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 A023554 Convolution of natural numbers >= 3 and (Fib(2), Fib(3), Fib(4), ...). 2
 3, 10, 22, 43, 78, 136, 231, 386, 638, 1047, 1710, 2784, 4523, 7338, 11894, 19267, 31198, 50504, 81743, 132290, 214078, 346415, 560542, 907008, 1467603, 2374666, 3842326, 6217051, 10059438, 16276552, 26336055, 42612674, 68948798, 111561543, 180510414 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS a(n) is the sum of row n in the triangle T(n,k) defined by: T(n,1) = T(n,n) = 2*n+1 for n>=1 and T(n,k) = 3*T(n-1,k-1) - 2*T(n-1,k) + T(n-2,k-1) for n>2, 2<=k<=n-1. - Lechoslaw Ratajczak, Nov 07 2020 Floretion Algebra Multiplication Program, FAMP code: (a(n)) = 4jesleftforcycseq[ - .25'i + .5'k - .25i' - .5j' + .5k' - .75'ii' + .75'jj' - .25'kk' + .25'jk' - .5'ki' + .25'kj' + .25e ], apart from initial terms. 4jesrightforcycseq = A022308; 2jesforcycseq(n+2) = n+2; identity: jesleft + jesright = jes; vesforcycseq was set to the constant sequence = (-1,-1,-1,-1,-1...). (Dement) LINKS Colin Barker, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1). FORMULA G.f.: x*(1+x)*(3-2*x) / ((1-x)^2*(1-x-x^2)). 2*(n+5) = A022308(n+4) - a(n+1) (conjectured). Note offset of A022308 is 0. - Creighton Dement, Feb 02 2005 From Colin Barker, Feb 20 2017: (Start) a(n) = -7 + (2^(-1-n)*((1-t)^n*(-19+9*t) + (1+t)^n*(19+9*t)))/t - 2*(1+n) where t=sqrt(5). a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) for n>4. (End) a(n) = Fibonacci(n+5) + 2*Fibonacci(n+3) - (2*n + 9). - G. C. Greubel, Jul 08 2019 a(n) = a(n-1) + a(n-2) + 2*n + 3 for n>2. - Lechoslaw Ratajczak, Nov 07 2020 MATHEMATICA Table[Fibonacci[n+5] + 2*Fibonacci[n+3] -2*n-9, {n, 40}] (* G. C. Greubel, Jul 08 2019 *) PROG (PARI) Vec(x*(1+x)*(3-2*x) / ((1-x)^2*(1-x-x^2)) + O(x^60)) \\ Colin Barker, Feb 20 2017 (PARI) vector(40, n, f=fibonacci; f(n+5)+2*f(n+3)-(2*n+9)) \\ G. C. Greubel, Jul 08 2019 (Magma) F:=Fibonacci; [F(n+5)+2*F(n+3)-(2*n+9): n in [1..40]]; // G. C. Greubel, Jul 08 2019 (SageMath) f=fibonacci; [f(n+5)+2*f(n+3)-(2*n+9) for n in (1..40)] # G. C. Greubel, Jul 08 2019 (GAP) F:=Fibonacci;; List([1..40], n-> F(n+5)+2*F(n+3)-(2*n+9)) # G. C. Greubel, Jul 08 2019 CROSSREFS Cf. A000045, A213584. Sequence in context: A140066 A006503 A248851 * A294414 A299336 A222629 Adjacent sequences: A023551 A023552 A023553 * A023555 A023556 A023557 KEYWORD nonn,easy AUTHOR Clark Kimberling STATUS approved

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Last modified June 23 10:25 EDT 2024. Contains 373629 sequences. (Running on oeis4.)