, 30115, 46281, 68223, 97177, 134499, 181665, 240271, 312033, 398787, 502489, 625215, 769161, 936643, 1130097, 1352079, 1605265, 1892451, 2216553, 2580607, 2987769, 3441315

Offset: 0

Mohammad K. Azarian, Jul 26 2007

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A084849, A056109, A056105, A056578, A056579.

Table[5n^4-4n^3+3n^2-2n+1,{n,0,30}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,3,57,319,1065},30] (* Harvey P. Dale, Nov 01 2020 *)

a(n)=5*n^4-4*n^3+3*n^2-2*n+1 \\ Charles R Greathouse IV, Oct 21 2022

From Chai Wah Wu, Nov 13 2018: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
G.f.: (-15*x^4 - 54*x^3 - 52*x^2 + 2*x - 1)/(x - 1)^5. (End)

A275673 List of numbers that are in a spoke of a hexagonal spiral.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 22, 25, 28, 31, 34, 37, 41, 45, 49, 53, 57, 61, 66, 71, 76, 81, 86, 91, 97, 103, 109, 115, 121, 127, 134, 141, 148, 155, 162, 169, 177, 185, 193, 201, 209, 217, 226, 235, 244, 253, 262, 271, 281, 291, 301, 311, 321

Offset: 1

Peter Kagey, Aug 04 2016

nonn

This sequence contains k if and only if k is in one of the following sequences: A056105, A056106, A056107, A056108, A056109, A003215.

Alternatively, this sequence consists of the numbers of the form 3k^2 + bk + 1 for nonnegative k and -2 <= b <= 3.

Peter Kagey, Table of n, a(n) for n = 1..10003

Cf. A056105, A056106, A056107, A056108, A056109, A003215.

a275673 n = a275673_list !! (n - 1)
a275673_list = scanl (+) 1 $ concatMap (replicate 6) [1..]

Conjectures from Colin Barker, Aug 05 2016: (Start)
a(n) = 2*a(n-1)-a(n-2)+a(n-6)-2*a(n-7)+a(n-8) for n>8.
G.f.: x*(1-x^6+x^7) / ((1-x)^3*(1+x)*(1-x+x^2)*(1+x+x^2)).
(End)
Conjecture: a(n) = (n+4-3*floor((n+4)/6)-2)*floor((n+4)/6)+1. - Luce ETIENNE, May 25 2017

cp's OEIS Frontend

A122430 Primes of the form 1+2*n+3*n^2.

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A131464 a(n) = 4*n^3 - 3*n^2 + 2*n - 1.

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A238377 Row sums of triangle in A204028.

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A113531 a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5.

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A113532 a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6.

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A130884 3n^3 + 2n^2 + n + 1.

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A130885 3n^3 - 2n^2 + n - 1.

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A130886 4n^4 + 3n^3 + 2n^2 + n + 1.

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A122430 Primes of the form 1+2n+3n^2.

A131464 a(n) = 4n^3 - 3n^2 + 2*n - 1.

A113531 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6*n^5.

A113532 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6n^5 + 7n^6.