A105770 Expansion of (x^2-x+1)*(4*x^2+x+1) / ((1+x+x^2)*(1-x)^3).

1, 2, 7, 10, 17, 28, 37, 50, 67, 82, 101, 124, 145, 170, 199, 226, 257, 292, 325, 362, 403, 442, 485, 532, 577, 626, 679, 730, 785, 844, 901, 962, 1027, 1090, 1157, 1228, 1297, 1370, 1447, 1522, 1601, 1684, 1765, 1850, 1939, 2026, 2117, 2212, 2305, 2402, 2503

Offset: 0

Author

Creighton Dement, Apr 18 2005

Keywords

• easy
• nonn

Comments

This sequence is "tesrokseq" at the link "Sequences in Context". The identity vesrok = jesrok + lesrok + tesrok holds.

Floretion Algebra Multiplication Program, FAMP Code: 4tesrokseq[ - .25'i + 1.25'j - .25'k - .25i' + 1.25j' - .25k' + 1.25'ii' + .25'jj' - .75'kk' + .75'ij' + .25'ik' + .75'ji' - .25'jk' + .25'ki' - .25'kj' + .25e] (Link to Sequences in Context contains further details on the "roktype" used).

Differs from A002522 (n^2+1) by two every third number.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Crossrefs

Cf. A105771, A105772, A002522.

Programs

Mathematica

LinearRecurrence[{2, -1, 1, -2, 1},{1, 2, 7, 10, 17},51] (* Ray Chandler, Sep 23 2015 *)

PARI

Vec((1 - x + x^2)*(1 + x + 4*x^2) / ((1 - x)^3*(1 + x + x^2)) + O(x^60)) \\ Colin Barker, May 19 2019

Formula

a(n) = n^2 + 1 + [0,0,2] (3-periodic). - Ralf Stephan, Nov 15 2010.

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>4. - Colin Barker, May 19 2019

3*a(n) = 3*n^2 +5 -2*A061347(n). - R. J. Mathar, Oct 25 2022