A276532 -id:A276532 - cp's OEIS frontend

A278706 a(n) = a(n-1) + a(n-3) + a(n-5) - a(n-6), a(0) = a(1) = a(2) = 1, a(3) = 2, a(4) = 3, a(5) = 5.

1, 1, 1, 2, 3, 5, 7, 10, 16, 24, 36, 54, 81, 123, 185, 278, 419, 631, 951, 1432, 2156, 3248, 4892, 7368, 11097, 16713, 25173, 37914, 57103, 86005, 129535, 195098, 293844, 442568, 666568, 1003942, 1512073, 2277387, 3430053, 5166126, 7780887, 11719071, 17650511

Offset: 0

Michael Somos, Nov 26 2016

nonn

If the recursion of A276532 is used to define b(n) = (b(n-1) * b(n-6) + b(n-2) * b(n-3) * b(n-4) * b(n-5)) / b(n-7), with b(0), ..., b(6) variables, then the denominator of b(n) = Product_{k=0..6} b(k)^a(n-k-7) if n>=8.

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 7*x^6 + 10*x^7 + 16*x^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..5623
Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, 0, 1, -1).

Cf. A276532.

I:=[1,1,1,2,3,5]; [n le 6 select I[n] else Self(n-1)+Self(n-3)+Self(n-5)-Self(n-6): n in [1..45]]; // Vincenzo Librandi, Nov 27 2016

LinearRecurrence[{1, 0, 1, 0, 1, -1}, {1, 1, 1, 2, 3, 5}, 45] (* Vincenzo Librandi, Nov 27 2016 *)

{a(n) = my(m=n, s=1); if( n<0, m=-6-n; s=-1); s * polcoeff( 1 / (1 - x - x^3 - x^5 + x^6) + x * O(x^m), m)};

G.f.: 1 / (1 - x - x^3 - x^5 + x^6).
0 = a(n) - a(n-1) - a(n-3) - a(n-5) + a(n-6) for all n in Z.
a(n) = - a(-6-n) for all n in Z.

A289980 Oblong (or pronic) numbers of the form x_n.y_n where z_n indicates the digit z repeated n times.

Original entry on oeis.org

12, 20, 30, 42, 56, 72, 90, 1122, 4422, 9900, 111222, 444222, 999000, 11112222, 44442222, 99990000, 1111122222, 4444422222, 9999900000, 111111222222, 444444222222, 999999000000, 11111112222222, 44444442222222, 99999990000000, 1111111122222222

Offset: 1

Bernard Schott, Jul 17 2017

There are only three infinite subsequences, that is, x_n.y_n is pronic for all n, and they occur with (x,y) = (1,2), (4,2), or (9,0). (See Crux Mathematicorum link.)

			6 * 7 = 42, 66 * 67 = 4422, 666 * 667 = 444222, 6666 * 6667 = 44442222.

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Charles W. Trigg, Problem 938, Crux Mathematicorum, page 199, Vol. 11, Jun. 1985.

Cf. A002378, A181718, A276532.

seq(coeff(series(2*x*(15000*x^14+22000*x^13+21000*x^12-1650*x^11-14420*x^10-17265*x^9-14865*x^8-8553*x^7-4374*x^6+1629*x^5+1082*x^4+645*x^3-15*x^2-10*x-6)/((x-1)*(100*x^3-1)*(10*x^3-1)*(x^2+x+1)),x,n+1), x, n), n = 1 .. 30); # Muniru A Asiru, Dec 02 2018

Table[Select[Map[FromDigits[Join @@ Map[ConstantArray[#, n] &, #]] &, Drop[Tuples[Range[0, 9], {2}], 10]], IntegerQ@ Sqrt[4 # + 1] &], {n, 7}] // Flatten (* Michael De Vlieger, Jul 17 2017, after Robert G. Wilson v at A002378 *)
CoefficientList[Series[2 (15000 x^14 + 22000 x^13 + 21000 x^12 - 1650 x^11 - 14420 x^10 - 17265 x^9 - 14865 x^8 - 8553 x^7 - 4374 x^6 + 1629 x^5 + 1082 x^4 + 645 x^3 - 15 x^2 - 10 x - 6) / ((x - 1) (100 x^3 - 1) (10 x^3 - 1) (x^2 + x + 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 02 2018 *)

3_n * (3_n + 1) = 1_n.2_n as 33 * 34 = 1122.
6_n * (6_n + 1)  = 4_n.2_n as 666 * 667 = 444222
9_n * (9_n + 1) = 9_n.0_n as 9999 * 10000 = 99990000.
G.f.: 2*x*(15000*x^14 + 22000*x^13 + 21000*x^12 - 1650*x^11 - 14420*x^10 - 17265*x^9 - 14865*x^8 - 8553*x^7 - 4374*x^6 + 1629*x^5 + 1082*x^4 + 645*x^3 - 15*x^2 - 10*x -6) / ((x-1) * (100*x^3-1) * (10*x^3-1) * (x^2+x+1)). - Alois P. Heinz, Jul 17 2017

cp's OEIS Frontend

A278706 a(n) = a(n-1) + a(n-3) + a(n-5) - a(n-6), a(0) = a(1) = a(2) = 1, a(3) = 2, a(4) = 3, a(5) = 5.

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