Terrel Trotter, Jr. - cp's OEIS frontend

A065595 a(n) = (sum of first n primes)^2 - sum of squares of first n primes.

0, 12, 62, 202, 576, 1304, 2698, 4902, 8444, 14244, 22242, 34082, 50236, 70704, 97118, 131886, 176844, 230524, 297658, 378314, 471608, 584104, 715410, 870982, 1057804, 1271924, 1511090, 1781586, 2080464, 2414944, 2819566, 3270206

Offset: 1

Terrel Trotter, Jr., Dec 01 2001

			a(4) = 202 because (2 + 3 + 5 + 7)^2 - (2^2 + 3^2 + 5^2 + 7^2) = 17^2 - (4 + 9 + 25 + 49) = 289 - 87 = 202

Harry J. Smith, Table of n, a(n) for n=1,...,500

Cf. A007504, A024450.

With[{prs=Prime[Range[40]]},Table[Total[Take[prs,n]]^2-Total[Take[ prs,n]^2], {n,40}]] (* Harvey P. Dale, Dec 04 2011 *)

{ a1=a2=0; for (n=1, 500, a1+=prime(n); a2+=prime(n)^2; write("b065595.txt", n, " ", a1^2 - a2) ) } \\ Harry J. Smith, Oct 23 2009

a(n) = (A007504(n))^2 - A024450(n). - R. J. Mathar, Oct 07 2011

a(26)-a(32) from Harry J. Smith, Oct 23 2009

A065762 a(n) = (sum of first n primes)^2 + sum of (squares of first n primes).

Original entry on oeis.org

8, 38, 138, 376, 992, 2058, 4030, 6956, 11556, 19038, 28958, 43536, 63052, 87218, 118050, 158436, 210356, 271478, 347590, 438328, 542280, 667258, 812342, 983756, 1189396, 1423918, 1684302, 1977696, 2300336, 2660354, 3097234, 3582196, 4126908, 4718214

Offset: 1

Terrel Trotter, Jr., Dec 04 2001

			a(4) = 376 because (2 + 3 + 5 + 7)^2 + (2^2 + 3^2 + 5^2 + 7^2) = 17^2 + (4 + 9 + 25 + 49) = 289 + 87 = 376.

Harry J. Smith, Table of n, a(n) for n = 1..500

Cf. A007504, A024450, A065595.

nn=50;With[{prs=Prime[Range[nn]]},Table[Total[Take[prs,n]]^2+ Total[Take[prs,n]^2],{n,nn}]] (* Harvey P. Dale, Aug 20 2011 *)

{ s=ss=0; for (n=1, 500, p=prime(n); s+=p; ss+=p^2; write("b065762.txt", n, " ", s^2 + ss) ) } \\ Harry J. Smith, Oct 30 2009

a(n) = A007504(n)^2 + A024450(n). - Michel Marcus, Oct 12 2015

More terms from Harvey P. Dale, Aug 20 2011

A069099 Centered heptagonal numbers.

Original entry on oeis.org

1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, 547, 638, 736, 841, 953, 1072, 1198, 1331, 1471, 1618, 1772, 1933, 2101, 2276, 2458, 2647, 2843, 3046, 3256, 3473, 3697, 3928, 4166, 4411, 4663, 4922, 5188, 5461, 5741, 6028, 6322, 6623, 6931, 7246

Offset: 1

Terrel Trotter, Jr., Apr 05 2002

Equals the triangular numbers convolved with [ 1, 5, 1, 0, 0, 0, ...]. - Gary W. Adamson and Alexander R. Povolotsky, May 29 2009
Number of ordered pairs of integers (x,y) with abs(x) < n, abs(y) < n and abs(x + y) < n, counting twice pairs of equal numbers. - Reinhard Zumkeller, Jan 23 2012; corrected and extended by Mauro Fiorentini, Jan 01 2018
The number of pairs without repetitions is a(n) - 2n + 3 for n > 1. For example, there are 19 such pairs for n = 3: (-2, 0), (-2, 1), (-2, 2), (-1, -1), (-1, 0), (-1, 1), (-1, 2), (0, -2), (0, -1), (0, 0), (0, 1), (0, 2), (1, -2), (1, -1), (1, 0), (1, 1), (2, -2), (2, -1), (2, 0). - Mauro Fiorentini, Jan 01 2018

			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   =         -(0) + (1).
8   =       -(0+1) + (2+3+4).
22  =     -(0+1+2) + (3+4+5+6+7).
43  =   -(0+1+2+3) + (4+5+6+7+8+9+10).
71  = -(0+1+2+3+4) + (5+6+7+8+9+10+11+12+13). (End)

T. D. Noe, Table of n, a(n) for n = 1..1000
Nicolas Bělohoubek and Viktor Javor, Centered heptagonal numbers appearing in aperiodic heptagonal tiling
Leo Tavares, Illustration: Crystal Numbers
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
Index entries for sequences related to centered polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000566 (heptagonal numbers).
Cf. A001263, A057655, A001106.
Cf. A003215, A000217.

a069099 n = length
   [(x,y) | x <- [-n+1..n-1], y <- [-n+1..n-1], x + y <= n - 1]
-- Reinhard Zumkeller, Jan 23 2012

FoldList[#1 + #2 &, 1, 7 Range@ 50] (* Robert G. Wilson v, Feb 02 2011 *)
LinearRecurrence[{3,-3,1},{1,8,22},50] (* Harvey P. Dale, Jun 04 2011 *)

a(n)=(7*n^2-7*n+2)/2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = (7*n^2 - 7*n + 2)/2.
a(n) = 1 + Sum_{k=1..n} 7*k. - Xavier Acloque, Oct 26 2003
Binomial transform of [1, 7, 7, 0, 0, 0, ...]; Narayana transform (A001263) of [1, 7, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007
a(n) = 7*n + a(n-1) - 7 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: x*(1+5*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(0)=1, a(1)=8, a(2)=22. - Harvey P. Dale, Jun 04 2011
a(n) = A024966(n-1) + 1. - Omar E. Pol, Oct 03 2011
a(n) = 2*a(n-1) - a(n-2) + 7. - Ant King, Jun 17 2012
From Ant King, Jun 17 2012: (Start)
Sum_{n>=1} 1/a(n) = 2*Pi/sqrt(7)*tanh(Pi/(2*sqrt(7))) = 1.264723171685652...
a(n) == 1 (mod 7) for all n.
The sequence of digital roots of the a(n) is period 9: repeat [1, 8, 4, 7, 8, 7, 4, 8, 1] (the period is a palindrome).
The sequence of a(n) mod 10 is period 20: repeat [1, 8, 2, 3, 1, 6, 8, 7, 3, 6, 6, 3, 7, 8, 6, 1, 3, 2, 8, 1] (the period is a palindrome).
(End)
E.g.f.: -1 +  (2 + 7*x^2)*exp(x)/2. - Ilya Gutkovskiy, Jun 30 2016
a(n) = A101321(7,n-1). - R. J. Mathar, Jul 28 2016
From Amiram Eldar, Jun 20 2020: (Start)
Sum_{n>=1} a(n)/n! = 9*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 9/(2*e) - 1. (End)
a(n) = A003215(n-1) + A000217(n-1). - Leo Tavares, Jul 19 2022

A069125 a(n) = (11n^2 - 11n + 2)/2.

Original entry on oeis.org

1, 12, 34, 67, 111, 166, 232, 309, 397, 496, 606, 727, 859, 1002, 1156, 1321, 1497, 1684, 1882, 2091, 2311, 2542, 2784, 3037, 3301, 3576, 3862, 4159, 4467, 4786, 5116, 5457, 5809, 6172, 6546, 6931, 7327, 7734, 8152, 8581, 9021, 9472, 9934, 10407, 10891, 11386, 11892

Offset: 1

Terrel Trotter, Jr., Apr 07 2002

Centered hendecagonal (11-gonal) numbers. - Omar E. Pol, Oct 03 2011

Numbers of the form (2*m+1)^2 + k*m*(m+1)/2: in this case is k=3. See also A254963. - Bruno Berselli, Feb 11 2015

			a(5)=111 because 111 = (11*5^2 - 11*5 + 2)/2 = (275 - 55 + 2)/2 = 222/2.

T. D. Noe, Table of n, a(n) for n = 1..1000
X. Acloque, Polynexus Numbers and other mathematical wonders. [broken link]
Leo Tavares, Illustration: Clipped Stars.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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

Cf. A000217, A003154, A152740, A254963.

FoldList[#1 + #2 &, 1, 11 Range@ 45] (* Robert G. Wilson v *)
Table[(11n^2-11n+2)/2,{n,60}] (* or *) LinearRecurrence[{3,-3,1},{1,12,34},60] (* Harvey P. Dale, Jun 25 2011 *)

a(n)=(11*n^2-11*n+2)/2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 1 + Sum_{j=0..n-1} (11*j). - Xavier Acloque, Oct 22 2003
Binomial transform of [1, 11, 11, 0, 0, 0, ...]; Narayana transform (A001263) of [1, 11, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007
a(n) = 11*n + a(n-1) - 11 with n > 1, a(1)=1. - Vincenzo Librandi, Aug 08 2010
G.f.: -x*(1+9*x+x^2)/(x-1)^3. - R. J. Mathar, Jun 05 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=12, a(2)=34. - Harvey P. Dale, Jun 25 2011
a(n) = A152740(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(3/11)*Pi/2)/sqrt(33).
Sum_{n>=1} a(n)/n! = 13*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 13/(2*e) - 1. (End)
a(n) = A003154(n) - A000217(n-1). - Leo Tavares, Mar 29 2022
E.g.f.: exp(x)*(1 + 11*x^2/2) - 1. - Elmo R. Oliveira, Oct 18 2024

More terms from Harvey P. Dale, Jun 25 2011

Name rewritten by Bruno Berselli, Feb 11 2015

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

Terrel Trotter, Jr., Apr 07 2002

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

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

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers
Index entries for sequences related to centered polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1)

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

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 *)

a(n)=13*n(n-1)/2+1 \\ Charles R Greathouse IV, Oct 07 2015

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

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

Terrel Trotter, Jr., Apr 07 2002

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

			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)

T. D. Noe, Table of n, a(n) for n = 1..1000
Leo Tavares, Illustration: Heptagonal Stars.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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

Cf. A163756, A193053.

FoldList[#1 + #2 &, 1, 14 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
Accumulate[14*Range[0,50]]+1 (* Harvey P. Dale, Apr 09 2012 *)

a(n)=7*n^2-7*n+1 \\ Charles R Greathouse IV, Oct 07 2015

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

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

Terrel Trotter, Jr., Apr 07 2002

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

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

T. D. Noe, Table of n, a(n) for n = 1..1000
E. Weisstein, Centered Polygonal Numbers
Index entries for sequences related to centered polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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

[(15*n^2 - 15*n + 2)/2 : n in [1..50]]; // Wesley Ivan Hurt, Nov 14 2014

A069128:=n->(15*n^2 - 15*n + 2)/2: seq(A069128(n), n=1..50); # Wesley Ivan Hurt, Nov 14 2014

FoldList[#1 + #2 &, 1, 15 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
LinearRecurrence[{3,-3,1},{1,16,46},50] (* Harvey P. Dale, Oct 22 2013 *)

a(n)=15*n*(n-1)/2+1 \\ Charles R Greathouse IV, Nov 15 2014

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

A069129 Centered 16-gonal numbers.

Original entry on oeis.org

1, 17, 49, 97, 161, 241, 337, 449, 577, 721, 881, 1057, 1249, 1457, 1681, 1921, 2177, 2449, 2737, 3041, 3361, 3697, 4049, 4417, 4801, 5201, 5617, 6049, 6497, 6961, 7441, 7937, 8449, 8977, 9521, 10081, 10657, 11249, 11857, 12481, 13121, 13777, 14449, 15137, 15841

Offset: 1

Terrel Trotter, Jr., Apr 07 2002

Also, sequence found by reading the line from 1, in the direction 1, 17, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A139098 in the same spiral. - Omar E. Pol, Apr 26 2008
The subsequence of primes begins: 17, 97, 241, 337, 449, 577, 881, 1249, 3041, 3361, 3697, 4049, 4801, 6961, 7937, 9521,  10657, 13121, 14449. See A184899: n such that the n-th centered 12-gonal number is prime. Indices of prime star numbers. - Jonathan Vos Post, Feb 27 2011
Binomial transform of [1, 16, 16, 0, 0, 0, ...] and Narayana transform (A001263) of [1, 16, 0, 0, 0, ...]. - Gary W. Adamson, Jul 28 2011
Centered hexadecagonal numbers or centered hexakaidecagonal numbers. - Omar E. Pol, Oct 03 2011
a(n) = m(n,n) for an array constructed by using the terms in A016813 as the antidiagonals; the first few antidiagonals are 1; 5,9; 13,17,21; 25,29,33,37. - J. M. Bergot, Jul 05 2013
[The first five rows begin: 1,9,21,37,57; 5,17,33,53,77; 13,29,49,73,101; 25,45,69,97,129; 41,65,93,125,161.]

			a(5) = 161 because 8*5^2 - 8*5 + 1 = 200 - 40 + 1 = 161.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Leo Tavares, Illustration: Octagonal Stars.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005448, A001844, A005891, A003215, A069099, A000217, A035008, A139098.

Bisection of A077221.

[8*n^2-8*n+1: n in [0..50]]; // Vincenzo Librandi, Feb 05 2013

FoldList[#1 + #2 &, 1, 16 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
Rest[CoefficientList[Series[-x(1+14x+x^2)/(x-1)^3,{x,0,50}],x]]  (* Harvey P. Dale, Apr 22 2011 *)

a(n)=8*n^2-8*n+1 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 8*n^2 - 8*n + 1.
a(n) = A035008(n-1) + 1. - Omar E. Pol, Apr 26 2008
a(n) = 16*n + a(n-1) - 16 with n > 1, a(1)=1. - Vincenzo Librandi, Aug 08 2010
G.f.: -x*(1+14*x+x^2) / (x-1)^3. - R. J. Mathar, Feb 04 2011
E.g.f.: (8*x^2 + 1)*exp(x). - G. C. Greubel, Jul 18 2017
a(n) = A056220(2n-1). - Bruce J. Nicholson, Aug 31 2017
Sum_{n>=1} 1/a(n) = Pi * tan(Pi/(2*sqrt(2))) / (4*sqrt(2)). - Vaclav Kotesovec, Jul 23 2019
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} a(n)/n! = 9*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 9/e - 1. (End)
Product_{n>=2} (a(n) - 1) / (a(n) + 1) = Pi/4. - Dimitris Valianatos, Jun 27 2020
a(n) = A016754(n-1) + 8*A000217(n-1). - Leo Tavares, Jul 19 2021

A069130 Centered 17-gonal numbers: (17n^2 - 17n + 2)/2.

Original entry on oeis.org

1, 18, 52, 103, 171, 256, 358, 477, 613, 766, 936, 1123, 1327, 1548, 1786, 2041, 2313, 2602, 2908, 3231, 3571, 3928, 4302, 4693, 5101, 5526, 5968, 6427, 6903, 7396, 7906, 8433, 8977, 9538, 10116, 10711, 11323, 11952, 12598, 13261, 13941, 14638, 15352

Offset: 1

Terrel Trotter, Jr., Apr 07 2002

Equals binomial transform of [1, 17, 17, 0, 0, 0, ...]. - Gary W. Adamson, Mar 26 2010

			a(5) = 171 because (17*5^2 - 17*5 + 2)/2 = (425 - 85 + 2)/2 = 342/2 = 171.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers
Index entries for sequences related to centered polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1)

Cf. centered polygonal numbers listed in A069190.

[ (17*n^2 - 17*n + 2)/2 : n in [1..50] ]; // Wesley Ivan Hurt, Jun 09 2014

A069130:=n->(17*n^2 - 17*n + 2)/2; seq(A069130(n), n=1..50); # Wesley Ivan Hurt, Jun 09 2014

FoldList[#1 + #2 &, 1, 17 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
Table[(17n^2-17n+2)/2,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,18,52},50] (* Harvey P. Dale, Jun 05 2011 *)

a(n)=17*binomial(n,2)+1 \\ Charles R Greathouse IV, Jun 05 2011

a(n) = (17*n^2 - 17*n + 2)/2.
a(n) = 17*n + a(n-1) - 17 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: x*(1+15*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(0)=1, a(1)=18, a(2)=52. - Harvey P. Dale, Jun 05 2011
Narayana transform (A001263) of [1, 17, 0, 0, 0, ...]. - Gary W. Adamson, Jul 28 2011
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = 2*Pi*tan(3*Pi/(2*sqrt(17)))/(3*sqrt(17)).
Sum_{n>=1} a(n)/n! = 19*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 19/(2*e) - 1. (End)
E.g.f.: exp(x)*(1 + 17*x^2/2) - 1. - Stefano Spezia, May 31 2022

Typo in formula fixed by Omar E. Pol, Dec 22 2008

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

Terrel Trotter, Jr., Apr 07 2002

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)

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

Ivan Panchenko, Table of n, a(n) for n = 1..1000
John Elias, Illustration of Initial Terms: Triangular & Hexagonal Configurations.
Lamine Ngom, An origin of A069131 (illustration).
Leo Tavares, Illustration: Tri-Hexagons.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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.

[9*n^2 - 9*n + 1 : n in [1..50]]; // Wesley Ivan Hurt, May 05 2021

FoldList[#1 + #2 &, 1, 18 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
LinearRecurrence[{3,-3,1},{1,19,55},50] (* Harvey P. Dale, Jan 20 2014 *)

a(n)=9*n^2-9*n+1 \\ Charles R Greathouse IV, Oct 07 2015

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

cp's OEIS Frontend

User: Terrel Trotter, Jr.

A065595 a(n) = (sum of first n primes)^2 - sum of squares of first n primes.

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A065762 a(n) = (sum of first n primes)^2 + sum of (squares of first n primes).

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A069099 Centered heptagonal numbers.

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A069125 a(n) = (11*n^2 - 11*n + 2)/2.

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A069126 Centered 13-gonal numbers.

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A069127 Centered 14-gonal numbers.

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

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A069125 a(n) = (11n^2 - 11n + 2)/2.

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

A069130 Centered 17-gonal numbers: (17n^2 - 17n + 2)/2.