A262221 -id:A262221 - cp's OEIS frontend

A080956 a(n) = (n+1)*(2-n)/2.

1, 1, 0, -2, -5, -9, -14, -20, -27, -35, -44, -54, -65, -77, -90, -104, -119, -135, -152, -170, -189, -209, -230, -252, -275, -299, -324, -350, -377, -405, -434, -464, -495, -527, -560, -594, -629, -665, -702, -740, -779, -819, -860, -902, -945, -989, -1034, -1080, -1127, -1175, -1224, -1274, -1325, -1377

Offset: 0

Paul Barry, Mar 01 2003

Coefficient of x in the polynomial C(n,0)+C(n+1,1)x+C(n+2,2)x(x-1)/2.
Equals A154990 * [1,2,3,...]. - Gary W. Adamson & Mats Granvik, Jan 19 2009
a(n) is essentially the case 1 of the polygonal numbers. The polygonal numbers are defined as P_k(n) = Sum_{i=1..n} ((k-2)*i-(k-3)). Thus P_1(n) = n*(3-n)/2 and a(n) = P_1(n+1). See A005563 for the case k=0. - Peter Luschny, Jul 08 2011
This is the case k=-1 of the formula (k*m*(m+1)-(-1)^k+1)/2. See similar sequences listed in A262221. - Bruno Berselli, Sep 17 2015

			a(5) = 6-(1+2+3+4+5). - _Stanislav Sykora_, Feb 19 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000027, A000096, A000217, A154990, A214292, A262221.

[(n+1)*(2-n)/2: n in [0..80]]; // Vincenzo Librandi, Jul 08 2011

G(x):=exp(x)*(x-x^2/2): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=1..54 ); # Zerinvary Lajos, Apr 05 2009

FoldList[#1 - #2 &, 1, Range[0, 44]] (* Arkadiusz Wesolowski, May 26 2013 *)
LinearRecurrence[{3,-3,1},{1,1,0},60] (* Harvey P. Dale, Nov 29 2019 *)

PARI
```
a(n)=(n+1)*(2-n)/2;
```

def A080956(n): return (2-n)*(n+1)//2 # G. C. Greubel, May 08 2025

a(n) = 2*(C(n+1, 1)-C(n+2, 2)) = (n+1)*(2-n)/2.
G.f.: (1-2*x)/(1-x)^3. - R. J. Mathar, Jun 11 2009
If we define f(n,i,a) = Sum_{k=0..n-i} (binomial(n,k)*stirling1(n-k,i)*Product_{j=0..k-1} (-a-j)), then a(n) = f(n,n-1,2), for n>=3. - Milan Janjic, Dec 20 2008
E.g.f.: exp(x)*(1-x^2/2). - Zerinvary Lajos, Apr 05 2009, R. J. Mathar, Jun 11 2009
a(n) = - A214292(n,1) for n > 0. - Reinhard Zumkeller, Jul 12 2012
Recurrence: a(0)=1, a(n+1) = a(n) - n. Also a(n)=(n+1)-Sum[k=1..n](k). Also a(n) = A000027(n+1) - A000217(n). Also, for n>1, a(n) = - A000096(n-2). - Stanislav Sykora, Feb 19 2014
Sum_{n>=3} 1/a(n) = -11/9. - Amiram Eldar, Sep 26 2022

Lajos e.g.f. adapted to offset zero by R. J. Mathar, Jun 11 2009

A069190 Centered 24-gonal numbers.

Original entry on oeis.org

1, 25, 73, 145, 241, 361, 505, 673, 865, 1081, 1321, 1585, 1873, 2185, 2521, 2881, 3265, 3673, 4105, 4561, 5041, 5545, 6073, 6625, 7201, 7801, 8425, 9073, 9745, 10441, 11161, 11905, 12673, 13465, 14281, 15121, 15985, 16873, 17785, 18721, 19681, 20665, 21673

Offset: 1

Terrel Trotter, Jr., Apr 10 2002

Sequence found by reading the line from 1, in the direction 1, 25, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Semi-axis opposite to A135453 in the same spiral. - Omar E. Pol, Sep 16 2011

			a(5) = 241 because 12*5^2 - 12*5 + 1 = 300 - 60 + 1 = 241.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
John Elias, Illustration: Odd Ordered Star Perimeters.
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 k-gonal numbers with k=3..25: A005448, A001844, A005891, A003215, A069099, A016754, A060544, A062786, A069125, A003154, A069126, A069127, A069128, A069129, A069130, A069131, A069132, A069133, A069178, A069173, A069174, A069190, A262221.

FoldList[#1 + #2 &, 1, 24 Range@ 45] (* Robert G. Wilson v *)
Table[12n^2-12n+1,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,25,73},50] (* Harvey P. Dale, Jul 17 2011 *)

a(n)=12*n^2-12*n+1 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 12*n^2 - 12*n + 1.
a(n) = 24*n + a(n-1) - 24 with a(1)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=25, a(3)=73. - Harvey P. Dale, Jul 17 2011
G.f.: x*(1+22*x+x^2)/(1-x)^3. - Harvey P. Dale, Jul 17 2011
Binomial transform of [1, 24, 24, 0, 0, 0, ...] and Narayana transform (cf. A001263) of [1, 24, 0, 0, 0, ...]. - Gary W. Adamson, Jul 26 2011
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi*tan(Pi/sqrt(6))/(4*sqrt(6)).
Sum_{n>=1} a(n)/n! = 13*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 13/e - 1. (End)
E.g.f.: exp(x)*(1 + 12*x^2) - 1. - Stefano Spezia, May 31 2022

More terms from Harvey P. Dale, Jul 17 2011

A064200 a(n) = 12n(n-1).

Original entry on oeis.org

0, 0, 24, 72, 144, 240, 360, 504, 672, 864, 1080, 1320, 1584, 1872, 2184, 2520, 2880, 3264, 3672, 4104, 4560, 5040, 5544, 6072, 6624, 7200, 7800, 8424, 9072, 9744, 10440, 11160, 11904, 12672, 13464, 14280, 15120, 15984, 16872, 17784, 18720, 19680, 20664, 21672

Offset: 0

Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Sep 22 2001

Luigi Berzolari, Allgemeine Theorie der Höheren Ebenen Algebraischen Kurven, Encyclopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, Band III_2, Heft 3, Leipzig: B. G. Teubner, 1906, p. 341.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Leo Tavares, Illustration: Twin Stars.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A262221, A003154.

Table[12n(n-1),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,0,24},40] (* Harvey P. Dale, Jul 22 2015 *)
Join[{0},24*Accumulate[Range[0,60]]] (* Harvey P. Dale, Dec 17 2022 *)

a(n)=12*n*(n-1) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 24*(n-1) + a(n-1) for n>0, with a(0)=0. - Vincenzo Librandi, Aug 07 2010
a(0)=0, a(1)=0, a(2)=24, a(n)=3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Jul 22 2015
G.f.: -(24*x^2)/(x-1)^3. - Harvey P. Dale, Jul 22 2015
a(n) = 2*A003154(n) - 2. See Twin Stars illustration. - Leo Tavares, Aug 23 2021
From Amiram Eldar, Feb 22 2023: (Start)
Sum_{n>=2} 1/a(n) = 1/12.
Sum_{n>=2} (-1)^n/a(n) = (2*log(2) - 1)/12.
Product_{n>=2} (1 - 1/a(n)) = -(12/Pi)*cos(Pi/sqrt(3)).
Product_{n>=2} (1 + 1/a(n)) = (12/Pi)*cos(Pi/sqrt(6)). (End)

A173307 a(n) = 13n(n+1).

Original entry on oeis.org

0, 26, 78, 156, 260, 390, 546, 728, 936, 1170, 1430, 1716, 2028, 2366, 2730, 3120, 3536, 3978, 4446, 4940, 5460, 6006, 6578, 7176, 7800, 8450, 9126, 9828, 10556, 11310, 12090, 12896, 13728, 14586, 15470, 16380, 17316, 18278, 19266, 20280, 21320, 22386, 23478, 24596

Offset: 0

Vincenzo Librandi, Feb 16 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A002378, A152741, A262221.

[13*n*(n+1): n in [0..40]]; // Vincenzo Librandi, Sep 28 2013

I:=[0, 26, 78]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Sep 28 2013

Table[13 n (n + 1), {n, 0, 50}] (* or *) CoefficientList[Series[26 x/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Sep 28 2013 *)
LinearRecurrence[{3,-3,1},{0,26,78},50] (* Harvey P. Dale, Apr 08 2014 *)

a(n)=13*n*(n+1) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 26*A000217(n).
From Vincenzo Librandi, Sep 28 2013: (Start)
G.f.: 26*x/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Feb 22 2023: (Start)
Sum_{n>=1} 1/a(n) = 1/13.
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*log(2) - 1)/13.
Product_{n>=1} (1 - 1/a(n)) = -(13/Pi)*cos(sqrt(17/13)*Pi/2).
Product_{n>=1} (1 + 1/a(n)) = (13/Pi)*cos(3*Pi/(2*sqrt(13))). (End)
From Elmo R. Oliveira, Dec 14 2024: (Start)
E.g.f.: 13*exp(x)*x*(2 + x).
a(n) = 13*A002378(n) = 2*A152741(n). (End)

Incorrect formulas and examples deleted by R. J. Mathar, Jan 04 2011

A010015 a(0) = 1, a(n) = 25*n^2 + 2 for n > 0.

Original entry on oeis.org

1, 27, 102, 227, 402, 627, 902, 1227, 1602, 2027, 2502, 3027, 3602, 4227, 4902, 5627, 6402, 7227, 8102, 9027, 10002, 11027, 12102, 13227, 14402, 15627, 16902, 18227, 19602, 21027, 22502, 24027, 25602, 27227, 28902, 30627, 32402, 34227, 36102, 38027, 40002

Offset: 0

N. J. A. Sloane

Subsequence of A160842. - Bruno Berselli, Feb 06 2012

The identity (25*n^2 + 1)^2 - (25*n^2 + 2)*(5*n)^2 = 1 can be written as (A016850(n+1) + 1)^2 - a(n+1)*A008587(n+1)^2 = 1. - Vincenzo Librandi, Feb 08 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A206399.

Join[{1}, 25 Range[40]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {27, 102, 227}, 50]] (* Vincenzo Librandi, Feb 08 2012 *)

A010015(n)=25*n^2+2-!n \\ M. F. Hasler, Feb 14 2012

G.f.: (1+x)*(1 + 23*x + x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*25 + 2)*e^x - 1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) =3/4+sqrt(2)/20*Pi*coth(Pi*sqrt(2)/5) = 1.062575323280590.. - R. J. Mathar, May 07 2024
a(n) = A262221(n)+A262221(n+1). - R. J. Mathar, May 07 2024

A270693 Alternating sum of centered 25-gonal numbers.

Original entry on oeis.org

1, -25, 51, -100, 151, -225, 301, -400, 501, -625, 751, -900, 1051, -1225, 1401, -1600, 1801, -2025, 2251, -2500, 2751, -3025, 3301, -3600, 3901, -4225, 4551, -4900, 5251, -5625, 6001, -6400, 6801, -7225, 7651, -8100, 8551, -9025, 9501, -10000, 10501

Offset: 0

Ilya Gutkovskiy, Mar 21 2016

The absolute value alternating sum of centered k-gonal numbers gives concentric k-gonal numbers.

More generally, the ordinary generating function for the alternating sum of centered k-gonal numbers is (1 - (k - 2)*x + x^2)/((1 - x)*(1 + x)^3).

OEIS Wiki, Centered polygonal numbers
Eric Weisstein's World of Mathematics, Centered Polygonal Number
Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Cf. A262221 (centered 25-gonal numbers).

Cf. A032527, A032528, A077043, A077221, A195041, A195042, A195045, A195046, A195047, A195048, A195049, A195058, A195142, A195043, A195143, A195145, A195146, A195147, A195148, A195149, A195158.

[((-1)^n*(50*n^2 + 100*n + 29) - 21)/8 : n in [0..40]]; // Wesley Ivan Hurt, Mar 21 2016

A270693:=n->((-1)^n*(50*n^2 + 100*n + 29) - 21)/8: seq(A270693(n), n=0..100); # Wesley Ivan Hurt, Sep 18 2017

LinearRecurrence[{-2, 0, 2, 1}, {1, -25, 51, -100}, 41]
Table[((-1)^n (50 n^2 + 100 n + 29) - 21)/8, {n, 0, 40}]

x='x+O('x^100); Vec((1-23*x+x^2)/((1-x)*(1+x)^3)) \\ Altug Alkan, Mar 21 2016

G.f.: (1 - 23*x + x^2)/((1 - x)*(1 + x)^3).
E.g.f.: (1/8)*(-21*exp(x) + (29 - 150*x + 50*x^2)*exp(-x)).
a(n) = -2*a(n-1) + 2*a(n-3) + a(n-4).
a(n) = ((-1)^n*(50*n^2 + 100*n + 29) - 21)/8.

A276264 Centered 25-gonal primes.

Original entry on oeis.org

151, 251, 701, 1951, 3001, 4751, 10151, 12401, 16651, 19501, 28201, 29401, 33151, 38501, 39901, 45751, 56951, 63901, 65701, 81001, 87151, 95701, 104651, 114001, 136501, 144451, 147151, 158201, 178501, 181501, 193751, 219451, 232901, 257401, 275651, 290701, 318001, 322001

Offset: 1

Ilya Gutkovskiy, Aug 26 2016

nonn

Primes of the form (25*k^2 + 25*k + 2)/2.

Numbers k such that (25*k^2 + 25*k + 2)/2 is prime: 3, 4, 7, 12, 15, 19, 28, 31, 36, 39, 47, 48, 51, 55, 56, 60, 67, 71, 72, 80, 83, 87, 91, ...

Robert Israel, Table of n, a(n) for n = 1..10000
OEIS Wiki, Centered polygonal numbers
Eric Weisstein's World of Mathematics, Centered Polygonal Number
Index entries for sequences related to centered polygonal numbers

Cf. A000040, A262221.

Cf. centered k-gonal primes listed in A276261.

select(isprime, [seq((25*k^2+25*k+2)/2, k=1..200)]); # Robert Israel, Sep 01 2016

Intersection[Table[(25 k^2 + 25 k + 2)/2, {k, 0, 1000}], Prime[Range[28000]]]

lista(nn) = for(n=1, nn, if(isprime(p=(25*n^2 + 25*n + 2)/2), print1(p, ", "))); \\ Altug Alkan, Aug 26 2016

cp's OEIS Frontend

A080956 a(n) = (n+1)*(2-n)/2.

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A069190 Centered 24-gonal numbers.

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A064200 a(n) = 12*n*(n-1).

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A173307 a(n) = 13*n*(n+1).

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A010015 a(0) = 1, a(n) = 25*n^2 + 2 for n > 0.

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A270693 Alternating sum of centered 25-gonal numbers.

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Magma

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A276264 Centered 25-gonal primes.

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A064200 a(n) = 12n(n-1).

A173307 a(n) = 13n(n+1).