A001932 Sum of Fibonacci (A000045) and Pell (A000129) numbers. (Formerly M0844 N0319)

0, 2, 3, 7, 15, 34, 78, 182, 429, 1019, 2433, 5830, 14004, 33694, 81159, 195635, 471819, 1138286, 2746794, 6629290, 16001193, 38624911, 93240069, 225087338, 543386088, 1311813146, 3166937355, 7645566463, 18457873863, 44560996378, 107579352390, 259718869118

Offset

0,2

Comments

In general, the sum of two Horadam sequences having signatures of (a,b) and (c,d) will be a fourth-order sequence with signature (a+c,d-a*c+b,-a*d-b*c,-b*d). - Gary Detlefs, Oct 13 2020

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 0..501

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Eric Weisstein's World of Mathematics, Horadam Sequence.

Index entries for linear recurrences with constant coefficients, signature (3,0,-3,-1).

Maple

gfpell := x/(1-2*x-x^2): gffib := x/(1-x-x^2): s := series(gfpell+gffib, x, 100): for i from 1 to 60 do printf(`%d, `, coeff(s, x, i)) od:
A001932:=-(z+2)*(2*z-1)/(z**2+z-1)/(z**2+2*z-1); # Conjectured (correctly) by Simon Plouffe in his 1992 dissertation
with (combinat):seq(sum((fibonacci(n, m)), m=1..2), n=1..30); # Zerinvary Lajos, Jun 19 2008

Mathematica

nn = 30; CoefficientList[Series[-x*(x + 2)*(2*x - 1)/(x^2 + x - 1)/(x^2 + 2*x - 1), {x, 0, nn}], x] (* T. D. Noe, Aug 17 2012 *)
LinearRecurrence[{3, 0, -3, -1}, {0, 2, 3, 7}, 30] (* T. D. Noe, Apr 16 2013 *)

Crossrefs

Sequence in context: A368410 A358734 A198683 * A213920 A248869 A005909

Adjacent sequences: A001929 A001930 A001931 * A001933 A001934 A001935

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Apr 06 2001

Status

approved