A083953 -id:A083953 - cp's OEIS frontend

A083951 Least increasing integer coefficients such that A(x)^(1/3) has only integer coefficients.

1, 3, 6, 7, 9, 12, 13, 15, 18, 21, 24, 27, 28, 30, 33, 34, 36, 39, 41, 42, 45, 47, 48, 51, 52, 54, 57, 60, 63, 66, 69, 72, 75, 77, 78, 81, 83, 84, 87, 88, 90, 93, 94, 96, 99, 100, 102, 105, 108, 111, 114, 116, 117, 120, 121, 123, 126, 127, 129, 132, 133, 135, 138, 139

Offset: 0

Paul D. Hanna, May 09 2003

nonn

a(k) == 1 (mod 3) at k=0,3,6,12,15,24,39,42,45,54,57,60,63,66,... a(k) == 2 (mod 3) at k=18,21,33,36,51,...

Robert G. Wilson v, Table of n, a(n) for n = 0..5000.

Cf. A083349, A083953.

a[0] = 1; a[n_] := a[n] = Block[{k = a[n - 1] + 1, s = Sum[ a[i]*x^i, {i, 0, n - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^n)^(1/3), {x, 0, n}], x]]] != True, k++ ]; k]; Array[ a, 70] (* Robert G. Wilson v, Sep 19 2008 *)

Three non-ascending values in the range 77 to 84 replaced with those from the b-file. - R. J. Mathar, Jan 14 2009

A111582 G.f.: A(x) = ( G(x)^3 - G(x^3) - 3x(1+x)/(1-x^3) )/(9*x^2) where G(x) is the g.f. of A110628.

Original entry on oeis.org

1, 3, 6, 11, 16, 22, 28, 33, 41, 51, 61, 76, 93, 107, 125, 144, 159, 178, 200, 215, 233, 256, 271, 288, 314, 332, 349, 377, 391, 407, 435, 450, 466, 497, 517, 538, 574, 595, 619, 653, 679, 705, 750, 784, 816, 859, 888, 919, 961, 1002, 1039, 1102, 1145, 1186

Offset: 0

Paul D. Hanna, Aug 28 2005

nonn

A110628 is the trisection of A083953; surprisingly, the self-convolution cube of A110628 is congruent modulo 9 to A083953, which consists entirely of 1's, 2's and 3's.

Cf. A110628, A083953.

cp's OEIS Frontend

A083951 Least increasing integer coefficients such that A(x)^(1/3) has only integer coefficients.

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A111582 G.f.: A(x) = ( G(x)^3 - G(x^3) - 3x(1+x)/(1-x^3) )/(9*x^2) where G(x) is the g.f. of A110628.

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cp's OEIS Frontend

A083951 Least increasing integer coefficients such that A(x)^(1/3) has only integer coefficients.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A111582 G.f.: A(x) = ( G(x)^3 - G(x^3) - 3*x*(1+x)/(1-x^3) )/(9*x^2) where G(x) is the g.f. of A110628.

Views

Author

Keywords

Comments

Crossrefs

A111582 G.f.: A(x) = ( G(x)^3 - G(x^3) - 3x(1+x)/(1-x^3) )/(9*x^2) where G(x) is the g.f. of A110628.