A069178 -id:A069178 - cp's OEIS frontend

A069190 Centered 24-gonal numbers.

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

A262221 a(n) = 25n(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

Bruno Berselli, Sep 15 2015

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, ...

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

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Table of numbers of the form (k*n*(n+1)-(-1)^k+1)/2.
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. 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.

Magma
```
[25*n*(n+1)/2+1: n in [0..50]];
```

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)]
```

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

A276261 Centered 21-gonal primes.

Original entry on oeis.org

127, 211, 757, 2521, 2857, 6301, 8527, 16381, 19867, 23689, 24697, 27847, 32341, 37171, 38431, 42337, 66361, 68041, 82237, 89839, 97777, 103951, 114661, 140071, 152461, 162751, 170689, 192781, 204331, 216217, 231547, 240997, 284131, 308827, 353557, 357421, 385057, 389089

Offset: 1

Ilya Gutkovskiy, Aug 26 2016

nonn

Primes of the form (21*k^2 + 21*k + 2)/2.

Numbers k such that (21*k^2 + 21*k + 2)/2 is prime: 3, 4, 8, 15, 16, 24, 28, 39, 43, 47, 48, 51, 55, 059, 60, 63, 79, 80, 88, 92, 96, 99, ...

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, A069178.

Cf. similar sequences of the centered k-gonal primes: A125602 (k = 3), A027862 (k = 4), A145838 (k = 5), A002407 (k = 6), A144974 (k = 7), A090562 (k = 10), A262344 (k = 11), A262493 (k = 13), A264821 (k = 14), A264822 (k = 15), A264823 (k = 16), A264824 (k = 17), A264825 (k = 18), A264844 (k = 19), A264845 (k = 20), A201715 (k = 24).

Intersection[Table[(21 k^2 + 21 k + 2)/2, {k, 0, 1000}], Prime[Range[33000]]]

lista(nn) = for(n=1, nn, if(isprime(p=(21*n^2 + 21*n + 2)/2), print1(p, ", "))); \\ Altug Alkan, Aug 26 2016

A010011 a(0) = 1, a(n) = 21*n^2 + 2 for n>0.

Original entry on oeis.org

1, 23, 86, 191, 338, 527, 758, 1031, 1346, 1703, 2102, 2543, 3026, 3551, 4118, 4727, 5378, 6071, 6806, 7583, 8402, 9263, 10166, 11111, 12098, 13127, 14198, 15311, 16466, 17663, 18902, 20183, 21506, 22871, 24278, 25727, 27218, 28751, 30326, 31943, 33602, 35303

Offset: 0

N. J. A. Sloane

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.

[1] cat [21*n^2+2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015

A010011:=n->`if`(n=0,1,21*n^2+2); seq(A010011(n), n=0..100); # Wesley Ivan Hurt, Nov 15 2013

Join[{1}, 21 Range[41]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {23, 86, 191}, 50]] (* Vincenzo Librandi, Aug 03 2015 *)

O.g.f.: 1-x*(23+17*x+2*x^2)/(-1+x)^3. - R. J. Mathar, Apr 12 2008
E.g.f.: (21*x*(x+1)+2)*e^x-1. - Gopinath A. R., Feb 13 2012
Sum_{n>=0} 1/a(n) = 3/4+sqrt(42)/84*Pi*coth( Pi*sqrt(42)/21) = 1.0738233857899... - R. J. Mathar, May 07 2024
a(n) = A069178(n)+A069178(n+1). - R. J. Mathar, May 07 2024

A081929 Sum of the n-th row of A077339.

Original entry on oeis.org

1, 22, 64, 127, 211, 316, 442, 589, 757, 946, 1156, 2467, 3979, 5692, 7606, 9721, 12037, 14554, 17272, 20191, 23311, 26632, 30154, 33877, 37801, 41926, 46252, 50779, 55507, 60436, 65566, 70897, 76429, 82162, 88096, 94231, 100567, 107104

Offset: 1

Amarnath Murthy, Apr 01 2003

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Not the same as A069178 (A069178(12)=1387 <> 2467=a(12) ). Cf. A077339, A077340.

For n>10, a(n) = 1/2 * (201n^2 - 2001n + 2).
From Chai Wah Wu, Jan 12 2023: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 13.
G.f.: x*(900*x^12 - 1080*x^11 - x^2 - 19*x - 1)/(x - 1)^3. (End)

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003

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A069190 Centered 24-gonal numbers.

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

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A276261 Centered 21-gonal primes.

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A010011 a(0) = 1, a(n) = 21*n^2 + 2 for n>0.

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A081929 Sum of the n-th row of A077339.

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