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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A096037 Triangle T(n,m) = (3*n+3*m-2)*(n+1-m)/2 read by rows.

Original entry on oeis.org

2, 7, 5, 15, 13, 8, 26, 24, 19, 11, 40, 38, 33, 25, 14, 57, 55, 50, 42, 31, 17, 77, 75, 70, 62, 51, 37, 20, 100, 98, 93, 85, 74, 60, 43, 23, 126, 124, 119, 111, 100, 86, 69, 49, 26, 155, 153, 148, 140, 129, 115, 98, 78, 55, 29, 187, 185, 180, 172, 161, 147, 130, 110, 87, 61, 32
Offset: 1

Views

Author

Gary W. Adamson, Jun 17 2004

Keywords

Examples

			The triangle starts in row n=1 as
2;
7,5;
15,13,8;
26,24,19,11;

Crossrefs

Cf. A094930, A024212, A011379, A002411, A096038, A095794.

Programs

  • Python
    def A096037(n,m):
    return (3*n+3*m-2)*(n+1-m)//2
print( [A096037(n,m) for n in range(20) for m in range(1,n+1)] )
# R. J. Mathar, Oct 11 2009

Formula

T(n,m) = (3*n+3*m-2)*(n+1-m)/2 .
T(n,m) = A094930(n,m)/(3*m-1).
T(n,1) = A005449(n).
T(n,n) = A016768(n-1).
Row sums: sum_{m=1..n} T(n,m) = n^2*(n+1) = A011379(n).

Extensions

Edited and extended, A-numbers corrected by R. J. Mathar, Oct 11 2009

A181968 a(n) = 54n^3 - 1.

Original entry on oeis.org

53, 431, 1457, 3455, 6749, 11663, 18521, 27647, 39365, 53999, 71873, 93311, 118637, 148175, 182249, 221183, 265301, 314927, 370385, 431999, 500093, 574991, 657017, 746495, 843749, 949103, 1062881, 1185407, 1317005, 1457999, 1608713, 1769471, 1940597, 2122415
Offset: 1

Views

Author

Arkadiusz Wesolowski, Apr 06 2012

Keywords

Comments

a(n) is coprime to 27*n^3*(27*n^3 - 1) - 2 = A016767(n)*(A016767(n)-1) - 2.
x^3 + y^3 + z^3 = w^3 has infinitely many solutions, where every pair of elements x, y and z are coprime.
This follows from the identity a(n)^3 + (A016767(n)+1)^3 + (A016768(n)-A008588(n))^3 = (A016768(n)+A008585(n))^3 for n >= 1.

References

  • Wacław Sierpiński, Czym sie zajmuje teoria liczb. Warsaw: PW "Wiedza Powszechna", 1957, pp. 59-60.

Links

Crossrefs

Cf. A008585, A016767.

Programs

  • Magma
    [ 54*n^3-1 : n in [1..34]];
  • Maple
    seq(54*n^3-1, n=1..34);
  • Mathematica
    Table[54*n^3 - 1, {n, 34}]
  • PARI
    vector(34, n, 54*n^3-1)

Formula

For n >= 1, a(n) = 54*A000578(n) - 1 = 2*A016767(n) - 1.
G.f.: (-1 + 57*x + 213*x^2 + 55*x^3)/(1 - x)^4.
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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