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

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

Author

Bruno Berselli, Sep 15 2015

Keywords

• nonn
• easy

Comments

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, ...

References

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

Links

• 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).

Crossrefs

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.

Programs

Magma

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

Mathematica

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)]

Formula

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