A105343 Elements of even index in the sequence gives A005893, points on surface of tetrahedron: 2n^2 + 2 for n > 1.

1, 3, 4, 7, 10, 15, 20, 27, 34, 43, 52, 63, 74, 87, 100, 115, 130, 147, 164, 183, 202, 223, 244, 267, 290, 315, 340, 367, 394, 423, 452, 483, 514, 547, 580, 615, 650, 687, 724, 763, 802, 843, 884, 927, 970, 1015, 1060, 1107, 1154, 1203, 1252, 1303, 1354, 1407

Offset: 0

Author

Creighton Dement, Apr 30 2005

Keywords

• easy
• nonn

Comments

Floretion Algebra Multiplication Program, FAMP Code: 2jesforrokseq[E*F*sig(E)] with E = + .5i' + .5j' + .5'ki' + .5'kj', F the sum of all floretion basis vectors and "sig" the swap-operator. RokType: Y[15] = Y[15] + Math.signum(Y[15])*p (internal program code)

May be seen as the jesforrok-transform of the zero-sequence (A000004) with respect to the floretion given in the program code.

Identical to A267459(n+1) for n > 0. - Guenther Schrack, Jun 01 2018

Examples

			G.f. = 1 + 3*x + 4*x^2 + 7*x^3 + 10*x^4 + 15*x^5 + 20*x^6 + 27*x^7 + ... - _Michael Somos_, Jun 26 2018
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A005893, A084099, A267459.

Programs

Magma

[1] cat [(2*n^2 + 9 - (-1)^n) div 4: n in [1..60]]; // Vincenzo Librandi, Oct 10 2011

Mathematica

Join[{1}, LinearRecurrence[{2, 0, -2, 1}, {3, 4, 7, 10}, 60]] (* Jean-François Alcover, Nov 13 2017 *)

PARI

{a(n) = if( n<1, n==0, (2*n^2 + 10)\4)}; /* Michael Somos, Jun 26 2018 */

Formula

G.f.: (1 + x - 2*x^2 + x^3 + x^4)/((x+1)*(1-x)^3); a(n+2) - 2*a(n+1) + a(n) = (-1)^(n+1)*A084099(n).

a(n) = (1/4)*(2*n^2 + 9 - (-1)^n ), n>1. - Ralf Stephan, Jun 01 2007

Sum_{n>=0} 1/a(n) = 3/4 + tanh(sqrt(5)*Pi/2)*Pi/(2*sqrt(5)) + coth(Pi)*Pi/4. - Amiram Eldar, Sep 16 2022