cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A069131 Centered 18-gonal numbers.

Original entry on oeis.org

1, 19, 55, 109, 181, 271, 379, 505, 649, 811, 991, 1189, 1405, 1639, 1891, 2161, 2449, 2755, 3079, 3421, 3781, 4159, 4555, 4969, 5401, 5851, 6319, 6805, 7309, 7831, 8371, 8929, 9505, 10099, 10711, 11341, 11989, 12655, 13339, 14041, 14761, 15499, 16255, 17029, 17821
Offset: 1

Views

Author

Terrel Trotter, Jr., Apr 07 2002

Keywords

Comments

Equals binomial transform of [1, 18, 18, 0, 0, 0, ...]. Example: a(3) = 55 = (1, 2, 1) dot (1, 18, 18) = (1 + 36 + 18). - Gary W. Adamson, Aug 24 2010
Narayana transform (A001263) of [1, 18, 0, 0, 0, ...]. - Gary W. Adamson, Jul 28 2011
From Lamine Ngom, Aug 19 2021: (Start)
Sequence is a spoke of the hexagonal spiral built from the terms of A016777 (see illustration in links section).
a(n) is a bisection of A195042.
a(n) is a trisection of A028387.
a(n) + 1 is promic (A002378).
a(n) + 2 is a trisection of A002061.
a(n) + 9 is the arithmetic mean of its neighbors.
4*a(n) + 5 is a square: A016945(n)^2. (End)

Examples

			a(5) = 181 because 9*5^2 - 9*5 + 1 = 225 - 45 + 1 = 181.

Links

Crossrefs

Cf. centered polygonal numbers listed in A069190.
Cf. A000217, A028387, A195042, A016945, A002378, A082040, A304163, A003215, A247792, A016777,A016778, A016790, A010008, A008600, A002061.
Cf. A000290, A139278, A069129, A062786, A033996, A060544, A027468, A016754, A124080, A069099, A152740, A049598, A005891, A152741, A001844, A163756, A005448, A194715.

Programs

Formula

a(n) = 9*n^2 - 9*n + 1.
a(n) = 18*n + a(n-1) - 18 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: ( x*(1+16*x+x^2) ) / ( (1-x)^3 ). - R. J. Mathar, Feb 04 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=19, a(3)=55. - Harvey P. Dale, Jan 20 2014
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(5)*Pi/6)/(3*sqrt(5)).
Sum_{n>=1} a(n)/n! = 10*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 10/e - 1. (End)
From Lamine Ngom, Aug 19 2021: (Start)
a(n) = 18*A000217(n) + 1 = 9*A002378(n) + 1.
a(n) = 3*A003215(n) - 2.
a(n) = A247792(n) - 9*n.
a(n) = A082040(n) + A304163(n) - a(n-1) = A016778(n) + A016790(n) - a(n-1), n > 0.
a(n) + a(n+1) = 2*A247792(n) = A010008(n), n > 0.
a(n+1) - a(n) = 18*n = A008600(n). (End)
From Leo Tavares, Oct 31 2021: (Start)
a(n)= A000290(n) + A139278(n-1)
a(n) = A069129(n) + A002378(n-1)
a(n) = A062786(n) + 8*A000217(n-1)
a(n) = A062786(n) + A033996(n-1)
a(n) = A060544(n) + 9*A000217(n-1)
a(n) = A060544(n) + A027468(n-1)
a(n) = A016754(n-1) + 10*A000217(n-1)
a(n) = A016754(n-1) + A124080
a(n) = A069099(n) + 11*A000217(n-1)
a(n) = A069099(n) + A152740(n-1)
a(n) = A003215(n-1) + 12*A000217(n-1)
a(n) = A003215(n-1) + A049598(n-1)
a(n) = A005891(n-1) + 13*A000217(n-1)
a(n) = A005891(n-1) + A152741(n-1)
a(n) = A001844(n) + 14*A000217(n-1)
a(n) = A001844(n) + A163756(n-1)
a(n) = A005448(n) + 15*A000217(n-1)
a(n) = A005448(n) + A194715(n-1). (End)
E.g.f.: exp(x)*(1 + 9*x^2) - 1. - Nikolaos Pantelidis, Feb 06 2023

A198017 a(n) = n*(7*n + 11)/2 + 1.

Original entry on oeis.org

1, 10, 26, 49, 79, 116, 160, 211, 269, 334, 406, 485, 571, 664, 764, 871, 985, 1106, 1234, 1369, 1511, 1660, 1816, 1979, 2149, 2326, 2510, 2701, 2899, 3104, 3316, 3535, 3761, 3994, 4234, 4481, 4735, 4996, 5264, 5539, 5821, 6110, 6406, 6709, 7019, 7336, 7660, 7991
Offset: 0

Views

Author

Bruno Berselli, Oct 21 2011 - based on remarks and sequences by Omar E. Pol

Keywords

Comments

First bisection of A193053 (see also the numerical spiral illustrated in the Links section).
The inverse binomial transform yields 1, 9, 7, 0, 0 (0 continued).

Links

Crossrefs

Cf. A195020 (vertices of the numerical spiral in link).
Cf. A017005 (first differences).
Cf. A069099, A001106.
Cf. A001106, A022264, A033572, A144555, A152760, A158482, A158485, A185019, A193053, A195021, A195023-A195030, A195320 [incomplete list].

Programs

  • Magma
    [n*(7*n+11)/2+1: n in [0..47]];
  • Mathematica
    Table[(n(7n+11))/2+1,{n,0,60}] (* or *) LinearRecurrence[{3,-3,1},{1,10,26},60] (* Harvey P. Dale, Mar 03 2013 *)
  • PARI
    for(n=0, 47, print1(n*(7*n+11)/2+1", "));

Formula

G.f.: (1 + 7*x - x^2)/(1-x)^3.
a(n) = A195020(2*n) + 2*n + 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = 2*a(n-1) - a(n-2) + 7.
From Elmo R. Oliveira, Dec 24 2024: (Start)
E.g.f.: exp(x)*(2 + 18*x + 7*x^2)/2.
a(n) = n + A001106(n+1). (End)

A069127 Centered 14-gonal numbers.

Original entry on oeis.org

1, 15, 43, 85, 141, 211, 295, 393, 505, 631, 771, 925, 1093, 1275, 1471, 1681, 1905, 2143, 2395, 2661, 2941, 3235, 3543, 3865, 4201, 4551, 4915, 5293, 5685, 6091, 6511, 6945, 7393, 7855, 8331, 8821, 9325, 9843, 10375, 10921, 11481, 12055, 12643, 13245, 13861, 14491
Offset: 1

Views

Author

Terrel Trotter, Jr., Apr 07 2002

Keywords

Comments

Binomial transform of [1, 14, 14, 0, 0, 0, ...] and Narayana transform (A001263) of [1, 14, 0, 0, 0, ...]. - Gary W. Adamson, Jul 29 2011
Centered tetradecagonal numbers or centered tetrakaidecagonal numbers. - Omar E. Pol, Oct 03 2011

Examples

			a(5) = 141 because 7*5^2 - 7*5 + 1 = 175 - 35 + 1 = 141.
a(5) = 71 because 71 = (7*5^2 - 7*5 + 2)/2 = (175 - 35 + 2)/2 = 142/2.
From _Bruno Berselli_, Oct 27 2017: (Start)
1   =         -(1) + (2).
15  =       -(1+2) + (3+4+5+6).
43  =     -(1+2+3) + (4+5+6+7+8+9+10).
85  =   -(1+2+3+4) + (5+6+7+8+9+10+11+12+13+14).
141 = -(1+2+3+4+5) + (6+7+8+9+10+11+12+13+14+15+16+17+18). (End)

Links

Crossrefs

Cf. A005448, A001844, A005891, A003215, A069099.
Cf. A163756, A193053.

Programs

Formula

a(n) = 7*n^2 - 7*n + 1.
a(n) = 14*n+a(n-1)-14 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: -x*(1+12*x+x^2) / (x-1)^3. - R. J. Mathar, Feb 04 2011
a(n) = A163756(n-1) + 1. - Omar E. Pol, Oct 03 2011
a(n) = a(-n+1) = A193053(2n-2) + A193053(2n-3). - Bruno Berselli, Oct 21 2011
Sum_{n>=1} 1/a(n) = Pi * tan(sqrt(3/7)*Pi/2) / sqrt(21). - Vaclav Kotesovec, Jul 23 2019
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} a(n)/n! = 8*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 8/e - 1. (End)
a(n) = A069099(n) + 7*A000217(n-1). - Leo Tavares, Jul 09 2021
E.g.f.: exp(x)*(1 + 7*x^2) - 1. - Stefano Spezia, Aug 01 2024

A086272 Rectangular array T(n,k) of central polygonal numbers, by antidiagonals.

Original entry on oeis.org

1, 3, 1, 7, 4, 1, 13, 10, 5, 1, 21, 19, 13, 6, 1, 31, 31, 25, 16, 7, 1, 43, 46, 41, 31, 19, 8, 1, 57, 64, 61, 51, 37, 22, 9, 1, 73, 85, 85, 76, 61, 43, 25, 10, 1, 91, 109, 113, 106, 91, 71, 49, 28, 11, 1, 111, 136, 145, 141, 127, 106, 81, 55, 31, 12, 1, 133, 166, 181, 181, 169
Offset: 1

Views

Author

Clark Kimberling, Jul 14 2003

Keywords

Comments

In the standard notation, the offset is different: the first row are the 2-gonal, the second row the 3-gonal numbers, etc. - R. J. Mathar, Oct 07 2011

Examples

			First rows:
1,3,7,13,21,31,43,57,73,91,111,..   A002061
1,4,10,19,31,46,64,85,109,136,166,...  A005448
1,5,13,25,41,61,85,113,145,181,221,..   A001844
1,6,16,31,51,76,106,141,181,226,276,...  A005891
1,7,19,37,61,91,127,169,217,271,331,...   A003215
1,8,22,43,71,106,148,197,253,316,386,...    A069099
1,9,25,49,81,121,169,225,289,361,441,...    A016754
1,10,28,55,91,136,190,253,325,406,496,...    A060544

Links

Crossrefs

Cf. A086270, A086271, A086273.

Formula

T(k, n)=(k+1)*binomial(n, 2)+1.

A195041 Concentric heptagonal numbers.

Original entry on oeis.org

0, 1, 7, 15, 28, 43, 63, 85, 112, 141, 175, 211, 252, 295, 343, 393, 448, 505, 567, 631, 700, 771, 847, 925, 1008, 1093, 1183, 1275, 1372, 1471, 1575, 1681, 1792, 1905, 2023, 2143, 2268, 2395, 2527, 2661, 2800, 2941, 3087, 3235, 3388, 3543
Offset: 0

Views

Author

Omar E. Pol, Sep 27 2011

Keywords

Comments

A033582 and A069127 interleaved.
Partial sums of A047336. - Reinhard Zumkeller, Jan 07 2012

Links

Crossrefs

Column 7 of A195040.
Cf. A032527, A032528, A033582, A047336, A069099, A069127, A077221, A195042.

Programs

  • Haskell
    a195041 n = a195041_list !! n
a195041_list = scanl (+) 0 a047336_list
-- Reinhard Zumkeller, Jan 07 2012
  • Magma
    [7*n^2/4+3*((-1)^n-1)/8: n in [0..50]]; // Vincenzo Librandi, Sep 29 2011
  • Mathematica
    CoefficientList[Series[-((x (1+5 x+x^2))/((-1+x)^3 (1+x))),{x,0,80}],x] (* or *) LinearRecurrence[{2,0,-2,1},{0,1,7,15},80] (* Harvey P. Dale, Jan 18 2021 *)
  • PARI
    a(n)=7*n^2\4 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = 7*n^2/4 + 3*((-1)^n - 1)/8.
From R. J. Mathar, Sep 28 2011: (Start)
G.f.: -x*(1+5*x+x^2) / ( (1+x)*(x-1)^3 ).
a(n) + a(n+1) = A069099(n+1). (End)
a(n) = n^2 + floor(3*n^2/4). - Bruno Berselli, Aug 08 2013
Sum_{n>=1} 1/a(n) = Pi^2/42 + tan(sqrt(3/7)*Pi/2)*Pi/sqrt(21). - Amiram Eldar, Jan 16 2023

A069126 Centered 13-gonal numbers.

Original entry on oeis.org

1, 14, 40, 79, 131, 196, 274, 365, 469, 586, 716, 859, 1015, 1184, 1366, 1561, 1769, 1990, 2224, 2471, 2731, 3004, 3290, 3589, 3901, 4226, 4564, 4915, 5279, 5656, 6046, 6449, 6865, 7294, 7736, 8191, 8659, 9140, 9634, 10141, 10661, 11194
Offset: 1

Views

Author

Terrel Trotter, Jr., Apr 07 2002

Keywords

Comments

Centered tridecagonal numbers or centered triskaidecagonal numbers. - Omar E. Pol, Oct 03 2011

Examples

			a(5) = 131 because 131 = (13*5^2 - 13*5 + 2)/2 = (325 - 65 + 2)/2 = 262/2 = 131.

Links

Crossrefs

Cf. A001263, A001844, A003215, A005448, A005891, A069099, A152741.

Programs

  • Mathematica
    FoldList[#1 + #2 &, 1, 13 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
LinearRecurrence[{3,-3,1},{1,14,40},60] (* Harvey P. Dale, Jan 20 2014 *)
With[{nn=50},Total/@Thread[{PolygonalNumber[13,Range[nn]],Range[0,nn-1]^2}]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Aug 29 2016 *)
  • PARI
    a(n)=13*n(n-1)/2+1 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = (13n^2 - 13n + 2)/2.
Binomial transform of [1, 13, 13, 0, 0, 0, ...]; Narayana transform (A001263) of [1, 13, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007
a(n) = 13*n+a(n-1)-13 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: -x*(1+11*x+x^2) / (x-1)^3. - R. J. Mathar, Feb 04 2011
a(n) = A152741(n-1) + 1. - Omar E. Pol, Oct 03 2011
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = 2*Pi*tan(sqrt(5/13)*Pi/2)/sqrt(65).
Sum_{n>=1} a(n)/n! = 15*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 15/(2*e) - 1. (End)
E.g.f.: exp(x)*(1 + 13*x^2/2) - 1. - Stefano Spezia, May 15 2022

A262221 a(n) = 25*n*(n + 1)/2 + 1.

Original entry on oeis.org

1, 26, 76, 151, 251, 376, 526, 701, 901, 1126, 1376, 1651, 1951, 2276, 2626, 3001, 3401, 3826, 4276, 4751, 5251, 5776, 6326, 6901, 7501, 8126, 8776, 9451, 10151, 10876, 11626, 12401, 13201, 14026, 14876, 15751, 16651, 17576, 18526, 19501, 20501, 21526, 22576, 23651
Offset: 0

Views

Author

Bruno Berselli, Sep 15 2015

Keywords

Comments

Also centered 25-gonal (or icosipentagonal) numbers.
This is the case k=25 of the formula (k*n*(n+1) - (-1)^k + 1)/2. See table in Links section for similar sequences.
For k=2*n, the formula shown above gives A011379.
Primes in sequence: 151, 251, 701, 1951, 3001, 4751, 10151, 12401, ...

References

  • E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 51 (23rd row of the table).

Links

Crossrefs

Cf. A000217, A008607 (first differences), A011379, A034856, A054254, A080956, A123296, A186349, A255184.
Cf. centered polygonal numbers listed in A069190.
Similar sequences of the form (k*n*(n+1) - (-1)^k + 1)/2 with -1 <= k <= 26: A000004, A000124, A002378, A005448, A005891, A028896, A033996, A035008, A046092, A049598, A060544, A064200, A069099, A069125, A069126, A069128, A069130, A069132, A069174, A069178, A080956, A124080, A163756, A163758, A163761, A164136, A173307.

Programs

  • Magma
    [25*n*(n+1)/2+1: n in [0..50]];
  • Mathematica
    Table[25 n (n + 1)/2 + 1, {n, 0, 50}]
25*Accumulate[Range[0,50]]+1 (* or *) LinearRecurrence[{3,-3,1},{1,26,76},50] (* Harvey P. Dale, Jan 29 2023 *)
  • PARI
    vector(50, n, n--; 25*n*(n+1)/2+1)
  • Sage
    [25*n*(n+1)/2+1 for n in (0..50)]

Formula

G.f.: (1 + 23*x + x^2)/(1 - x)^3.
a(n) = a(-n-1) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A123296(n) + 1.
a(n) = A000217(5*n+2) - 2.
a(n) = A034856(5*n+1).
a(n) = A186349(10*n+1).
a(n) = A054254(5*n+2) with n>0, a(0)=1.
a(n) = A000217(n+1) + 23*A000217(n) + A000217(n-1) with A000217(-1)=0.
Sum_{i>=0} 1/a(i) = 1.078209111... = 2*Pi*tan(Pi*sqrt(17)/10)/(5*sqrt(17)).
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=0} a(n)/n! = 77*e/2.
Sum_{n>=0} (-1)^(n+1) * a(n)/n! = 23/(2*e). (End)
E.g.f.: exp(x)*(2 + 50*x + 25*x^2)/2. - Elmo R. Oliveira, Dec 24 2024

A069128 Centered 15-gonal numbers: a(n) = (15*n^2 - 15*n + 2)/2.

Original entry on oeis.org

1, 16, 46, 91, 151, 226, 316, 421, 541, 676, 826, 991, 1171, 1366, 1576, 1801, 2041, 2296, 2566, 2851, 3151, 3466, 3796, 4141, 4501, 4876, 5266, 5671, 6091, 6526, 6976, 7441, 7921, 8416, 8926, 9451, 9991, 10546, 11116, 11701, 12301, 12916, 13546, 14191, 14851, 15526
Offset: 1

Views

Author

Terrel Trotter, Jr., Apr 07 2002

Keywords

Comments

Centered pentadecagonal numbers or centered quindecagonal numbers or centered pentakaidecagonal numbers. - Omar E. Pol, Oct 03 2011

Examples

			a(5) = 151 because (15*5^2 - 15*5 + 2)/2 = 151.

Links

Crossrefs

Cf. A005448, A001844, A005891, A003215, A069099.

Programs

Formula

a(n) = (15*n^2 - 15*n + 2)/2.
a(n) = 15*n+a(n-1)-15 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: -x*(1+13*x+x^2) / (x-1)^3. - R. J. Mathar, Feb 04 2011
Binomial transform of [1, 15, 15, 0, 0, 0, ...] and Narayana transform (A001263) of [1, 15, 0, 0, 0, ...]. - Gary W. Adamson, Jul 28 2011
a(n) = A194715(n-1) + 1. - Omar E. Pol, Oct 03 2011
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = 2*Pi*tan(sqrt(7/15)*Pi/2)/sqrt(105).
Sum_{n>=1} a(n)/n! = 17*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 17/(2*e) - 1. (End)
E.g.f.: exp(x)*(1 + 15*x^2/2) - 1. - Nikolaos Pantelidis, Feb 07 2023

A242239 T(n,k)=Number of length n+k+1 0..k arrays with every value 0..k appearing at least once in every consecutive k+2 elements, and new values 0..k introduced in order.

Original entry on oeis.org

3, 6, 5, 10, 12, 8, 15, 22, 22, 13, 21, 35, 43, 40, 21, 28, 51, 71, 82, 74, 34, 36, 70, 106, 139, 157, 136, 55, 45, 92, 148, 211, 271, 304, 250, 89, 55, 117, 197, 298, 416, 531, 586, 460, 144, 66, 145, 253, 400, 592, 821, 1047, 1129, 846, 233, 78, 176, 316, 517, 799
Offset: 1

Views

Author

R. H. Hardin, May 08 2014

Keywords

Comments

Table starts
...3....6...10...15....21....28....36....45....55....66....78....91...105
...5...12...22...35....51....70....92...117...145...176...210...247...287
...8...22...43...71...106...148...197...253...316...386...463...547...638
..13...40...82..139...211...298...400...517...649...796...958..1135..1327
..21...74..157..271...416...592...799..1037..1306..1606..1937..2299..2692
..34..136..304..531...821..1174..1590..2069..2611..3216..3884..4615..5409
..55..250..586.1047..1626..2332..3165..4125..5212..6426..7767..9235.10830
..89..460.1129.2059..3231..4642..6308..8229.10405.12836.15522.18463.21659
.144..846.2176.4047..6411..9256.12587.16429.20782.25646.31021.36907.43304
.233.1556.4195.7955.12716.18442.25138.32821.41527.51256.62008.73783.86581

Examples

			Some solutions for n=5 k=4
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..1....1....1....1....1....1....1....1....1....1....1....1....1....0....1....1
..2....2....0....2....2....1....0....0....2....0....2....2....0....1....2....0
..1....3....2....3....3....2....2....2....3....2....3....3....2....2....3....2
..3....4....3....4....4....3....3....3....0....3....0....0....3....3....4....3
..4....0....4....1....0....4....4....4....4....4....4....4....4....4....1....4
..0....2....2....0....1....0....2....0....2....1....2....1....0....2....0....1
..2....1....1....0....2....1....1....1....1....0....1....2....1....0....2....0
..2....3....0....2....3....0....0....1....0....0....3....4....2....1....3....4
..1....0....0....3....0....2....2....2....3....2....2....3....3....1....0....2

Links

Crossrefs

Column 1 is A000045(n+3)
Column 2 is A196700(n+3)
Row 1 is A000217(n+1)
Row 2 is A000326(n+1)
Row 3 is A069099(n+1)
Row 4 is A220083

Formula

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = a(n-1) +a(n-2) +a(n-3)
k=3: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4)
k=4: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5)
k=5: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5) +a(n-6)
k=6: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5) +a(n-6) +a(n-7)
k=7: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5) +a(n-6) +a(n-7) +a(n-8)
k=8: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5) +a(n-6) +a(n-7) +a(n-8) +a(n-9)
k=9: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5) +a(n-6) +a(n-7) +a(n-8) +a(n-9) +a(n-10)
Empirical for row n:
n=1: a(n) = (1/2)*n^2 + (3/2)*n + 1
n=2: a(n) = (3/2)*n^2 + (5/2)*n + 1
n=3: a(n) = (7/2)*n^2 + (7/2)*n + 1
n=4: a(n) = (15/2)*n^2 + (9/2)*n + 1
n=5: a(n) = (31/2)*n^2 + (11/2)*n + 1 for n>1
n=6: a(n) = (63/2)*n^2 + (13/2)*n + 1 for n>2
n=7: a(n) = (127/2)*n^2 + (15/2)*n + 1 for n>3
n=8: a(n) = (255/2)*n^2 + (17/2)*n + 1 for n>4
n=9: a(n) = (511/2)*n^2 + (19/2)*n + 1 for n>5
n=10: a(n) = (1023/2)*n^2 + (21/2)*n + 1 for n>6
n=11: a(n) = (2047/2)*n^2 + (23/2)*n + 1 for n>7
n=12: a(n) = (4095/2)*n^2 + (25/2)*n + 1 for n>8
n=13: a(n) = (8191/2)*n^2 + (27/2)*n + 1 for n>9
n=14: a(n) = (16383/2)*n^2 + (29/2)*n + 1 for n>10
n=15: a(n) = (32767/2)*n^2 + (31/2)*n + 1 for n>11
Empirical large-k generalization, for k>n-4: T(n,k) = ((2^n-1)/2)*k^2 + ((2*n+1)/2)*k + 1
Empirical recurrence generalization, for column k: a(n) = sum {i in 1..k+1} a(n-i)

A285811 Primes equal to a centered heptagonal number plus 1.

Original entry on oeis.org

2, 23, 107, 149, 317, 1619, 2459, 3257, 3929, 5189, 6029, 6323, 7247, 15017, 19427, 21023, 21569, 26189, 42737, 45887, 55127, 56009, 63317, 66173, 67139, 70079, 82469, 101747, 105359, 110273, 125687, 136523, 137909, 149249, 155087, 159539, 167099, 171719
Offset: 1

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Author

Colin Barker, Apr 27 2017

Keywords

Comments

Primes in A209294. - Omar E. Pol, Apr 27 2017

Links

Crossrefs

Cf. A000040, A069099, A285809, A285810, A285812.

Programs

  • PARI
    cpg(m, n) = m*n*(n-1)/2+1 \\ n-th centered m-gonal number
maxk=600; L=List(); for(k=1, maxk, if(isprime(p=cpg(7, k) + 1), listput(L, p))); Vec(L)
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