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 A135364 First column of a triangle - see Comments lines. 11
 1, 2, 3, 7, 17, 40, 93, 216, 502, 1167, 2713, 6307, 14662, 34085, 79238, 184206, 428227, 995507, 2314273, 5380032, 12507057, 29075380, 67592058, 157132471, 365288677, 849193147, 1974134558, 4589306057, 10668842202 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS ...1; ...2,...1; ...3,...3,...1; ...7,...5,...4,...1; ..17,..10,...7,...5,...1; ..40,..24,..13,...9,...6,...1; ..93,..57,..31,..16,..11,...7,...1; From the second, the sum of a row gives the first term of the following one. Diagonal differences are the first term upon. First column is a(n). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Richard Choulet, Transformation à la Curtz. Curtz like Transformation, March 2008. Index entries for linear recurrences with constant coefficients, signature (3,-2,1). FORMULA From Richard Choulet, Jan 06 2008: (Start) a(n+1) = a(n) + a(n-1) + (n-1)*a(1) + (n-2)*a(2) + ... + 2*a(n-2) for n>=3. O.g.f.: 1 + x*(2 - 3*x + 2*x^2) / (1 - 3*x + 2*x^2 - x^3). a(n+3) = 3*a(n+2) - 2*a(n+1) + a(n). (End) a(n) = A034943(n) + A034943(n+1). - R. J. Mathar, Apr 09 2008 a(0) = 1, a(n) = term (1,3) in the 1 X 3 matrix [7,3,2].[3,1,0; -2,0,1; 1,0,0]^(n-1) (n>0). - Alois P. Heinz, Jul 24 2008 a(n) = 2*A095263(n-1) -3*A095263(n-2) +2*A095263(n-3) with a(0) = 1. - G. C. Greubel, Apr 19 2021 MAPLE a:= n-> `if`(n=0, 1, (<<7|3|2>> .<<3|1|0>, <-2|0|1>, <1|0|0>>^(n-1))[1, 3]): seq(a(n), n=0..50); # Alois P. Heinz, Jul 24 2008 MATHEMATICA LinearRecurrence[{3, -2, 1}, {1, 2, 3, 7, 17}, 51] (* G. C. Greubel, Oct 11 2016; Apr 19 2021 *) PROG (Magma) I:=[3, 7, 17]; [1, 2] cat [n le 3 select I[n] else 3*Self(n-1) -2*Self(n-2) +Self(n-3): n in [1..51]]; // G. C. Greubel, Apr 19 2021 (Sage) @CachedFunction def A095263(n): return sum( binomial(n+j+2, 3*j+2) for j in (0..n//2) ) def A135364(n): return 1 if n==0 else 2*A095263(n-1) -3*A095263(n-2) +2*A095263(n-3) [A135364(n) for n in (0..50)] # G. C. Greubel, Apr 19 2021 CROSSREFS Cf. A034943, A097550, A136302, A136303, A136304, A136305, A137229, A137234, A137249. Sequence in context: A191033 A105554 A145230 * A051291 A178178 A333499 Adjacent sequences: A135361 A135362 A135363 * A135365 A135366 A135367 KEYWORD nonn AUTHOR Paul Curtz, Dec 09 2007 EXTENSIONS More terms from Richard Choulet, Jan 06 2008 STATUS approved

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Last modified June 13 22:21 EDT 2024. Contains 373391 sequences. (Running on oeis4.)