A204542 Numbers that are congruent to {1, 4, 11, 14} mod 15.

1, 4, 11, 14, 16, 19, 26, 29, 31, 34, 41, 44, 46, 49, 56, 59, 61, 64, 71, 74, 76, 79, 86, 89, 91, 94, 101, 104, 106, 109, 116, 119, 121, 124, 131, 134, 136, 139, 146, 149, 151, 154, 161, 164, 166, 169, 176, 179, 181, 184, 191, 194, 196, 199, 206, 209, 211, 214, 221, 224

Offset: 1

Author

Michael Somos, Jan 16 2012

Keywords

• nonn
• easy

Comments

The exponents in the q-series for A204220 are the squares of the numbers of this sequence.

The product of any two terms belongs to the sequence and therefore also a(n)^2, a(n)^3, a(n)^4, etc. - Bruno Berselli, Nov 28 2012

Examples

			G.f. = x + 4*x^2 + 11*x^3 + 14*x^4 + 16*x^5 + 19*x^6 + 26*x^7 + 29*x^8 + 31*x^9 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• John Greene and James A. Sellers, Extending recent parity results of Nyirenda and Mugwangwavari for partitions with initial repetitions, Integers (2025), Vol. 25, Art. No. A32. See p. 8.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A204220, A204221.

Programs

Magma

[n : n in [0..100] | n mod 15 in [1, 4, 11, 14]]; // Wesley Ivan Hurt, Jun 07 2016

Maple

A204542:=n->floor(15 * n / 4) - ((n + 1) mod 4): seq(A204542(n), n=1..100); # Wesley Ivan Hurt, Jun 07 2016

Mathematica

Select[Range[250], MemberQ[{1,4,11,14}, Mod[#,15]]&] (* or *) LinearRecurrence[{1,0,0,1,-1}, {1,4,11,14,16}, 60] (* Harvey P. Dale, Apr 15 2015 *)

PARI

{a(n) = (n * 15) \ 4 - (n + 1) % 4};

PARI

{a(n) = if( n<1, -a(1-n), polcoeff( x * (1 + 3*x + 7*x^2 + 3*x^3 + x^4) / ((1-x) * (1-x^4)) + x * O(x^n), n))};

Formula

G.f.: x * (1 + 3*x + 7*x^2 + 3*x^3 + x^4) / ((1-x) * (1-x^4)).

a(n) = -a(1-n), a(n) = 15 + a(n-4), a(n) = floor(15 * n / 4) - ((n + 1) mod 4) for all n in Z.

a(n) = (30*n+10*i^(n*(n+1))-3*(-1)^n+1)/8 -2, where i=sqrt(-1). - Bruno Berselli, Nov 28 2012

a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - Wesley Ivan Hurt, Jun 07 2016

E.g.f.: (4 - 5*(sin(x) - cos(x)) + 3*(5*x - 2)*sinh(x) + 3*(5*x - 3)*cosh(x))/4. - Ilya Gutkovskiy, Jun 07 2016

Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2*(sqrt(5)+5))*Pi/15. - Amiram Eldar, Dec 30 2021