A135895 Triangle, read by rows, equal to R^2, the matrix square of R = A135894.

1, 2, 1, 7, 6, 1, 34, 39, 10, 1, 215, 300, 95, 14, 1, 1698, 2741, 990, 175, 18, 1, 16220, 29380, 11635, 2296, 279, 22, 1, 182714, 363922, 154450, 32865, 4410, 407, 26, 1, 2378780, 5135894, 2302142, 517916, 74319, 7524, 559, 30, 1, 35219202, 81557270

Offset

0,2

Comments

Triangle P = A135880 is defined by: column k of P^2 equals column 0 of P^(2k+2) such that column 0 of P^2 equals column 0 of P shift left.

Links

Table of n, a(n) for n=0..46.

Formula

Column k of R^2 = column 1 of P^(2k+1) for k>=0 where triangle P = A135880; column 0 of R^2 = column 1 of P; column 1 of R^2 = column 1 of P^3; column 2 of R^2 = column 1 of P^5.

Example

Triangle R^2 begins:
1;
2, 1;
7, 6, 1;
34, 39, 10, 1;
215, 300, 95, 14, 1;
1698, 2741, 990, 175, 18, 1;
16220, 29380, 11635, 2296, 279, 22, 1;
182714, 363922, 154450, 32865, 4410, 407, 26, 1;
2378780, 5135894, 2302142, 517916, 74319, 7524, 559, 30, 1;
35219202, 81557270, 38229214, 8980944, 1353522, 145805, 11830, 735, 34, 1;
where R = A135894 begins:
1;
1, 1;
2, 3, 1;
6, 12, 5, 1;
25, 63, 30, 7, 1;
138, 421, 220, 56, 9, 1;
970, 3472, 1945, 525, 90, 11, 1; ...
where column k of R = column 0 of P^(2k+1)
and P = A135880 begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
where column k of P equals column 0 of R^(k+1).

Prog

(PARI) {T(n, k)=local(P=Mat(1), R=Mat(1), PShR); if(n>0, for(i=0, n, PShR=matrix(#P, #P, r, c, if(r>=c, if(r==c, 1, if(c==1, 0, P[r-1, c-1])))); R=P*PShR; R=matrix(#P+1, #P+1, r, c, if(r>=c, if(r<#P+1, R[r, c], if(c==1, (P^2)[ #P, 1], (P^(2*c-1))[r-c+1, 1])))); P=matrix(#R, #R, r, c, if(r>=c, if(r<#R, P[r, c], (R^c)[r-c+1, 1]))))); (R^2)[n+1, k+1]}

Crossrefs

Cf. A135882 (column 0), A135890 (column 1); A135894 (R), A135880 (P), A135888 (P^3), A135892 (P^5).

Sequence in context: A019668 A091700 A157743 * A039814 A178120 A180568

Adjacent sequences: A135892 A135893 A135894 * A135896 A135897 A135898

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 15 2007

Status

approved