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

1,2

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

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).

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

Example

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 + ...

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 *)

Prog

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

Crossrefs

Cf. A204220, A204221.

Sequence in context: A369615 A095797 A205846 * A247521 A285979 A386287

Adjacent sequences: A204539 A204540 A204541 * A204543 A204544 A204545

Keyword

nonn,easy

Author

Michael Somos, Jan 16 2012

Status

approved