A069174 -id:A069174 - 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

A010013 a(0) = 1, a(n) = 23*n^2 + 2 for n>0.

Original entry on oeis.org

1, 25, 94, 209, 370, 577, 830, 1129, 1474, 1865, 2302, 2785, 3314, 3889, 4510, 5177, 5890, 6649, 7454, 8305, 9202, 10145, 11134, 12169, 13250, 14377, 15550, 16769, 18034, 19345, 20702, 22105, 23554, 25049, 26590, 28177, 29810, 31489, 33214, 34985, 36802, 38665

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.

Join[{1}, 23 Range[41]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)

G.f.: (1+x)*(1+21*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*23+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4 + sqrt(46)/92*Pi*coth( Pi*sqrt(46)/23) = 1.0677349581... - R. J. Mathar, May 07 2024
a(n) = A069174(n)+A069174(n+1). - R. J. Mathar, May 07 2024

A195039 23 times triangular numbers.

Original entry on oeis.org

0, 23, 69, 138, 230, 345, 483, 644, 828, 1035, 1265, 1518, 1794, 2093, 2415, 2760, 3128, 3519, 3933, 4370, 4830, 5313, 5819, 6348, 6900, 7475, 8073, 8694, 9338, 10005, 10695, 11408, 12144, 12903, 13685, 14490, 15318, 16169, 17043, 17940, 18860

Offset: 0

Omar E. Pol, Sep 12 2011

Related to the primitive Pythagorean triple [15, 8, 17].

Sequence found by reading the line from 0, in the direction 0, 23, ..., and the same line from 0, in the direction 0, 69, ..., in the Pythagorean spiral whose edges have length A195035 and whose vertices are the numbers A195036. This is the main diagonal of the square spiral.

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

Bisection of A195036.

Cf. A000217, A069174, A195035.

23*Accumulate[Range[0,40]] (* or *) LinearRecurrence[{3,-3,1},{0,23,69},50] (* Harvey P. Dale, Aug 28 2012 *)

a(n)=23*n*(n+1)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (23*n^2 + 23*n)/2 = 23*n*(n+1)/2 = 23*A000217(n).
a(0)=0, a(1)=23, a(2)=69, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Aug 28 2012
From Elmo R. Oliveira, Dec 15 2024: (Start)
G.f.: 23*x/(1-x)^3.
E.g.f.: 23*exp(x)*x*(2 + x)/2.
a(n) = A069174(n+1) - 1. (End)

A276263 Centered 23-gonal primes.

Original entry on oeis.org

139, 829, 4831, 15319, 36709, 53959, 58789, 65551, 74521, 107089, 142969, 198859, 227011, 278071, 292561, 727399, 750721, 804541, 879199, 957169, 1181281, 1325491, 1364821, 1519519, 1700161, 1835401, 1881631, 2111539, 2231461, 2396509, 2778079, 2926981, 3067879

Offset: 1

Ilya Gutkovskiy, Aug 26 2016

nonn

Primes of the form (23*k^2 + 23*k + 2)/2.

Numbers k such that (23*k^2 + 23*k + 2)/2 is prime: 3, 8, 20, 36, 56, 68, 71, 75, 80, 96, 111, 131, 140, 155, 159, 251, 255, 264, 276, ...

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, A069174.

Cf. centered k-gonal primes listed in A276261.

Intersection[Table[(23 k^2 + 23 k + 2)/2, {k, 0, 1000}], Prime[Range[230000]]]
Select[Table[(23k^2+23k+2)/2,{k,600}],PrimeQ] (* Harvey P. Dale, Jun 17 2021 *)

lista(nn) = for(n=1, nn, if(isprime(p=(23*n^2 + 23*n + 2)/2), print1(p, ", "))); \\ Altug Alkan, Aug 26 2016

cp's OEIS Frontend

A069190 Centered 24-gonal numbers.

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

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A010013 a(0) = 1, a(n) = 23*n^2 + 2 for n>0.

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A195039 23 times triangular numbers.

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A276263 Centered 23-gonal primes.

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