A062786 Centered 10-gonal numbers.

1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, 781, 911, 1051, 1201, 1361, 1531, 1711, 1901, 2101, 2311, 2531, 2761, 3001, 3251, 3511, 3781, 4061, 4351, 4651, 4961, 5281, 5611, 5951, 6301, 6661, 7031, 7411, 7801, 8201, 8611, 9031, 9461, 9901, 10351, 10811

Offset: 1

Author

Jason Earls, Jul 19 2001

Keywords

• easy
• nonn
• changed

Comments

Deleting the least significant digit yields the (n-1)-st triangular number: a(n) = 10*A000217(n-1) + 1. - Amarnath Murthy, Dec 11 2003

All divisors of a(n) are congruent to 1 or -1, modulo 10; that is, they end in the decimal digit 1 or 9. Proof: If p is an odd prime different from 5 then 5n^2 - 5n + 1 == 0 (mod p) implies 25(2n - 1)^2 == 5 (mod p), whence p == 1 or -1 (mod 10). - Nick Hobson, Nov 13 2006

Centered decagonal numbers. - Omar E. Pol, Oct 03 2011

The partial sums of this sequence give A004466. - Leo Tavares, Oct 04 2021

The continued fraction expansion of sqrt(5*a(n)) is [5n-3; {2, 2n-2, 2, 10n-6}]. For n=1, this collapses to [2; {4}]. - Magus K. Chu, Sep 12 2022

Numbers m such that 20*m + 5 is a square. Also values of the Fibonacci polynomial y^2 - x*y - x^2 for x = n and y = 3*n - 1. This is a subsequence of A089270. - Klaus Purath, Oct 30 2022

All terms can be written as a difference of two consecutive squares a(n) = A005891(n-1)^2 - A028895(n-1)^2, and they can be represented by the forms (x^2 + 2mxy + (m^2-1)y^2) and (3x^2 + (6m-2)xy + (3m^2-2m)y^2), both of discriminant 4. - Klaus Purath, Oct 17 2023

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Leo Tavares, Illustration: Pentagonal Stars.
• Leo Tavares, Illustration: Mid-section Stars.
• Leo Tavares, Illustration: Mid-section Star Pillars.
• Leo Tavares, Illustration: Trapezoidal Rays.
• 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).

Crossrefs

Cf. A001263, A124080, A101321, A028387, A016861, A003154, A005891, A000217, A004466, A144390, A000326, A060544.

Cf. also A016754, A002378, A069099, A045943, A003215, A046092, A001844, A028896, A005448, A024966, A082970.

Programs

GAP

List([1..50], n-> 1+5*n*(n-1)); # G. C. Greubel, Mar 30 2019

Magma

[1+5*n*(n-1): n in [1..50]]; // G. C. Greubel, Mar 30 2019

Mathematica

FoldList[#1+#2 &, 1, 10Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
1+5*Pochhammer[Range[50]-1, 2] (* G. C. Greubel, Mar 30 2019 *)

PARI

j=[]; for(n=1,75,j=concat(j,(5*n*(n-1)+1))); j

PARI

for (n=1, 1000, write("b062786.txt", n, " ", 5*n*(n - 1) + 1) ) \\ Harry J. Smith, Aug 11 2009

Python

def a(n): return(5*n**2-5*n+1) # Torlach Rush, May 10 2024

Sage

[1+5*rising_factorial(n-1, 2) for n in (1..50)] # G. C. Greubel, Mar 30 2019

Formula

a(n) = 5*n*(n-1) + 1.

From Gary W. Adamson, Dec 29 2007: (Start)

Binomial transform of [1, 10, 10, 0, 0, 0, ...];

Narayana transform (A001263) of [1, 10, 0, 0, 0, ...]. (End)

G.f.: x*(1+8*x+x^2) / (1-x)^3. - R. J. Mathar, Feb 04 2011

a(n) = A124080(n-1) + 1. - Omar E. Pol, Oct 03 2011

a(n) = A101321(10,n-1). - R. J. Mathar, Jul 28 2016

a(n) = A028387(A016861(n-1))/5 for n > 0. - Art Baker, Mar 28 2019

E.g.f.: (1+5*x^2)*exp(x) - 1. - G. C. Greubel, Mar 30 2019

Sum_{n>=1} 1/a(n) = Pi * tan(Pi/(2*sqrt(5))) / sqrt(5). - Vaclav Kotesovec, Jul 23 2019

From Amiram Eldar, Jun 20 2020: (Start)

Sum_{n>=1} a(n)/n! = 6*e - 1.

Sum_{n>=1} (-1)^n * a(n)/n! = 6/e - 1. (End)

a(n) = A005891(n-1) + 5*A000217(n-1). - Leo Tavares, Jul 14 2021

a(n) = A003154(n) - 2*A000217(n-1). See Mid-section Stars illustration. - Leo Tavares, Sep 06 2021

From Leo Tavares, Oct 06 2021: (Start)

a(n) = A144390(n-1) + 2*A028387(n-1). See Mid-section Star Pillars illustration.

a(n) = A000326(n) + A000217(n) + 3*A000217(n-1). See Trapezoidal Rays illustration.

a(n) = A060544(n) + A000217(n-1). (End)

From Leo Tavares, Oct 31 2021: (Start)

a(n) = A016754(n-1) + 2*A000217(n-1).

a(n) = A016754(n-1) + A002378(n-1).

a(n) = A069099(n) + 3*A000217(n-1).

a(n) = A069099(n) + A045943(n-1).

a(n) = A003215(n-1) + 4*A000217(n-1).

a(n) = A003215(n-1) + A046092(n-1).

a(n) = A001844(n-1) + 6*A000217(n-1).

a(n) = A001844(n-1) + A028896(n-1).

a(n) = A005448(n) + 7*A000217(n).

a(n) = A005448(n) + A024966(n). (End)

From Klaus Purath, Oct 30 2022: (Start)

a(n) = a(n-2) + 10*(2*n-3).

a(n) = 2*a(n-1) - a(n-2) + 10.

a(n) = A135705(n-1) + n.

a(n) = A190816(n) - n.

a(n) = 2*A005891(n-1) - 1. (End)

Extensions

Better description from Terrel Trotter, Jr., Apr 06 2002