cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A262221 a(n) = 25*n*(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

Views

Author

Bruno Berselli, Sep 15 2015

Keywords

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

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

A272871 Imaginary part of (n + i)^4.

Original entry on oeis.org

0, 0, 24, 96, 240, 480, 840, 1344, 2016, 2880, 3960, 5280, 6864, 8736, 10920, 13440, 16320, 19584, 23256, 27360, 31920, 36960, 42504, 48576, 55200, 62400, 70200, 78624, 87696, 97440, 107880, 119040, 130944, 143616, 157080, 171360, 186480, 202464, 219336
Offset: 0

Views

Author

Colin Barker, May 08 2016

Keywords

Examples

			a(5) = 480 because (5 + i)^4 = 476 + 480*i.

Links

Crossrefs

Cf. A002378, A007531, A064200.
Cf. A272870.

Programs

  • Mathematica
    Table[Im[(n + I)^4], {n, 0, 38}] (* or *)
Table[4 (n - 1) n (n + 1), {n, 0, 38}] (* or *)
CoefficientList[Series[24 x^2/(1 - x)^4, {x, 0, 38}], x] (* Michael De Vlieger, May 08 2016 *)
  • PARI
    a(n) = 4*(n-1)*n*(n+1)
  • PARI
    vector(50, n, n--; imag((n+I)^4))
  • PARI
    concat(vector(2), Vec(24*x^2/(1-x)^4 + O(x^50)))

Formula

a(n) = 4*A007531(n+1).
a(n) = 4*(n-1)*n*(n+1).
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3.
G.f.: 24*x^2 / (1-x)^4.
a(n) = b(n+1)*b(n-1)-b(n)*b(n-2), where b(n) is A002378(n). - Anton Zakharov, Aug 15 2016
From Ilya Gutkovskiy, Aug 15 2016: (Start)
E.g.f.: 4*x^2*(3 + x)*exp(x).
a(n) = 24*binomial(n+1,3).
a(n) = Sum_{k=0..n} A064200(k). (End)
Showing 1-2 of 2 results.

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