

A103145


a(n) = (1/6)*(n^3 + 21*n^2 + 74*n + 18).


1



3, 19, 43, 76, 119, 173, 239, 318, 411, 519, 643, 784, 943, 1121, 1319, 1538, 1779, 2043, 2331, 2644, 2983, 3349, 3743, 4166, 4619, 5103, 5619, 6168, 6751, 7369, 8023, 8714, 9443, 10211, 11019, 11868, 12759, 13693, 14671, 15694, 16763
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OFFSET

0,1


COMMENTS

A floretiongenerated sequence relating truncated triangle and pyramidal numbers. The following reasoning suggests that (a(n)) may not be the result of some "arbitrary" addition of these sequencesit may possess some geometric meaning of its own: The FAMP identity: "jesrightfor + jesleftfor = jesfor" holds and was used to find the relation a(n) = 2*A051936(n+4)_4 + A051937(n+4)_4 . In the above case, "jesfor" returns the truncated triangular numbers (times 1); "jesrightfor" returns the truncated pyramidal numbers; and (a(n)) is given by "jesleftfor" (times 1). All sequences result from a Force transform of the sequence c(n) = n + 5 (c was not chosen arbitrarily, for details see program code). Specifically, the sequence (a(n)) is the (ForType 1A) jesleftfortransform of the sequence c(n) = n + 5 with respect to the floretion given in the program code.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,6,4,1).


FORMULA

a(n) = 2*A051936(n+4)_4 + A051937(n+4)_4 (for n = 0, 1, 2, 3) or a(m) = (1/6)*(m^3 + 9m^2  46m  6) = 2*A051936(m) + A051937(m) (for m = 4, 5, 6).
G.f.: (32*x)*(1 + 3*x  3*x^2)/(1x)^4.  Colin Barker, Apr 30 2012
a(n) = 4*a(n1)  6*a(n2) + 4*a(n3)  a(n4).  Vincenzo Librandi, Jun 26 2012


MATHEMATICA

CoefficientList[Series[(32*x)*(1+3*x3*x^2)/(1x)^4, {x, 0, 40}], x] (* or *) LinearRecurrence[{4, 6, 4, 1}, {3, 19, 43, 76}, 50] (* Vincenzo Librandi, Jun 26 2012 *)


PROG

Floretion Algebra Multiplication Program, FAMP Code: 1jesleftfor[A*B] with A = .25'i  .25i'  .25'ii' + .25'jj' + .25'kk' + .25'jk' + .25'kj'  .25e and B = + 'i + .5j' + .5k' + .5'ij' + .5'ik'; 1vesfor[A*B](n) = n + 5. ForType: 1A Alternative description: 1jesleftfor[A*B], ForType: 1A, LoopType: tes (first iteration after transforming the zerosequence A000004).
(Magma) I:=[3, 19, 43, 76]; [n le 4 select I[n] else 4*Self(n1)6*Self(n2)+4*Self(n3)Self(n4): n in [1..45]]; // Vincenzo Librandi, Jun 26 2012
(PARI) a(n) = (n^3+21*n^2+74*n+18)/6; \\ Altug Alkan, Sep 23 2018


CROSSREFS

Cf. A051936, A051937.
Sequence in context: A031393 A146672 A146704 * A100694 A146664 A028880
Adjacent sequences: A103142 A103143 A103144 * A103146 A103147 A103148


KEYWORD

easy,nonn


AUTHOR

Creighton Dement, Mar 17 2005


STATUS

approved



